text:
\section{Introduction}
\label{intro}
Stochastic differential equations play an important role in modelling various phenomena arising in fields as diverse as finance, physics, chemistry, engineering, biology, neuroscience and others, see e.g.\ \cite{allen}, \cite{hindriks}, \cite{musiela} and \cite{wong}.\ These equations usually depend on parameters, which are often unknown. On the other hand knowledge of these parameters is critical for the study of the process at hand and hence their estimation based on the observational data on the process under study is of great importance in practical applications. A formal setup that we consider in this paper is as follows: let $(\Omega, \mathcal{F}, \mathbb{P})$ be a  probability space. Consider a Brownian motion $W=(W_t)_{t\geq 0}$ and a random variable $\xi$ independent of $W$ that are defined on $(\Omega, \mathcal{F}, \mathbb{P})$ and let $\mathfrak{F}=(\mathfrak{F}_t)_{t \geq 0}$ be the augmented filtration generated by $\xi$ and $W.$ Consider a stochastic differential equation driven by $W,$
\begin{equation}
\label{sde}
\begin{cases}
dX_t=\mu(X_t;\theta)dt+\sigma(X_t;\theta)dW_t,\\
X_0=\xi,
\end{cases}
\end{equation}
where $\theta\in\Theta\subset\mathbb{R}$ is an unknown parameter and $X_0=\xi$ defines the initial condition. Assume that there exists a unique strong solution to \eqref{sde} on $(\Omega, \mathcal{F}, \mathbb{P})$ with respect to the Brownian motion $W$ and initial condition $\xi.$ Let $\theta_0$ denote the true parameter value. Furthermore, let $X$ be ergodic with invariant density $\pi(\cdot;\theta_0)$ and let $\xi\sim \pi(\cdot;\theta_0).$ The solution $X$ is thus a strictly stationary process. Given a discrete time sample $X_0,X_{\Delta},X_{2\Delta},\ldots,X_{n\Delta}$ from the process $X,$ our goal is to estimate the parameter $\theta_0.$ Hence here we consider a parametric inference problem for a stochastic differential equation. There is also a rich body of literature on nonparametric inference for stochastic differential equations, see e.g.\ \cite{comte}, \cite{reiss} and \cite{jacod} and references therein. A general reference on statistical inference for ergodic diffusion processes is \cite{kutoyants}.

A natural approach to estimation of $\theta_0$ is the maximum likelihood method. Assume that the transition density $p(\Delta,x,y;\theta)$ of $X$ exists. Then the likelihood function associated with observations $X_{0},X_{\Delta},\ldots,X_{n\Delta}$ can be written as
\begin{equation*}
p(X_0,X_{\Delta},X_{2\Delta},\ldots,X_{n\Delta};\theta)=\pi(X_{0};\theta)\prod_{j=0}^{n-1}p(\Delta,X_{j\Delta},X_{(j+1)\Delta};\theta),
\end{equation*}
and the maximum likelihood estimator can be computed by maximising the right-hand side of this expression over $\theta,$ provided both the invariant density and the transition density are known explicitly. Unfortunately, for many realistic and practically useful models transition densities are not available in explicit form, which makes exact computation of the maximum likelihood estimator impossible. In those cases when the likelihood cannot be evaluated analytically, a number of alternative estimators have been proposed in the literature, which try to emulate the maximum likelihood method and rely upon some approximation of the likelihood, whence their name, the approximate maximum likelihood estimators, derives. For an overview and relevant references see e.g.\ Section 5 in \cite{h_sorensen}. Although successful in a significant number of examples, these methods typically suffer from a considerable computational burden, see a brief discussion on pp.\ 350--351 in \cite{h_sorensen}.  We also remark that in general in statistical problems if the likelihood is a nonlinear function of the parameter of interest, computation of the maximum likelihood estimator is often far from straightforward, see e.g.\ \cite{barnett}. Returning then to diffusion processes, even if the transition densities are explicitly known, they still might be highly nonlinear functions of the parameter $\theta,$ which might render maximisation of the log-likelihood a difficult task. This is in particluar true for the Cox-Ingersoll-Ross (CIR) process (see e.g.\ pp.\ 356--358 in \cite{musiela} for more information on the CIR process), where the transition densities are noncentral chi-square densities, already numerical evaluation of which, saying nothing about the optimisation process itself, is a nontrivial task, see \cite{dyrting}.

A popular alternative to approximate maximum likelihood methods is furnished by Z-estimators, which are defined as zeroes in $\theta$ of estimating equations
\begin{equation*}
F_n(X_0,X_{\Delta},\ldots,X_{n\Delta};\theta)=0
\end{equation*}
for some given functions $F_n.$   For a general introduction to Z-estimators see e.g.\ Chapter 5 in \cite{vdvaart}. Z-estimators are often faster to compute than approximate maximum likelihood estimators, but the question of the choice of the estimating equations is a subtle one with no readily available recipes in many cases. For instance, the existing methods at times yield choices of $F_n$ that might give rise to numerical problems or that are infeasible in practice, see remarks on pp.\ 343--344 in \cite{h_sorensen}. For additional information on this approach to parameter estimation for diffusion processes and references see \cite{m_sorensen}, \cite{jacobsen}, \cite{kessler} and Section 4 in \cite{h_sorensen}.

In the present work we study an approach alternative to the ones surveyed above. In particular, we will use a characterisation of the invariant density $\pi(\cdot;\cdot)$ of \eqref{sde} as a solution of the ordinary differential equation (here a prime denotes a derivative with respect to $x$)
\begin{equation}
\label{ode}
\mu(x;\theta){\pi}(x;\theta) - \frac{1}{2}\left[ \sigma^2(x;\theta) {\pi}(x;\theta) \right]^{\prime}=0,
\end{equation}
to motivate an estimator $\hat{\theta}_n$ of $\theta_0$ defined as
\begin{equation}
\label{est_theta}
\hat{\theta}_n = \operatorname{argmin}_{\theta\in\Theta} R_n(\theta),
\end{equation}
where
\begin{equation}
\label{crf}
R_n(\theta)=\int_{\mathbb{R}}\left( \mu(x;\theta)\hat{\pi}(x) - \frac{1}{2}\left[ \sigma^2(x;\theta) \hat{\pi}(x) \right]^{\prime} \right)^2 w(x)dx.
\end{equation}
Here $w(\cdot)$ is a weight function chosen beforehand and $\hat{\pi}(\cdot)$ is a nonparametric estimator of ${\pi}(\cdot;\theta_0).$ In particular, in the latter capacity we will use a kernel density estimator. The intuition for $\hat{\theta}_n$ is that for $\hat{\pi}(\cdot)$ that is `close' to ${\pi}(\cdot;\theta_0),$ in view of \eqref{ode} the same must be true for $\hat{\theta}_n$ and $\theta_0.$

The estimator $\hat{\theta}_n$ will be called a smooth and match estimator. Its name reflects the fact that it is obtained through a two-step procedure: in the first step, which is referred to as a smoothing step, the data $Z_0,Z_1,\ldots,Z_n$ are smoothed in order to obtain a nonparametric estimator $\hat{\pi}(\cdot)$ of the stationary density $\pi(\cdot;\theta_0).$ In the second step, which is referred to as a matching step, a characterisation of $\pi(\cdot;\theta_0)$ as a solution of \eqref{ode} is used and an estimator of $\theta_0$ is obtained in such a way that the left-hand side of \eqref{ode} with $\pi(\cdot;\theta_0)$ replaced by $\hat{\pi}(\cdot)$  approximately matches zero. The construction of the estimator $\hat{\theta}_n$ is motivated by a similar construction used in parameter estimation problems for ordinary differential equations, see \cite{gugu} for additional information and references. Approaches to parameter estimation for stochastic differential equations that are close in spirit to the one considered in the present work, in that they rely on matching a parametric function to its nonparametric estimator, are studied in \cite{ait2}, \cite{bandi}, \cite{kristensen} and \cite{h_soren}. We remark that our approach differs from the approaches in these papers either by the type of asymptotics or by the criterion function.

The estimator $\hat{\theta}_n$ is especially straightforward to compute when the drift coefficient $\mu(\cdot;\cdot)$ is linear in the components of the parameter $\theta,$ see Remark \ref{rem_estex} below. Obviously, ease of computation cannot be a sole justification for the use of any  particular estimator and hence in order to provide more motivation for the use of our estimator in the present work we will study its asymptotic properties. Since the estimator $\hat{\theta}_n$ is ultimately motivated by a characterisation of the marginal density of $X,$ in the most general setting when both the drift and the dispersion coefficients  in \eqref{sde} depend on the parameter $\theta,$ the full parameter vector $\theta$ will typically be impossible to estimate due to identifiability problems. We hence have to specialise to some particular case, and we do this for the case when the dispersion coefficient $\sigma(\cdot;\theta)$ does not depend on $\theta$ and is a known function $\sigma(\cdot).$ Thus the stochastic differential equation underlying our model is
\begin{equation}
\label{sde0}
\begin{cases}
dX_t=\mu(X_t;\theta)dt+\sigma(X_t)dW_t,\\
X_0=\xi.
\end{cases}
\end{equation}

The structure of the paper is as follows: in Section \ref{assumptions} for the reader's convenience we list together the assumptions on our model. Detailed remarks on these assumptions are given in Section \ref{remarks}. When reading the paper, a reader can either browse through Section \ref{assumptions} or refer to it as need arises in the subsequent sections. A reader who finds the assumptions in Section \ref{assumptions} believable can skip Section \ref{remarks} at first reading. In Sections \ref{results} and \ref{normality} we state the main results of the paper, namely $\sqrt{n}$-consistency and asymptotic normality of $\hat{\theta}_n.$ In Section \ref{onestep} we discuss a further asymptotic improvement of the estimator $\hat{\theta}_n$ through a Newton-Raphson type procedure. Results of a small simulation study are presented in Section \ref{simulations}. Section \ref{proofs} contains proofs of the results from Sections \ref{results} and \ref{normality}. Finally, Appendix \ref{appendix} contains several technical lemmas used in the proofs of the results from Sections \ref{results} and \ref{normality}.

We remark that in the present work we do not strive for maximal generality. Rather, our goal is to explore asymptotic properties of an intuitively appealing estimator of $\theta_0,$ and to show that this estimator leads to reasonable results in a number of examples.

Throughout the paper we use the following notation for derivatives: a dot denotes a derivative of an arbitrary function $q(x;\theta)$ with respect to $\theta,$ while a prime denotes its derivative with respect to $x.$ We also define the strong mixing coefficient $\alpha_{\Delta}(k)$ as
\begin{equation*}
\alpha_{\Delta}(k)=\sup_{m \geq 0}\sup_{A\in \mathfrak{F}_{\leq m},B\in \mathfrak{F}_{\geq m+k} } |\mathbb{P}(AB)-\mathbb{P}(A)\mathbb{P}(B)|,
\end{equation*}
where $\mathfrak{F}_{\leq m}=\sigma(Z_j,j\leq m)$ and $\mathfrak{F}_{\geq m}=\sigma(Z_j,j\geq m)$ for $m\in\mathbb{N}\cup\{0\}.$ Here $Z_j=X_{j \Delta}$ for $j\in\mathbb{N}\cup\{0\}.$ We call the sequence $Z_j$ $\alpha$-mixing (or strongly mixing), if $\alpha_{\Delta}(k)\rightarrow 0$ as $k\rightarrow\infty.$
When comparing two sequences $\{ a_n \}$ and $\{ b_n \}$ of real numbers, we will use the notation $a_n\lesssim b_n$ to denote the fact that $\exists C>0,$ such that for $\forall n\in\mathbb{N}$ the inequality $a_n\leq C b_n$ holds. A similar convention will be used for $a_n \gtrsim b_n.$ The notation $a_n \asymp b_n$ will denote the fact that the sequences $\{ a_n \}$ and $\{ b_n \}$ are asymptotically of the same order.

\section{Assumptions}
\label{assumptions}

In this section we list the assumptions under which the theoretical results of the paper are proved.

\begin{assump}
\label{cnd_theta}
The parameter space $\Theta$ is a compact subset of $\mathbb{R}$: $\Theta=[a,b]$ for $a<b.$
\end{assump}

\begin{assump}
\label{assump_sol}
The drift coefficient $\mu(\cdot;\theta)$ is known up to the parameter $\theta$ and the dispersion coefficient $\sigma(\cdot)$ is a known function. Furthermore, there exists a unique strong solution $X=(X_t)_{t\geq 0}$ to \eqref{sde0} on $(\Omega, \mathcal{F}, \mathbb{P})$ with respect to the Brownian motion $W$ and initial condition $\xi.$ It is a homogeneous Markov process with transition density $p(t,x,y;\theta).$ Moreover, this solution is ergodic with bounded invariant density $\pi(\cdot;\cdot)$ that has a bounded, continuous and integrable derivative $\pi^{\prime}(\cdot;\cdot),$ and for $\xi\sim\pi(\cdot;\theta)$ the solution $X$ is a strictly stationary process. Also, $\dot{\pi}(\cdot,\cdot)$ exists. Finally, for all $\theta\in\Theta$ it holds that the support of $\pi(\cdot;\cdot),$ i.e.\ the state space of $X,$ equals $\mathbb{R}.$
\end{assump}

\begin{assump}
\label{cnd_mixing}
A sample $X_0,X_{\Delta},\ldots$ (here $\Delta>0$ is fixed) from $X$ corresponding to the true parameter value $\theta_0$ is $\alpha$-mixing with strong mixing coefficients $\alpha_{\Delta}(k)$ satisfying the condition $\sum_{k=0}^{\infty}\alpha_{\Delta}(k)<\infty.$
\end{assump}

\begin{assump}
\label{cnd_smoothness}
The stationary density $\pi(\cdot;\cdot)$ satisfies the condition
\begin{equation*}
\forall t\in\mathbb{R}, \quad \left\{ \int_{\mathbb{R}} ( \pi^{(\alpha)}(x+t;\theta) - \pi^{(\alpha)}(x;\theta) )^2dx \right\}^{1/2} \leq L_{\theta} ,
\end{equation*}
for some constant $L_{\theta}>0$ (that may depend on $\theta$) and some integer $\alpha>3.$
\end{assump}

\begin{assump}
\label{cnd_kernel}
The kernel $K$ is symmetric and continuously differentiable, has support $[-1,1]$ and
satisfies the conditions
\begin{equation*}
\int_{-1}^1 K(u)du=1, \quad \int_{-1}^1 u^{l}K(u)du=0, \quad l=1,\ldots,\alpha.
\end{equation*}
Here $\alpha$ is the same as in Assumption \ref{cnd_smoothness}.
\end{assump}

\begin{assump}
\label{cnd_h}
The bandwidth $h=h_n$ depends on $n$ and $h\downarrow 0$ as $n\rightarrow\infty$ in such a way that
$nh^4\rightarrow\infty.$
\end{assump}

\begin{assump}
\label{cnd_w}
The weight function $w$ is nonnegative, continuously differentiable, bounded and integrable.
\end{assump}

\begin{assump}
\label{assump_characterisation}
The invariant density $\pi(\cdot;\cdot)$ solves the differential equation
\begin{equation}
\label{diffeq0}
\mu(x;\theta_0){\pi}(x) - \frac{1}{2}\left[ \sigma^2(x) {\pi}(x) \right]^{\prime}=0,
\end{equation}
where $\pi(\cdot)$ is the unknown function. Differentiability of $\sigma(\cdot)$ is also assumed.
\end{assump}

\begin{assump}
\label{cnd_drift_volatility}
The drift coefficient $\mu(\cdot;\cdot)$ is three times differentiable with respect to $\theta.$ The drift and dispersion coefficients and the corresponding derivatives are continuous functions of $x$ and $\theta.$ Furthermore, there exist functions $\widetilde{\mu}_j(\cdot),j=1,\ldots,4,$ such that
\begin{equation}
\label{mu_tilde}
\sup_{\theta\in\Theta} |\stackrel{(i)}{\mu}(x;\theta)| \leq \widetilde{\mu}_{i+1}(x), \quad \forall x\in\mathbb{R},
\end{equation}
for $i=0,1,2,3,$ and a function $\widetilde{\mu}_5(\cdot),$ such that
\begin{equation*}
\sup_{\theta\in\Theta} |\dot{\mu}^{\prime}(x;\theta)| \leq \widetilde{\mu}_{5}(x), \quad \forall x\in\mathbb{R}.
\end{equation*}
Here $\stackrel{(i)}{\mu}$ denotes the $i$th derivative of a function $\mu$ with respect to $\theta$ and $\stackrel{(0)}{\mu}(\cdot;\cdot)=\mu(\cdot;\cdot).$ Moreover, the functions
\begin{gather*}
\widetilde{\mu}_1^2(\cdot)w(\cdot), \quad \sigma^4(\cdot)w(\cdot), \quad \sigma^2(\cdot)(\sigma^{\prime}(\cdot))^2w(\cdot),\\
\widetilde{\mu}_2(\cdot)\widetilde{\mu}_1(\cdot)w(\cdot), \quad \widetilde{\mu}_5(\cdot)\sigma^2(\cdot)w(\cdot), \quad \widetilde{\mu}_2^2(\cdot)w(\cdot), \\
\sigma^2(\cdot)w(\cdot), \quad \sigma(\cdot)\sigma^{\prime}(\cdot)w(\cdot), \quad \widetilde{\mu}_2(\cdot)\sigma^2(\cdot)w^{\prime}(\cdot),\\
\widetilde{\mu}_3(\cdot)\widetilde{\mu}_1(\cdot)w(\cdot), \quad \widetilde{\mu}_3(\cdot)\sigma^4(\cdot)w(\cdot), \quad \widetilde{\mu}_3(\cdot)\widetilde{\mu}_1(\cdot)w(\cdot),\\
\widetilde{\mu}_3(\cdot)\sigma(\cdot)\sigma^{\prime}(\cdot)w(\cdot), \quad \widetilde{\mu}_3(\cdot)\sigma^2(\cdot)w(\cdot), \quad \widetilde{\mu}_3(\cdot)\widetilde{\mu}_2(\cdot)w(\cdot),\\
\widetilde{\mu}_4(\cdot)\widetilde{\mu}_1(\cdot)w(\cdot), \quad
\widetilde{\mu}_4(\cdot)\sigma(\cdot)\sigma^{\prime}(\cdot)w(\cdot), \quad \widetilde{\mu}_4(\cdot)\sigma^2(\cdot)w(\cdot),\\
\widetilde{\mu}_2(\cdot)\sigma^2(\cdot)w(\cdot), \quad \widetilde{\mu}_2(\cdot)\sigma(\cdot)\sigma^{\prime}(\cdot)w(\cdot), \quad \widetilde{\mu}_4(\cdot)\sigma^2(\cdot)w(\cdot)
\end{gather*}
are bounded and integrable. Finally,
$\lim_{|x|\rightarrow\infty}\widetilde{\mu}_2(x)w(x)\sigma^2(x)=0.$
\end{assump}

\section{Remarks on assumptions}
\label{remarks}

In this section we provide remarks on the assumptions made in Section \ref{assumptions}.

\begin{rem}
\label{univ} In Assumption \ref{cnd_theta} we assume that the parameter $\theta$ is univariate. This assumption is made for simplicity of the proofs only and the results of the paper can also be generalised to the case when $\theta$ is multivariate. Compactness of the parameter space $\Theta$ guarantees existence of our estimator $\hat{\theta}_n.$
\end{rem}

\begin{rem}
\label{rem_sol} In this remark we deal with Assumption \ref{assump_sol}. A standard condition that guarantees existence and uniqueness of a strong solution to \eqref{sde} is a Lipschitz and linear growth condition on the coefficients $\mu(\cdot;\theta)$ and $\sigma(\cdot)$ together with an assumption that $\ex[\xi^2]<\infty,$ see e.g.\ Theorem 1 on p.\ 40 in \cite{gikhman} or Theorem 2.9 on p.\ 289 in \cite{karatzas}. The same condition also implies that $X$ will be a Markov process, see e.g.\ Theorem 1 on p.\ 66 in \cite{gikhman}, time-homogeneity of which can be shown as on pp.\ 106--107 in \cite{gikhman}. Moreover, $X$ will be a diffusion process, see Theorem 2 on p.\ 67 in \cite{gikhman}. Conditions for ergodicity of $X$ and existence of the invariant density are given e.g.\ in Theorem 3 on p.\ 143 in \cite{gikhman}, while those for existence of the transition density $p(\Delta,x,y;\theta),$ as well as its characterisations can be found in \S 13 of Chapter 3 of \cite{gikhman}. Ergodicity is a standard assumption in parameter estimation problems for diffusion processes from discrete time observations, at least in the problems with $\Delta$ fixed. The condition in Assumption \ref{assump_sol} that the support of $\pi(\cdot;\theta)$ for every $\theta\in\Theta$ equals $\mathbb{R}$ is a purely technical one and is needed only in order to avoid extra technicalities when dealing with boundary bias effects characteristic of kernel density estimators. This condition is for instance satisfied in case when the process $X$ is an Ornstein-Uhlenbeck process,
\begin{equation}
\label{ou}
dX_t=-\theta X_t dt+\sigma dW_t,
\end{equation}
with $\theta>0$ and known $\sigma,$ because in this case
$\pi(x;\theta)$ is a normal density with mean $0$ and variance $\sigma^2/(2\theta),$ see Proposition 5.1 on p.\ 219  in \cite{karlin} or Example 4 on p.\ 221 there. For more information on the Ornstein-Uhlenbeck process see Example 6.8 on p.\ 358 in \cite{karatzas} or results on the Ornstein-Uhlembeck process scattered throughout \cite{karlin}. In the financial literature a slight generalisation of the Ornstein-Uhlenbeck process is used to model the dynamics of the short interest rate and the corresponding model is known under the name of the Vasicek model, see for instance pp.\ 350--355 in \cite{musiela}.
A general case when the support of $\pi(\cdot;\theta)$ does not coincide with $\mathbb{R}$ as for instance for the CIR process, where it is equal to $(0,\infty),$ can be dealt with using the same approach as in the present work in combination with a boundary bias correction method that uses a kernel with special properties, see e.g.\ \cite{gasser}. An alternative in the case when the state space of $X$ is $(0,\infty)$ is to use the transformation $Y_t=\log X_t.$ The process $Y$ will have the state space $\mathbb{R}$ and its governing stochastic differential equation can be obtained through It\^{o}'s formula. \qed
\end{rem}

\begin{rem}
\label{rem_mixing}
Assumption \ref{cnd_mixing} implies certain restrictions on the rate of decay of $\alpha$-mixing coefficients $\alpha_{\Delta}(k).$ Conditions yielding information on their rate of decay can be obtained for instance from the corresponding results for $\beta$-mixing coefficients $\beta(s)$ for the process $X.$ A $\beta$-mixing coefficient $\beta(s)$ (attributed to Kolmogorov in \cite{rozanov} and alternatively called the absolute regularity coefficient) for the process $X$ is defined as follows,
\begin{equation*}
\beta(s)=\sup_{t\geq 0}\ee\left[ \esssup_{B\in \mathfrak{F}_{\geq t+s}} |\pp(B|\mathfrak{F}_{\leq t})-\pp(B)| \right],
\end{equation*}
where $\mathfrak{F}_{\geq s+t}=\sigma(X_u,u\geq s+t),$ $\mathfrak{F}_{\leq t}=\sigma(X_u,u\leq t)$ and $\pp(\cdot|\mathfrak{F}_{\leq t})$ is the regular conditional probability on $\mathfrak{F}_{\geq t+s}$ given $\mathfrak{F}_{\leq t}$ (the latter will exist in our context by Theorem 3.19 on pp.\ 307--308 in \cite{karatzas}). Theorem 1 in \cite{veretennikov} gives a sufficient condition  on the drift coefficient (satisfied for instance in the case of the Ornstein-Uhlenbeck process), which entails a bound
\begin{equation}
\label{polbound}
\beta(s) \leq C \frac{1}{(1+s)^{\kappa+1}},
\end{equation}
where $C$ is a constant independent of $s$ and $\kappa$ depends in a simple way on the drift coefficient. An $\alpha$-mixing coefficient $\alpha(s)$ (introduced in \cite{rosenblatt}) is defined as
\begin{equation*}
\alpha(s)=\sup_{t\geq 0} \sup_{A\in \mathfrak{F}_{\leq t},B\in \mathfrak{F}_{\geq t+s} } |\mathbb{P}(AB)-\mathbb{P}(A)\mathbb{P}(B)|.
\end{equation*}
The following inequality is well-known: $2\alpha(s)\leq\beta(s),$ see Proposition 1 on p.\ 4 in \cite{doukhan}. Since one trivially has $\alpha_{\Delta}(k)\leq\alpha(k\Delta),$ it follows that $\alpha_{\Delta}(k)\leq(1/2)\beta(k\Delta).$ Therefore, by \eqref{polbound} in this case $\sum_{k=0}^{\infty} \alpha_{\Delta}(k)<\infty,$ i.e.\ the requirement in Assumption \ref{cnd_mixing} will hold. \qed
\end{rem}

\begin{rem}
\label{rem_smoothness} This remark deals with Assumption \ref{cnd_smoothness}. Viewing $\theta$ as fixed, conditions under which the invariant density $\pi(x;\theta)$ is infinitely differentiable with respect to $x$ can be found in Theorem 3 of \cite{kusuoka}. In simple cases like that of the Ornstein-Uhlenbeck \eqref{ou}, the regularity assumptions can and have to be checked by a direct calculation. The requirement that $\alpha>3$ is needed in order to establish Theorem \ref{rootn}. Under Assumption \ref{cnd_smoothness} the stationary density $\pi(\cdot;\theta)$ belongs to the Nikol'ski class of functions $\mathcal{H}(\alpha,L)$ as defined e.g.\ in Definition 1.4 in \cite{tsybakov}.  Another possibility is to assume that the invariant density $\pi(\cdot;\theta)$ is $\alpha$ times differentiable with continuous, bounded and square integrable derivative of order $\alpha,$ see e.g.\ paragraph VI.4 on p.\ 79 and Theorem VI.5 on p.\ 80 in \cite{bosq}. In case the weight function $w$ has a compact support, Lemma \ref{mise} (which is a basic result used in the proofs of the main statements of the paper) can also be proved under the assumption
\begin{equation*}
\forall x,t\in\mathbb{R}, \quad |\pi^{(\alpha)}(x+t;\theta) - \pi^{(\alpha)}(x;\theta) | \leq L_{\theta} ,
\end{equation*}
i.e.\ the assumption that the density $\pi(\cdot;\theta)$ belongs to the H\"{o}lder class $\Sigma(\alpha,L)$ as defined e.g.\ in Definition 1.2 in \cite{tsybakov}. However, if $w$ has compact support, in our analysis we will not be using all the information supplied by the stationary density. This might require stronger conditions on the drift and dispersion coefficients $\mu(\cdot;\cdot)$ and $\sigma(\cdot)$ in order for the identifiability condition \eqref{identifiability} in the statement of Theorem \ref{rootn} hold true and hence it is preferable to keep $w$ general. \qed
\end{rem}

\begin{rem}
\label{rem_kernel}
Assumption \ref{cnd_kernel} is a standard condition in kernel estimation, see e.g.\ p.\ 13 in \cite{tsybakov}. The kernel $K$ satisfying Assumption \ref{cnd_kernel} is called a kernel of order $\alpha.$ For a method of its construction see Section 1.2.2 in \cite{tsybakov}. \qed
\end{rem}

\begin{rem}
\label{rem_h}
Assumption \ref{cnd_h} is needed in order to establish consistency of the estimators $\hat{\pi}(\cdot)$ and $\hat{\pi}^{\prime}(\cdot),$ see Lemma \ref{mise}. \qed
\end{rem}

\begin{rem}
\label{rem_w}
This remark deals with Assumption \ref{cnd_w}. In practice when implementing the estimator of $\hat{\theta}_n,$ one would typically use $w$ with compact support. See Section \ref{simulations} for details. \qed
\end{rem}

\begin{rem}
\label{rem_banon}
Sufficient conditions guaranteeing \eqref{diffeq0} in Assumption \ref{assump_characterisation} can be gleaned from \cite{banon}, see Lemma 3.2 there, and involve regularity conditions on the drift coefficient $\mu(\cdot;\cdot)$ and dispersion coefficient $\sigma(\cdot).$ Note that for simple cases like the Ornstein-Uhlenbeck process \eqref{ou}, where an explicit formula for the invariant density is available, Assumption \ref{assump_characterisation} can also be verified directly. \qed
\end{rem}

\begin{rem}
\label{rem_drift_dispersion}
Conditions on the drift and dispersion coefficients made in Assumption \ref{cnd_drift_volatility} are used to prove asymptotic results of the paper. With an appropriate choice of the weight function $w(\cdot)$ they are satisfied in a number of interesting examples, for instance in the case of the Ornstein-Uhlenbeck process \eqref{ou} with $\theta>0$ unknown and $\sigma$ known. Examination of the proofs shows that complicated conditions in Assumption \ref{cnd_drift_volatility} can be significantly simplified if the weight function $w$ is taken to have a compact support. Note also that because of a great flexibility in selection of the weight function $w,$ Assumption \ref{cnd_drift_volatility} will be satisfied in a large number of examples. \qed
\end{rem}

\section{Consistency}
\label{results}

Let $K$ be a kernel function and a number $h>0$ (that depends on $n$) be a bandwidth. To construct our estimator of $\theta_0,$ we first need to construct a nonparametric estimator of the stationary density $\pi(\cdot;\theta_0).$ The stationary density $\pi(\cdot;\theta_0)$ will be estimated by a kernel density estimator
\begin{equation*}
\hat{\pi}(x)=\frac{1}{(n+1)h} \sum_{j=0}^n K\left( \frac{x-Z_j}{h} \right),
\end{equation*}
while $\hat{\pi}^{\prime}(\cdot)$ will serve as an estimator of $\pi^{\prime}(\cdot;\theta_0)$ (we assume that $K(\cdot)$ is differentiable). Kernel density estimators are among the most popular nonparametric density estimators, see e.g.\ Chapter 1 in \cite{tsybakov} for an introduction in the i.i.d.\ case and Section 2 in Chapter 4 of \cite{gyorfi} for the case of dependent identically distributed observations.

In the sequel we will need to know the convergence rate of the estimator $\hat{\pi}(\cdot)$ and its derivative $\hat{\pi}^{\prime}(\cdot)$ in the weighted $L_2$-norm with weight function $w.$ As usual in nonparametric density estimation, to that end some degree of smoothness of the stationary density $\pi(\cdot;\cdot),$ as well as appropriate conditions on the kernel $K,$ bandwidth $h$ and weight function $w$ are needed. These are supplied in Section \ref{assumptions}. Furthermore, to establish useful asymptotic properties of the estimators $\hat{\pi}(\cdot)$ and its derivative $\hat{\pi}^{\prime}(\cdot),$ some further assumptions on the observations have to be made. We will assume that the sequence $Z_j=X_{j\Delta}$ is strongly mixing with mixing coefficients satisfying a condition spelled out in Section \ref{assumptions}.

The following result holds true.

\begin{lemma}
\label{mise}
Under Assumptions \ref{cnd_theta}--\ref{cnd_w} we have
\begin{equation}
\label{mise1}
\ex\left[ \int_{\mathbb{R}} (\hat{\pi}(x)-\pi(x;\theta_0))^2 w(x) dx \right] \lesssim h^{2\alpha}+\frac{1}{nh^{2}},
\end{equation}
and
\begin{equation}
\label{mise3}
\ex\left[ \int_{\mathbb{R}} (\hat{\pi}^{\prime}(x)-\pi^{\prime}(x;\theta))^2 w(x) dx \right] \lesssim h^{2(\alpha-1)}+\frac{1}{nh^{4}}.
\end{equation}
\end{lemma}

\begin{rem}
\label{rem_mise}
The bound in inequality \eqref{mise1}, and by extension in inequality \eqref{mise3}, can be sharpened by using more refined arguments in the proof of Lemma \ref{mise}, such as Theorem 3 on p.\ 9 in \cite{doukhan}. However, the `usual' order bound on the mean integrated squared error in kernel density estimation for i.i.d.\ observations when the unknown density is `smooth of order $\alpha$', i.e.\
\begin{equation}
\label{mise_usual}
\ex\left[ \int_{\mathbb{R}} (\hat{\pi}(x)-\pi(\theta,x))^2 w(x) dx \right] \lesssim h^{2\alpha}+\frac{1}{nh},
\end{equation}
see e.g.\ Theorem 1.3 in \cite{tsybakov}, does not seem to be obtainable without further conditions. For dependent observations the bound \eqref{mise_usual} is true by Theorem 3.3 in \cite{viennet}, which, however, is proved under $\beta$-mixing assumption on observations (which is stronger than $\alpha$-mixing) and some extra condition on the $\beta$-mixing coefficients (see also \cite{gourieroux} and \cite{krist} for related results). The proof of a similar result in \cite{vieu} under $\alpha$-mixing assumption and some complicated conditions on the mixing coefficients, see Theorem 2.2 there, is unfortunately incorrect: the assumption (2.3b) in that paper is impossible to satisfy unless the observations are independent, formula (A.9) contains a mistake and formula (9.2) requires some further conditions in order to hold. \qed
\end{rem}

Let the estimator $\hat{\theta}_n$ of $\theta_0$ be defined by \eqref{est_theta}.

\begin{rem}
\label{rem_minimiser}
Under our assumptions in Section \ref{assumptions} the criterion function $R_n(\theta)$ from \eqref{crf} is a continuous function of $\theta$ and hence by compactness of $\Theta$ a minimiser of $R_n(\theta)$ over $\theta\in\Theta$ exists. Consequently, so does the estimator $\hat{\theta}_n,$ although it might be non-unique. Moreover, the estimator $\hat{\theta}_n$ will be a measurable function of the observations $Z_0,Z_1,\ldots,Z_n$ and hence when dealing with convergence properties of $\hat{\theta}_n,$ the use of outer probability, will not be needed. Observe that $\hat{\theta}_n,$ being defined through a minimisation procedure, is an M-estimator, see e.g.\ Chapter 5 in \cite{vdvaart}. \qed
\end{rem}

\begin{rem}
\label{rem_esteq}
An approach to parameter estimation for stochastic differential equations that is based on estimating equations as described in Section \ref{intro} in practice might suffer from non-uniqueness of a parameter estimate, i.e.\ non-uniqueness of a root of the estimating equations. `Wrong' selection of a root of the estimating equations might even render the estimator inconsistent, see e.g.\ remarks on pp.\ 70--71 in \cite{vdvaart}. For a thorough discussion of the multiple root problems and possible remedies for them see \cite{small}. On the other hand, an approach based on maximisation of a criterion function, such as the one advocated in the present work, is less prone to failures of this type. \qed
\end{rem}

\begin{rem}
\label{rem_estex}
In many interesting models, in particular in those where the drift coefficient $\mu(\cdot;\cdot)$ is linear in $\theta,$ the estimator $\hat{\theta}_n$ will have a simple expression. For instance, one can check that for the Ornstein-Uhlenbeck process \eqref{ou} with $\theta>0$ unknown and and $\sigma=1$ known, the estimator $\hat{\theta}_n$ of the true parameter value $\theta_{0}$ is given by
\begin{equation}
\label{thetaou}
\hat{\theta}_n = - \frac{1}{2} \frac{ \int_{\mathbb{R}} x\hat{\pi}(x)  \hat{\pi}^{\prime}(x) w(x) dx }{ \int_{\mathbb{R}} x^2 \hat{\pi}^2(x) w(x)dx  }.
\end{equation}
Compare this expression to the rather complex and nonlinear score function for the same model as given on p.\ 77 in \cite{kessler}, which is used as an estimating function when $\theta$ is estimated by the maximum likelihood method and which requires use of some numerical root finding technique for the computation of the estimator. A general conclusion that can be drawn from this and other similar examples is that our approach in many interesting examples will provide explicit estimators. However, it should be noted that from the point of view of numerical stability, evaluation of the estimator through expressions such as in \eqref{thetaou} cannot be recommended in practice. Rather, one should approximate the criterion function $R_n(\cdot)$ through a Riemann sum and next compute from this approximation the estimator $\hat{\theta}_n$ as a weighted least squares estimator. When $\mu(\cdot;\cdot)$ is linear in $\theta,$ the problem will further reduce to a standard task of computing the weighted least squares estimator in the linear regression model. Finally, we remark that with a proper implementation of the nonparametric kernel estimators $\hat{\pi}_n(\cdot)$ and $\hat{\pi}_n^{\prime}(\cdot),$ computational effort for their evaluation is very modest; see e.g.\ \cite{marron}. \qed
\end{rem}

\begin{rem}
\label{rem_kessler}
A desire to have simple expressions for estimators based on estimating equations in \cite{kessler} at times leads to unnatural assumptions on the parameter space $\Theta.$ For instance, in Section 6.4 in \cite{kessler} in the model
\begin{equation*}
\begin{cases}
dX_t=-\theta X_t dt+\sqrt{\theta+X_t^2}dW_t,\\
X_0=\xi,
\end{cases}
\end{equation*}
in order to accommodate a simple looking estimator of the true parameter $\theta_0,$ $\theta_0>7/2$ has to be assumed, while the more general condition $\theta_0>0$ appears to be more natural here. On the other hand, the assumption $\theta_0>7/2$ is not needed for our estimator $\hat{\theta}_n$ and $\theta_0>0$ suffices (this model formally does not fit into our framework, because the unknown parameter $\theta$ is also included in the dispersion coefficient of the stochastic differential equation. However, our asymptotic analysis holds for this model as well). \qed
\end{rem}

It can be expected that as $n\rightarrow\infty,$ for every $\theta\in\Theta$ the criterion function $R_n(\theta)$ converges in some appropriate sense  to the limit criterion function
\begin{equation*}
R(\theta)=\int_{\mathbb{R}}\left( \mu(x;\theta){\pi}(x;\theta_0) - \frac{1}{2}\left[ \sigma^2(x) {\pi}(x;\theta_0) \right]^{\prime} \right)^2 w(x)dx.
\end{equation*}
Note that by our assumptions $R(\theta_0)=0$ and that $R(\theta)\geq 0$ for $\theta\in\Theta.$ Hence the parameter value $\theta_0$ is a minimiser of the asymptotic criterion function $R(\theta)$ over $\theta\in\Theta.$ Under suitable identifiability conditions it can be ensured that $\theta_0$ is the unique minimiser of $R(\theta).$ Next, if convergence of $R_n(\theta)$ to $R(\theta)$ is strong enough, a minimiser of $R_n(\theta)$ will converge to a minimiser of $R(\theta)$. Said another way, $\hat{\theta}_n$ will be consistent for $\theta_0.$ This is a standard approach to prove consistency of M-estimators, see e.g.\ Section 5.2 in \cite{vdvaart}.

In order to carry out the above programme for the proof of consistency of $\hat{\theta}_n,$ we need that the drift coefficient $\mu(\cdot;\cdot)$ and the dispersion coefficient $\sigma(\cdot)$ satisfy certain regularity conditions. These are listed in Section \ref{assumptions}. Then the following theorem holds true.

\begin{thm}
\label{consistency}
Under Assumptions \ref{cnd_theta}--\ref{cnd_drift_volatility} and the additional identifiability condition
\begin{equation}
\label{identifiability}
\forall \varepsilon>0, \quad \inf_{\theta:|\theta-\theta_0|\geq\varepsilon} R(\theta) > R(\theta_0),
\end{equation}
the estimator $\hat{\theta}_n$ is weakly consistent: $\hat{\theta}_n \convp \theta_0.$
\end{thm}

\begin{rem}
\label{rem_identifiability}
The identifiability condition \eqref{identifiability} is standard in M-estimation, see e.g.\ a discussion in Section 5.2 in \cite{vdvaart}. It means that a point of minimum of the asymptotic criterion function is a well-separated point of minimum. Since under our conditions the asymptotic criterion function $R(\theta)$ is a continuous function of $\theta$ and $\Theta$ is compact, uniqueness of a global minimiser of $R(\theta)$ over $\theta$ will imply \eqref{identifiability}, cf.\ Problem 5.27 on p.\ 84 in \cite{vdvaart}. As one  particular example, one can check that condition \eqref{identifiability} is satisfied for the Ornstein-Uhlenbeck process \eqref{ou}, assuming that $\theta$ is unknown, while $\sigma$ is known. \qed
\end{rem}

\begin{thm}
\label{rootn}
Let the assumptions of Theorem \ref{consistency} hold and let additionally $\theta_0$ be an interior point of $\Theta.$ If $h\asymp n^{-\gamma}$ with $\gamma=1/(2\alpha)$ and
$\ddot{R}(\theta_0)\neq 0,$
then
$\sqrt{n}(\hat{\theta}_n-\theta_0)=O_{\mathbb{P}}(1).$
\end{thm}

\begin{rem}
\label{rem_secondder}
The assumption $\ddot{R}(\theta_0)\neq 0$ is satisfied in a number of important examples, for instance in the case of the Ornstein-Uhlenbeck process \eqref{ou} with $\theta>0$ unknown and known $\sigma.$ \qed
\end{rem}

\begin{rem}
\label{rem_switch} Under appropriate conditions, by the same method as studied in the present work, one can also handle the case when the drift coefficient $\mu(\cdot;\theta)$ does not depend on parameter $\theta,$ while the dispersion coefficient does. \qed
\end{rem}

\begin{rem}
\label{rem_extension}
In the present paper we assumed that the dispersion coefficient $\sigma(\cdot)$ was known. In practice this is not always a realistic assumption. A possible extension of the smooth and match method to this more general setting is to assume that $\sigma(\cdot)$ is a totally unknown function, to estimate it nonparametrically and next to define an estimator of the parameter of interest $\theta_0$ again via an expression \eqref{est_theta}, but with $\sigma(\cdot)$ replaced by its nonparametric estimator $\hat{\sigma}(\cdot)$ in $R_n(\theta).$ Under appropriate assumptions this approach should again yield a $\sqrt{n}$-consistent estimator of $\theta_0,$ although some nontrivial technicalities can be anticipated. \qed
\end{rem}

\begin{rem}
\label{rem_dband} Theorem \ref{rootn} holds also for bandwidth sequences $h\asymp n^{-\gamma}$ with $\gamma$ other than $1/(2\alpha).$ However, $\gamma$ cannot be arbitrary, for this might lead to violation of consistency of $\hat{\pi}(\cdot)$ and $\hat{\pi}^{\prime}(\cdot),$ see Lemma \ref{mise}. The condition on the bandwidth sequence in the statement of Theorem \ref{rootn} is of an asymptotic nature and is not directly applicable in practice. In practical applications a simple method called the quasi-optimality method is likelily to produce reasonable results, see e.g.\ \cite{reiss2} for more information. See also the results of the simulation examples considered in Section \ref{simulations}. \qed
\end{rem}

\section{Asymptotic normality}
\label{normality}
Examination of the proof of Lemma \ref{t1t2t3t4} in Appendix \ref{appendix}, on which the proof of Lemma \ref{lemma_rootn1} and eventually that of Theorem \ref{rootn} relies, shows that under appropriate extra conditions not only $\sqrt{n}$-consistency of the estimator $\hat{\theta}_n,$ but also its asymptotic normality can be established.

Let
\begin{equation*}
v(x)=2\dot{\mu}(x;\theta_0)\mu(x;\theta_0)\pi(x;\theta_0)w(x)
+\left[ \dot{\mu}(x;\theta_0) \pi(x;\theta_0) w(x) \right]^{\prime} \sigma^2(x).
\end{equation*}
The following result holds true.

\begin{thm}
\label{thm_an}
Let the assumptions of Theorem \ref{consistency} hold (with Assumption \ref{rem_smoothness} strengthened to the requirement $\alpha>4$) and let additionally $\theta_0$ be an interior point of $\Theta.$ Assume that $h\asymp n^{-\gamma}$ with
\begin{equation*}
\frac{1}{2\alpha}<\gamma<\frac{1}{8}.
\end{equation*}
If
\begin{equation}
\label{secondder2}
\ddot{R}(\theta_0)\neq 0, \quad \var[v(Z_0)]+2\sum_{j=1}^{\infty}\cov[v(Z_0),v(Z_j)]>0, \quad \|v^{(\alpha)}\|_{\infty}<\infty,
\end{equation}
and for some $\delta>0,$
\begin{equation}
\label{an_mixing}
\ex[|Z_j|^{2+\delta}]<\infty, \quad \sum_{k=1}^{\infty} (\alpha_{\Delta}(k))^{\delta/(2+\delta)}<\infty,
\end{equation}
then
\begin{equation*}
\sqrt{n+1}(\hat{\theta}_n-\theta_0)\convd \mathcal{N}(0,s^2).
\end{equation*}
Here
\begin{equation*}
s^2=\frac{ \var[v(Z_0)]+2\sum_{j=1}^{\infty}\cov[v(Z_0),v(Z_j)] }{(\ddot{R}(\theta_0))^2}.
\end{equation*}
\end{thm}

\section{One-step Newton-Raphson type procedure}
\label{onestep}

Although according to Theorems \ref{rootn} and \ref{thm_an} the estimator $\hat{\theta}_n$ is $\sqrt{n}$-consistent and even asymptotically normal, it is obviously not necessarily asymptotically the best one, which in the present model and observation scheme is typically the case for Z-estimators based on martingale estimating equations as well. Here we interpret asymptotically the best estimator as the one that is regular and has the smallest possible asymptotic variance among all regular estimators, see e.g.\ Chapter 8 in \cite{vdvaart} for an exposition of the asymptotic efficiency theory in the i.i.d.\ setting. Under regularity conditions the maximum likelihood estimator achieves the efficiency bound. As far as Z-estimators in diffusion models are concerned, a line of research in the literature is to try to choose estimating equations within a certain class of functions in an optimal way, see e.g.\ \cite{m_sorensen}, \cite{jacobsen} and \cite{kessler}. However, most of the work in this direction deals with the high frequency data setting where $\Delta=\Delta_n\rightarrow 0$ as $n\rightarrow\infty.$ In our case optimal choice of the estimating equations would correspond to the problem of an optimal choice of the weight function $w(\cdot)$ within a certain class of weight functions. This is not an easy problem to solve and it is a priori not clear whether this approach would lead to a simple and feasible optimal weight function $w_{opt}.$ A possibly better and more direct approach to improving asymptotic performance of the estimator $\hat{\theta}_n$ is to use it as a starting point of a one-step Newton-Rapshon type procedure. The idea is well-known in statistics, see e.g.\ Section 5.7 in \cite{vdvaart}, and is as follows: consider an estimating equation $\Psi_n(\theta)=0.$ Given a preliminary estimator $\hat{\theta}_n,$ define a one-step estimator $\overline{\theta}_n$ of $\theta_0$ as a solution in $\theta$ to the equation
\begin{equation}
\label{eq1step}
\Psi_n(\hat{\theta}_n)+ \dot{\Psi}_n(\hat{\theta}_n) (\theta-\hat{\theta}_n)=0,
\end{equation}
where $\dot{\Psi}_n(\cdot)$ is the derivative of $\Psi_n(\cdot)$ with respect to $\theta.$ This corresponds to replacing $\Psi_n(\theta)$ with its tangent at $\hat{\theta}_n$ and when iterated several times, each time using as a new starting point the previously found solution to \eqref{eq1step}, is known in numerical analysis under the name of the Newton (or Newton-Raphson) method, see e.g.\ Section 2.3 in \cite{burden}. This method is used to find zeroes of nonlinear equations. In statistics, on the other hand, just one such iteration suffices to obtain an estimator that is as good asymptotically as the one defined by the estimating equation $\Psi_n(\theta)=0,$ provided the preliminary estimator is already $\sqrt{n}$-consistent (a precise result can be found in Theorem 5.45 in \cite{vdvaart}). A computational advantage of a one-step approach over a more direct maximum likelihood approach is that often a preliminary $\sqrt{n}$-consistent estimator is easy to compute, while the computational time required for one Newton-Raphson type iteration step is negligible.

Under suitable conditions one can use in the capacity of $\Psi_n$ the martingale estimating functions, see e.g.\ \cite{m_sorensen}, or even the score function $S_n(\theta)$ (i.e.\ a gradient of the likelihood function with respect to the unknown parameter $\theta$), provided the required derivatives of $\Psi_n$ can be evaluated either analytically or numerically in a quick and numerically stable way. The estimator $\hat{\theta}_n$ can thus be upgraded to an asymptotically efficient one. We omit a detailed discussion and a precise statement to save space and will simply note that the regularity conditions required to justify the one-step method are mild enough in our case (as an example, they are satisfied in the case of the Ornstein-Uhlenbeck process).

\section{Simulations}
\label{simulations}

In this section we present results of a small simulation study that we performed using the Ornstein-Uhlenbeck process \eqref{ou} as a test model. This study is in no way exhaustive and the results obtained merely serve as an illustration of the theoretical results from Sections \ref{results}--\ref{onestep}.

Three required components for the construction of our estimator $\hat{\theta}_n$ from \eqref{est_theta} are the weight function $w(\cdot),$ the kernel $K$ and the bandwidth $h.$ As a weight function we used a suitably rescaled version of the function
\begin{equation*}
\lambda_{c,\beta}(x)=
\begin{cases}
1, & \text{if $|x|\leq c,$}\\
\exp[-\beta \exp[-\beta/(|x|-c)^2]/(|x|-1)^2], & \text{if $c<|t|<1,$}\\
0, & \text{if $|x| \geq 1,$}
\end{cases}
\end{equation*}
with constants $c$ and $\beta$ equal to $0.7$ and $0.5,$ respectively. This weight function was already used in simulation examples in \cite{gugu}. The rationale for its use is simple: $w$ will be equal to one on a greater part of its support, which comes in handy in computations, while at the same time being smooth. As a kernel we used
\begin{equation*}
K(x)=\left( \frac{105}{64} - \frac{315}{64} x^2 \right) (1-x^2)^2 1_{[|x|\leq 1]},
\end{equation*}
which was also employed in simulation examples in \cite{gugu} and yielded good results there. Finally, in all our examples the bandwidth was selected through the so-called quasi-optimality approach by computing the estimates $\hat{\theta}_n=\hat{\theta}_{n,h}$ for a range of different bandwidths $h$ and then picking the one that brought the least change to the next estimate. In greater detail, for a sequence of bandwidths $\{h^{(i)}\}$ we chose the bandwidth $\hat{h}$ such that
\begin{equation*}
\hat{h}=\operatorname{argmin}_{h^{(i)}} \|\hat{\theta}_{n,h^{(i+1)}}-\hat{\theta}_{n,h^{(i)}}\|
\end{equation*}
and next computed the estimate $\hat{\theta}_{n,\hat{h}}.$ In order not to clutter the notation, in the sequel we will omit the dependence of $\hat{\theta}_{n,\hat{h}}$ on $\hat{h}$ and will simply write $\hat{\theta}_n.$ \cite{reiss2} contains theoretical justification for this method of smoothing parameter selection in nonparametric estimation problems.

Our goal was to compare the behaviour of our estimator $\hat{\theta}_n,$  the one-step estimator $\overline{\theta}_n$ which was using $\hat{\theta}_n$ as a preliminary estimator, the estimator based on a simple estimating function from formula (29) in \cite{kessler} given by the expression
\begin{equation*}
\theta^{*}_n=\frac{n}{2 \sum_{j=0}^{n-1} X_{j\Delta}^2},
\end{equation*}
and the maximum likelihood estimator $\widetilde{\theta}_n.$ Since the practical performance of the maximum likelihood estimator $\widetilde{\theta}_n$ in the case of the Ornstein-Uhlenbeck process is quite good, while the loss in asymptotic efficiency for the estimator $\theta^{*}_n$ in comparison to $\widetilde{\theta}_n$ is small, the competition with these two estimators was a tough task for our estimator $\hat{\theta}_n.$

All the computations were performed in Wolfram Mathematica 8.0, see \cite{mathematica}. Simulating samples from the Ornstein-Uhlenbeck process is straightforward, since it is an AR(1) process. We took $\theta_0=2$ and $\sigma=1$ and simulated from the process $X$ samples of sizes $100$ and $200$ (thus $n=99$ and $199$) with intervals between successive observations $\Delta=0.01,0.05,0.1$ and $1.$

As a criterion for comparison of different estimators the mean squared error was used. For fixed $\Delta$ and $n$ and for $k=200$ different samples we computed the estimates $\hat{\theta}_n,\overline{\theta}_n,\theta^{*}_n$ and $\widetilde{\theta}_n$ and then for each of $k=200$ estimates $\hat{\theta}_n,\overline{\theta}_n,\theta^{*}_n,\widetilde{\theta}_n$ we evaluated the corresponding mean squared error, that is the sum of the sample variance and sample bias squared (sample mean minus the true parameter value $\theta_0=2$ squared). The support of the weight function $w(\cdot)$ was taken to be the interval $[-1.4,1.4],$ which roughly corresponds to the interval $[-3s_n,3s_n],$ where $s_n$ is the sample standard deviation of the observations. The results obtained from our simulations are reported in Table \ref{table1}, where we also included the theoretical optimal value $\text{EB}$ for the mean squared error that can be obtained from the asymptotic efficiency bound, see Example 3.2 and formula (12) in \cite{kessler}.
\begin{table}
\caption{Mean squared errors for the estimates $\hat{\theta}_n,\overline{\theta}_n,\theta^{*}_n,\widetilde{\theta}_n$ together with the optimal value $\text{EB}$ obtained from the asymptotic efficiency bound in the case of the Ornstein-Uhlenbeck process \eqref{ou} with $\theta_0=2$ and $\sigma=1.$}
\label{table1}
\begin{center}
\begin{tabular}{rr|rrrrr}
$\Delta$ & $n$ & $\hat{\theta}_n$ & $\overline{\theta}_n$ & $\theta^{*}_n$ & $\widetilde{\theta}_n$ & $\text{EB}$\\
\hline
$0.01$ & $99$ & $1.900$ & $11.24$ & $8.545$ & $11.28$ & $4.001$\\
       &$199$ & $2.152$ & $3.774$ & $3.474$ & $3.776$ & $2.000$\\
       \hline
$0.05$ & $99$ & $1.061$ & $1.384$ & $1.371$ & $1.394$ & $0.803$\\
       &$199$ & $0.578$ & $0.647$ & $0.615$ & $0.651$ & $0.401$\\
       \hline
$0.1$  & $99$ & $0.663$ & $0.697$ & $0.677$ & $0.701$ & $0.405$\\
       &$199$ & $0.291$ & $0.204$ & $0.206$ & $0.205$ & $0.203$\\
       \hline
$1$    & $99$ & $0.155$ & $0.067$ & $0.079$ & $0.070$ & $0.080$\\
       &$199$ & $0.093$ & $0.040$ & $0.042$ & $0.040$ & $0.040$
\end{tabular}
\end{center}
\end{table}
A conclusion (modulo the Monte Carlo simulation errors) that lends itself from this table is that for small $\Delta$ the estimator $\hat{\theta}_n$ seems to either outperform other estimators, or to perform just as well as other estimators, but once $\Delta$ and $n$ are sufficiently large, it is itself outperformed by other estimators (our conclusions are also supported by some other simulations not reported here). Curiously enough, for $\Delta=0.01$ and $n=99$ the estimator $\hat{\theta}_n$ beats the asymptotic efficiency bound, although of course its performance is not (and cannot be) particularly good in this case. It is also interesting to note that the maximum likelihood estimator is not the best estimator in all the cases, which should not be surprising, for its superiority over other estimators is in the asymptotic sense only (it is also known to be strongly biased for small $n\Delta,$ see e.g.\ \cite{tang}). Note that whenever the maximum likelihood estimator $\widetilde{\theta}_n$ performs well, so does the one-step estimator $\overline{\theta}_n,$ which in general seems to yield virtually indistinguishable results. Another general remark is that for $n$ fixed all the estimators tend to perform better for larger values of $\Delta.$ An intuitive explanation of this fact is that increasing $\Delta$ decreases the degree of dependence between different observations, which coupled with the fact that in the case of the Ornstein-Uhlenbeck process the marginal distributions of the process $X$ contain enough information on the parameter $\theta_0,$ improves the estimation quality.

In conclusion, keeping in mind that in our simulation study we used a very simple bandwidth selector and a weight function $w(\cdot),$ the choice of which was primarily motivated by simplicity considerations, the performance of our estimator $\hat{\theta}_n$ can be deemed as satisfactory.

\section{Proofs}
\label{proofs}

\begin{proof}[Proof of Lemma \ref{mise}]
We will only prove \eqref{mise1}, as the proof of \eqref{mise3} uses similar arguments. By a standard decomposition of the weighted mean integrated square error into the sum of the weighted integrated square bias and weighted integrated variance we have
\begin{equation}
\label{mise_decomp}
\begin{aligned}
\ex\left[ \int_{\mathbb{R}} (\hat{\pi}(x)-\pi(x;\theta_0))^2 w(x) dx \right] & =  \int_{\mathbb{R}} (\ex[\hat{\pi}(x)]-\pi(x;\theta_0))^2 w(x) dx\\
&+\ex\left[\int_{\mathbb{R}} (\hat{\pi}(x)-\ex[\hat{\pi}(x)])^2 w(x) dx\right]\\
&=T_1+T_2.
\end{aligned}
\end{equation}
By assumptions of the lemma combined with Proposition 1.5 in \cite{tsybakov} it holds that
\begin{equation}
\label{t1}
T_1 \leq \| w \|_{\infty} \left( \frac{L_{\theta_0}}{\ell !} \int_{\mathbb{R}} |u|^{\alpha} |K(u)| du \right)^2 h^{2\alpha}.
\end{equation}
Next, denote
\begin{equation}
\label{yz}
Y(Z_j,x)=\frac{1}{h}K\left( \frac{x-Z_j}{h} \right)-\ex\left[ \frac{1}{h}K\left( \frac{x-Z_j}{h}\right) \right].
\end{equation}
Then
\begin{align*}
T_2 & = \frac{1}{(n+1)^2} \ex\left[ \int_{\mathbb{R}} \left( \sum_{j=0}^n Y(Z_j,x) \right)^2 w(x)dx \right]\\
& = \frac{1}{(n+1)^2} \sum_{j=0}^n \ex\left[ \int_{\mathbb{R}}    Y^2(Z_j,x)  w(x)dx \right]\\
& + \frac{1}{(n+1)^2} \sum_{i\neq j} \ex\left[ \int_{\mathbb{R}}  Y(Z_i,x)Y(Z_j,x) w(x) dx \right]\\
& = \frac{1}{n+1} \ex\left[ \int_{\mathbb{R}}    Y^2(Z_1,x)  w(x)dx \right]\\
& + \frac{1}{(n+1)^2} \sum_{i\neq j} \ex\left[ \int_{\mathbb{R}}  Y(Z_i,x)Y(Z_j,x) w(x) dx \right]\\
& = T_3+T_4
\end{align*}
holds. By Proposition 1.4 in \cite{tsybakov} we have
\begin{equation}
\label{t3}
\begin{aligned}
T_3 \leq \frac{1}{(n+1)h} \|w\|_{\infty} \int_{\mathbb{R}} K^2(u) du.
\end{aligned}
\end{equation}
Now note that
\begin{equation*}
\|Y(\cdot,\cdot)\|_{\infty} \leq 2 \|K\|_{\infty} \frac{1}{h}.
\end{equation*}
Consequently, by Lemma 3 on p.\ 10 in \cite{doukhan},
\begin{equation*}
|\ex[Y(Z_i,x)Y(Z_j,x)]| \leq 16 \|K\|_{\infty}^2 \frac{1}{h^2} \alpha_{\Delta}(|i-j|).
\end{equation*}
Thus
\begin{equation}
\label{t4}
\begin{aligned}
T_4 & \leq \frac{1}{(n+1)^2 h^2} 16 \|K\|_{\infty}^2 \| w \|_1 \sum_{i \neq j } \alpha_{\Delta}(|i-j|)\\
& = \frac{1}{(n+1)^2 h^2} 32 \|K\|_{\infty}^2 \| w \|_1 \sum_{0\leq i < j \leq n } \alpha_{\Delta}(j-i).
\end{aligned}
\end{equation}
Working out the sum on the right-hand side, we get
\begin{align*}
\sum_{0\leq i < j \leq n } \alpha_{\Delta}(j-i) & = \sum_{k=1}^{n} (n+1-k)\alpha_{\Delta}(k)\\
& \leq (n+1) \sum_{k=1}^{\infty} \alpha_{\Delta}(k),
\end{align*}
which can be seen by counting the corresponding possibilities and the trivial observation that $n+1-k\leq n+1$ for $k=1,\ldots,n.$ Note that the sum on the right-hand side of the last display is finite by Assumption \ref{cnd_mixing}. The above display, the fact that $T_2=T_3+T_4$ and the bounds \eqref{t3} and \eqref{t4} imply that
\begin{equation}
\label{t2}
T_2 \lesssim \frac{1}{nh^2}.
\end{equation}
The statement \eqref{mise1} follows from decomposition \eqref{mise_decomp} combined with formulae \eqref{t1} and \eqref{t2}. In view of the remark made at the beginning of the proof, this completes the proof of the lemma.
\end{proof}

\begin{proof}[Proof of Theorem \ref{consistency}]
We first settle the issue of measurability of $\hat{\theta}_n.$ By Lemma 2 in \cite{jennrich} to that end it is enough to have that for each fixed $\theta$ the criterion function $R_n(\theta)$ is a measurable function of the sample $Z_0,\ldots,Z_n,$ and that for $(Z_0,\ldots,Z_n)\in\mathbb{R}^{n+1}$ viewed as fixed, the function $R_n(\theta)$ is continuous in $\theta.$ However, measurability follows easily from our assumptions, while continuity of $R_n(\theta)$ in $\theta$ is a consequence of the fact that under our conditions by the corollary on p.\ 74 in \cite{whittaker} and by de la Vall\'ee Poussin's test on p.\ 72 there the function $R_n(\theta)$ is in fact three times differentiable with respect to $\theta$ (this follows by a tedious but easy verification of the assumptions made in the corollary on p.\ 74 in \cite{whittaker}). Thus in the convergence considerations we do not need to appeal to outer probability.

We will prove that
\begin{equation}
\label{convergence}
\sup_{\theta\in\Theta} |R_n(\theta)-R(\theta)|\convp 0.
\end{equation}
The statement of the theorem will then follow from this fact and assumption \eqref{identifiability} by Theorem 5.7 in \cite{vdvaart} (the fact that Chapter 5 in \cite{vdvaart} largely deals with the i.i.d.\ setting is immaterial in this case).

By the Cauchy-Schwarz inequality we have
\begin{multline*}
|R_n(\theta)-R(\theta)|\\
=\Biggl|\int_{\mathbb{R}} \left( \mu(x;\theta) \hat{\pi}(x) - \frac{1}{2} \left[ \sigma^2(x) \hat{\pi}(x) \right]^{\prime} - \mu(x;\theta)\pi(x;\theta_0) + \frac{1}{2} \left[ \sigma^2(x)\pi(x;\theta_0) \right]^{\prime} \right)\\
\times
\left( \mu(x;\theta)\hat{\pi}(x) - \frac{1}{2}\left[ \sigma^2(x) \hat{\pi}(x)\right]^{\prime} + \mu(x;\theta)\pi(x;\theta_0) - \frac{1}{2}\left[ \sigma^2(x) \hat{\pi}(x) \right]^{\prime} \right)\\
\times w(x)dx\Biggr|\\
\leq \left\{ \int_{\mathbb{R}} \left( \mu(x;\theta) (\hat{\pi}(x)-\pi(x;\theta_0)) - \frac{1}{2} \left[ \sigma^2(x)(\hat{\pi}(x)-\pi(x;\theta_0)) \right]^{\prime} \right)^2 w(x)dx \right\}^{1/2}\\
\times
\left\{  \int_{\mathbb{R}} \left( \mu(x;\theta) (\hat{\pi}(x)+\pi(x;\theta_0)) - \frac{1}{2} \left[ \sigma^2(x)(\hat{\pi}(x)+\pi(x;\theta_0)) \right]^{\prime} \right)^2 w(x)dx \right\}^{1/2}\\
=\sqrt{T_1(\theta)}\sqrt{T_2(\theta)}
\end{multline*}
with obvious definitions of $T_1(\theta)$ and $T_2(\theta).$ This inequality and Lemma \ref{lemma_t1_t2} from Appendix \ref{appendix} then yield \eqref{convergence}, which in view of the remarks we made at the beginning of this proof completes the proof of the theorem.
\end{proof}

\begin{proof}[Proof of Theorem \ref{rootn}]
Introduce the set
\begin{equation}
\label{gnepsilon}
G_{n,\varepsilon}=\{ |\hat{\theta}_n-\theta_0| \leq \varepsilon \},
\end{equation}
where $\varepsilon>0$ is some fixed number. Since $\theta_0$ is an interior point of $\Theta,$ by choosing $\varepsilon$ small enough one can achieve that on the set $G_n$ the estimator $\hat{\theta}_n$ belongs to the interior of $\Theta$ too. By the fact that $\hat{\theta}_n$ is a point of minimum of $R_n(\theta)$ it then follows that $1_{G_{n,\varepsilon}} \dot{R}_n(\hat{\theta}_n)=0.$ From this and from the mean-value theorem we have
\begin{align*}
1_{G_{n,\varepsilon}}\dot{R}_n(\theta_0) & = 1_{G_{n,\varepsilon}} \left(\dot{R}_n(\theta_0) - \dot{R}_n(\hat{\theta}_n) \right)\\
&=1_{G_{n,\varepsilon}}\int_0^1 \ddot{R}_n (\hat{\theta}_n+\lambda(\theta_0-\hat{\theta}_n))d\lambda(\theta_0-\hat{\theta}_n).
\end{align*}
The statement of the theorem follows by multiplication of the leftmost and rightmost terms of the above equality with $\sqrt{n}$ and application of Lemmas \ref{lemma_rootn1} and \ref{lemma_rootn2} from Appendix \ref{appendix}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm_an}]
From the proofs of Theorem \ref{rootn} and Lemmas \ref{lemma_rootn1}--\ref{lemma_rootn2} from Appendix \ref{appendix} (note that our assumptions on $h$ and $\gamma$ are also used here), as well as Slutsky's lemma (Lemma 2.8 in \cite{vdvaart}) it follows that in order to establish the theorem, it is sufficient to establish asymptotic normality of
\begin{align*}
\sqrt{n+1}\int_{\mathbb{R}}v(x)(\hat{\pi}_n(x)-\pi(x;\theta_0))dx&=
\sqrt{n+1}\int_{\mathbb{R}}v(x)(\ex[\hat{\pi}_n(x)]-\pi(x;\theta_0))dx\\
&+\sqrt{n+1}\int_{\mathbb{R}}v(x)(\hat{\pi}_n(x)-\ex[\hat{\pi}_n(x)])dx.
\end{align*}
By a standard argument, cf.\ the proof of Proposition 1.2 in \cite{tsybakov}, and by our assumption on $h,$ the first term on the right-hand side of the above display converges to zero. As far as the second term is concerned, by a change of the integration variable to $u=(x-Z_j)/h$ and a simple rearrangement of the terms it can be rewritten as
\begin{multline*}
\frac{1}{\sqrt{n+1}}\sum_{j=0}^n \left\{ v(Z_j) - \ex[v(Z_j)] \right\}\\
+\frac{1}{\sqrt{n+1}}\sum_{j=0}^n \int_{-1}^{1} \{ v(Z_j+hu) - v(Z_j) \}K(u)du\\
-\sqrt{n+1} \ex\left[ \int_{-1}^{1} \{ v(Z_j+hu) - v(Z_j) \}K(u)du\right].
\end{multline*}
We want to show that the last two terms on the right-hand side of the above display vanish in probability as $n\rightarrow\infty.$ By Chebyshev's inequality it is sufficient to prove that
\begin{equation*}
\sqrt{n+1} \ex\left[ \int_{-1}^{1} \{ v(Z_j+hu) - v(Z_j) \}K(u)du\right]=o(1).
\end{equation*}
This, however, can be done through a standard argument (cf.\ the proof of Proposition 1.2 in \cite{tsybakov}) by expanding $v(Z_j+hu)$ into the Taylor polynomial of order $\alpha$ and next using the fact that $K$ is a kernel of order $\alpha,$ which yields that the left-hand side of the above display is of order $n^{1/2}h^{\alpha}=o(1).$ On the other hand, by Theorem 18.5.3 in \cite{ibragimov},
\begin{multline*}
\left\{(n+1)\left[\var[v(Z_0)]+2\sum_{j=1}^{\infty}\cov(v(Z_0),v(Z_j))\right]\right\}^{-1/2}\\
\times\sum_{j=0}^n \left\{ v(Z_j) - \ex[v(Z_j)] \right\}\convd \mathcal{N}\left(0,1\right).
\end{multline*}
Combination of the above results and Slutsky's lemma yield the statement of the theorem.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Hi all,well I just got back from the doctor. The appointment went well. I told him all what was going on with me. So he did a complete exam. He noticed the brown blood/discharge. He told me I may have had a m/c. I got a faint postive in the office and the brown blood point to m/c. the u/s showed a sac but no baby. He also couldn't find a heartbeat. He said he was sorry and it looked like I suffered a m/c. I didn't have a healthy pregnancy.It look like I was 3 months along. Which would explain alot. Last night bright red blood came. Hopefully I won't haveto have a D/C. But the pregnancy seem to be coming out on its own......Right now i'm happy I got answers. Its just not the answers I wanted. I feel ok. A little cramps(painful last night) and bleeding. I am sad, but feel like something had to have been wrong with the pregnancy. My body wasn't ready for pregnancy yet. Mother nature did her job....A m/c is a part of life. I had one at 3 months. A least the baby wasn't born stillborn or unhealthy. Thank god for that. I took the news well. I had a feeling something was wrong. That feeling was right. There was nothing that could have been done to stop the m/c. Miscarriages happen..........Good news: I am able to have more babies if I want to. One day DH and I will make that decision. Right now I'm enjoying my two beautiful boys. I am lucky to have two kids already. I'm very blessed and loved................Smooches!!!!
Thanks y'all for all your prayers and nice comments.
Hi KEEKEE, I've read some of your post and I'm so so sorry about the m/c. I believe in karma, and you have been a real pillar of strength and support for a lot of these women. I believe the universe will give you everything you want and need. STAY STRONG!! I hope you will still visit the forums. Good wishes!
Keekee, See you didn't belong in the Loony Bin!!! You knew something was going on with your body. I'm sorry for the news. I hope you don't have to do a D/C also. And I will pray for you too that the pregnancy continues to comes out on it's own and your body returns to normal quickly. I'm glad you can still have kids! Maybe down the road you will have a little girl, so your boys will have someone to pick on and of course play big brother to. :) I wish you well!
AF arrived this morning. I'm gutted.
Hey Kee kee did the dr. do a urine test or a blood test to tell?
Hi everyone, I'm up early. Its 7:00am here. That early for me. I get to bed late......First I say to say thank you for all your kind words. You all made me smile.Dh and I decided not to tell our family. So you and Dh have been my support. Thanks again........... Amiee, I had a OB exam to find out what was going on. No, I still didn't get a postive.................Estee, I'm so sorry to hear about your Af. I know how much you want a baby. I know right now you are hurting. I wish I can take the pain all away. We all know you deserve a baby. You need a baby. You will have a baby. I know you are sick/ tired of hearing this. Maybe the next time will work. I'm on meds, I just don't know what to say right now. I been hoping and praying for you. I wish I can give you a hug. It sounds like you need one. Please try to get some rest. I know thats hard. Don't hold back tears. Just let it out. You will feel a little better. I'm sending you tons of kisses............I'm on meds. I'm gonna try to post later. I had to give in to the pain meds. I'm having cramps, but the bleeding is slowing down. It remind me of my past painful periods. I'm handling it well................Smooches!!!!!


text:
Aviva in UK Russian mission
2 December, 2010 - 23:24 By Staff Reporter
Norwich-based insurance giant, Aviva, was among a powerful British trade delegation to Russia, led by Business Secretary, Vince Cable.
The three-day visit to Moscow was designed to strengthen ties between the UK and Russia and to promote trade worth billions to the UK's economy every year.He accompanied the largest ever UK business delegation, comprising 37 businesses that have enjoyed success in Russia in sectors ranging from pharmaceuticals to finance and from energy to food and drink.More than six hundred British businesses now have a base in Russia including many household names such as Marks & Spencer and Costa Coffee.Vince Cable said: "So far this year, we have exported nearly £2 billion of goods and services to Russia and I hope that, through delegations such as this, trade relations will continue to grow between our two countries."UK Trade and Investment (UKTI) International Trade Director for the East of England, Vic Annells, added: "Without question Russia has enormous potential for businesses in the East of England – however companies can sometimes be deterred by the complexities of this market and the challenges they face in exporting there."My message to companies is not to let this stop you. There is a wealth of opportunities on offer in this market. UKTI, through its local teams, can help you prepare and research the market thoroughly."• PHOTOGRAPH SHOWS: Vince Cable (left) during his three-day visit to Moscow
Medical furniture business to reign in Spain
Swedish deal for Sepura
Bee friendly insecticide creates a buzz around Europe
Nap hand for East of England region at IBC Amsterdam
Amino at the double with European deals

redpajama_set_name: RedPajamaCommonCrawl


text:
ReetFeed » Viral » 'what! I have got Padma Shri? Raveena Tandon was shocked to hear the news, happy on Republic Day
'what! I have got Padma Shri? Raveena Tandon was shocked to hear the news, happy on Republic Day
This time Republic Day became special for Raveena Tandon. Received the honor for commendable work in the entertainment world.
Mumbai. The Padma Awards were announced by the Government of India before the 74th Republic Day. The name of beautiful actress Raveena Tandon was also included in this list of awards. After getting Padma Shri to Raveena, she is getting congratulatory messages continuously. Raveena has dedicated this honor to her father Ravi Tandon. When Raveena got information about this honour, she was shocked.
When Raveena got a call and was told that she had been awarded the Padma Shri, her first reaction was, 'What, I have got the Padma Shri?'. Raveena could not believe this news at once. When he believed this news, he talked to Hindustan Times about his career and journey so far.
(pc: instagram@officialraveenatandon)
Everyone's love is the real reason
In the 90s, Raveena gave many hit films in Bollywood. She is still active in the world of entertainment and likes to play roles out of the box. Regarding the award, he had to say, 'What can I say? I am feeling very honored and grateful. Actually, all this has happened because of the love of the people. I have been a part of the industry for a long time and everyone has given me opportunities, opportunities to do good work. I was not expecting this honor at all. I do not understand how to react to this.
Raveena Tandon's daughter Rasha will be seen with Ajay Devgan! Abhishek Kapoor will launch, Nyasa's special relationship with lead actor
Significantly, recently Raveena has also been selected as a delegate in W20. This is the Women Empower Engagement wing of the G20. Raveena has been included in this wing for her social work. Talking about the work front, she was last seen in South's blockbuster 'KGF 2' in an important role.
Tags: Padma Shri Award, Raveena Tandon, Republic day


The Britannia Inn taken in the late 1890's; pictured stood outside is probably Peter O'Malia.
The Britannia no longer exists today but was situated on the main road heading North out of Chorley and is first recorded as early as 1818.
Landlords listed through the years were Thomas Riddihough (1818), M Morton (1822), Thomas Bradshaw (1824), Thomas Hattern (1828), Peter Hunter (1835-39), Jeremiah Porter (1851), Alfred Whalley (1861), William Riding (1861), Alfred & James Whalley (1865-76), Edward Tattersall (1879-1882), Joseph Bond (1889), Margaret Bond (1891), Peter O'Malia (1896-1900) and Joseph O'Malia (1900-).
Joseph O'Malia appeared in the papers in 1900 following a request to open a new hotel in Chorley at the junction of Hamilton Road and Devonshire Road. His application was turned down but his license was renewed nonetheless at the Britannia.
By 1911 Joseph O'Malia was living to the north of the town on Beaconsfield Terrace and working at the Gas Works on Water Street, which would have been adjacent to the Britannia. Joseph's father and previous landlord Peter O'Malia was still living close by at 33 Parker Street and working as a Grocer in 1905.
There is no trace of the Britannia on the 1911 census or commercial directories after 1901. The 1911 census goes from no.48 Water Street to no.144 so I can only assume that by this time the building had been demolished.


Commercial-grade table features a blow-molded, high-density polyethylene top.
Legs are constructed of sturdy, 1" tubular steel with a charcoal-gray powder-coated finish.
Commercial-grade table features a blow-molded, high-density polyethylene top. Heavy-duty top with radius edges is dent, scratch, and stain resistant!
Legs are constructed of sturdy, 1" tubular steel with a charcoal-gray powder-coated finish. They also feature non-marring feet and fold into the underside for easy storage and protection.


Appetizer recipes is pretty easy to make. The recipe is easy to memorize, easy to make larger or smaller, and has infinite variations. There are different ways to make Appetizer recipe and these can prepare according to your choice—to make Appetizer recipe you can make different recipes, but there are several Appetizer recipe which rely on flavors that aren't baked "in" to the cheesecake.
To 180 degrees Celcius, preheat your oven!
Place butter in a bowl and crush in the 2 cloves of Garlic!
Mix very well until smooth!
Slice your bread into diagonal shapes and spread the butter evenly on top!
Place in your preheated oven and cook until Golden!
Remove from oven and place on a warm serving dish!
It's a great way to help feed a lot of hungry people, and easy to pull together, even at the last minute. Better yet, teach your kids how to make these goodies and they'll be able to feed themselves.


The 2016 United States presidential election in the District of Columbia was held on November 8, 2016, as part of the 2016 general election in which all fifty states and the District of Columbia participated. D.C. voters chose electors to represent them in the Electoral College via a popular vote.
On March 12 and June 14, 2016, voters chose delegates to the Republican and Democratic National Conventions respectively.
Clinton won the election with 282,830 votes or 90.9%, the third strongest win for a Democrat after Barack Obama's victories in 2008 and 2012. Donald Trump received 12,723 votes or 4.1% of the vote, which is the lowest share of the popular vote received by any Republican candidate since voters in the District of Columbia were granted the right to vote in presidential elections. As a result of this, Hillary Clinton received the widest margin of victory, 86.8%, of any winning presidential candidate, breaking the record of 86% set by Barack Obama in 2008. The District of Columbia has voted for the Democratic ticket in every election since 1964, which was the first election in which D.C. voters were allowed to participate.
The incumbent President of the United States, Barack Obama, a Democrat and former U.S. Senator from Illinois, was first elected president in the 2008 election, running with former Senator Joe Biden of Delaware. Defeating the Republican nominee, Senator John McCain of Arizona, with 52.9% of the popular vote and 68% of the electoral vote, Obama succeeded two-term Republican President George W. Bush, the former Governor of Texas. Obama and Biden were reelected in the 2012 presidential election, defeating former Massachusetts Governor Mitt Romney with 51.1% of the popular vote and 61.7% of electoral votes. Although Barack Obama's approval rating in the RealClearPolitics poll tracking average remained between 40 and 50 percent for most of his second term, it experienced a surge in early 2016 and reached its highest point since 2012 during June of that year. Analyst Nate Cohn noted that a strong approval rating for President Obama would equate to a strong performance for the Democratic candidate, and vice versa.
Following his second term, President Obama was not eligible for another reelection. In October 2015, his running-mate and two-term Vice President Biden decided not to enter the race for the Democratic presidential nomination either. With their term expiring on January 20, 2017, the electorate was asked to elect a new president, the 45th president and 48th vice president of the United States, respectively.
Main article: Political party strength in Washington, D.C.
The Twenty-third Amendment to the United States Constitution, ratified in 1961, grants the District of Columbia the right to choose presidential electors equal to the number from the least populous state (currently Wyoming's three). Since the amendment's ratification, the District of Columbia has cast its electoral votes for the Democratic candidate in every election. A Republican has never been the District's Mayor, and the current Council has 10 Democrats and two Independents.
Due to the small geographical size of the District of Columbia and the very small number of Republicans in the District, the local Republican party decided go directly to a "state convention", which took place at the Loews Madison Hotel at 1177 15th St NW from 10 a.m. - 4 p.m. The Convention/Caucus method was chosen because the June 14th primary was deemed too late, and DC would be penalized and only get 16 delegates.
The Democratic primary was held June 14. The date was chosen because it was thought that by then the race would be over and the voters could then concentrate on local races.
On March 30, ten weeks ahead of the Washington D.C. primary, NBC affiliate News 4 reported that the Democratic Party's D.C. State Committee had submitted registration paperwork for listing presidential candidate Bernie Sanders on the primary ballots a day late, even though the Sanders campaign had correctly and timely registered with the state party. After a voter filed a challenge, this would possibly lead to Sanders' name being missing on the ballots. As the D.C. Council announced it would hold an emergency vote to put Sanders back on the ballots, and with Clinton's campaign chairman John Podesta asking to make sure an administrative error wouldn't exclude a candidate, D.C. Democratic Party chairwoman Anita Bonds told CNN that "Bernie will be on the ballot." She further explained that the party has always notified the D.C. board of elections a day after the deadline, with the only difference being that this time, someone challenged the inclusion of Sanders.
^ "General Election 2016 - Unofficial Results". Retrieved November 10, 2016.
^ "District Of Columbia Presidential Election Voting History". Retrieved 2018-10-08.
^ Tom Sherwood (March 30, 2016). "Bernie Sanders May Be Off DC Ballot After Democratic Party Filing". NBC 4 Washington. Retrieved April 1, 2016.
^ Tom Sherwood (March 31, 2016). "DC to Hold Emergency Vote to Get Bernie Sanders on Democratic Primary Ballot". NBC 4 Washington. Retrieved April 1, 2016.
^ Tom LoBlanco (March 31, 2016). "Sanders likely on D.C. ballot despite challenge". CNN. Retrieved April 1, 2016.


text:
Try Try Again, Sixth Is A Charm For Mercedes
International Marathon Swimming Hall of Fame inductees Kevin Murphy, Tim Johnson and Montserrat Tresserras Dou all contributed a variety of interesting information about pioneering marathon swimmer Mercedes Gleitze.
Johnson recalled, "Mercedes' 1928 swim was terrific, pioneering a swim across the Strait of Gibraltar. It was an unimaginable and unthinkable swim at the time. She made sure the press knew why she was doing it.
Most telling is the swim wasn't duplicated for twenty years."
Montserrat added, "Mercedes swam the English Channel and her swim was recognized by Channel Swimming Association after she and her coach George Henry Allen accepted the affidavit before a commissioner of oaths.
On page 63 of my book Nadando El Estrecho, Sus Orígenes Y Su Historia (Swimming the Strait, Its Origin and History), the affidavit is reproduced. Mercedes was the first European lady and third lady in the Channel Swimming Association files."
Mercedes was successful on her sixth attempt, after three attempts from Africa and three attempts from Spain. She finally swam from Tarifa, Spain on April 5th 1928 to Punta Leona on the Moroccan coast in 12 hours 50 minutes. And she kept on swimming.
Murphy noted, "I have a typed copy of a report of her 1929 Wash swim which makes clear that she swam from Butterwick to Heacham after being dissuaded from attempting Skegness to Hunstanton."
Johnson added an interesting note of a focused marathon swimmer, "Consider what Mercedes did on her honeymoon. After marrying her sweetheart, Patrick Carey, she went to Turkey and swam the Hellespont at its widest point in 2 hours 45 minutes. She was married on August 9th 1930, and by August 16th she was off stroking in the Hellespont."
Sources for this article include the New York Times from April 7th 1928, the Asociación de cruce a nado del Estrecho de Gibraltar website, the NY Times from August 17th 1930 entitled Woman Swims Hellespont, the Dover Museum, the Dover Express, the Madrid Library Town Council, the Dover Library, and the Nadando El Estrecho, Sus Orígenes Y Su Historia.
Copyright © 2008-2018 by World Open Water Swimming Association


text:
Singapore Football
Tampines want to be first team to beat Home this season
Tampines Rovers skipper Madhu Mohana challenging for the high ball during the season-opening Charity Shield against Albirex Niigata. TNP PHOTO: GAVIN FOO
Tampines fired up to end Home's unbeaten run and boost their own title hopes
Shamir Osman
Sports Correspondent
TAMPINES ROVERS HOME UNITED
They conceded four goals the last time they played at the Jalan Besar Stadium, as their Asian Football Confederation (AFC) Cup dream came to an end.
Tampines Rovers ended their Group G campaign with a whopping 17 goals conceded, letting in four goals or more in half of their six games.
However, it is a different story in the Great Eastern-Hyundai S.League and Home United, their opponents tomorrow night, had better be wary.
Tampines have conceded only four goals in seven S.League games this season - the best defensive record in the league.
And they want to be the first team to beat Home in the league this year to fuel their own title hopes.
"The AFC Cup is played at a different pace, it's faster," Tampines skipper Madhu Mohana told The New Paper.
Players' extra training among reasons for Albirex dominance
Gombak, Tanjong Pagar apply to rejoin S.League in 2019
DPMM release Kean and imports
Albirex hungry for more after defending S.League title
CHASE THE GAME
"But, in those games, we conceded an early goal, had to chase the game, and we were forced to open up at the back. That doesn't happen very often."
This is an important game, we don't want Home and Albirex to run away from the rest of us. Tampines Rovers skipper Madhu Mohana, on tonight's match against home United
Tampines are third on 15 points in the nine-team standings.
They are two points behind Home, with defending champions Albirex Niigata perched atop the ladder on 19 points.
"I have always wanted to be in a team that concede few goals, because that's how you win the league - with a solid defence," said Tampines' Singapore international Daniel Bennett.
"Both Home and us need the three points and it's a hard one to predict.
"They've improved from last year for sure, but we're getting back to a full squad, something we've not had all year."
Bennett, 39, was referring to the injury troubles that saw the Stags play with just 12 senior outfield players on matchday at one point this season.
They've improved from last year for sure, but we're getting back to a full squad, something we've not had all year. Tampines Rovers defender Daniel Bennett on home United
"It's still early in the season and, to be honest, I never look at the league table," he added.
"I just want the team to do what we can on the pitch.
"Both teams have several national players, and also players who have that will to succeed.
"Obviously, Albirex will be looking at this game where one of us (Tampines and Home) will drop points."
Tampines are keen to put their AFC Cup woes behind them, and there is no better way than to inflict on Home their first defeat in this season's league - and get right into the S.League title race.
"This is an important game, we don't want Home and Albirex to run away from the rest of us," said Madhu.
"That was what happened last year, Albirex just ran away (with the title)."
"It's important to get points tomorrow, because I think we have a good chance of winning the league," added the 26-year-old defender.
"We've been training hard, and I think it's a really good chance that we have.
"We just can't drop points from games like this. Both are good teams with several national players, and it'll come down to who wants it more on the day."
s.leaguetampines roversHome United
shamiro@sph.com.sg
Read articles by Shamir Osman


text:
Colin Firth's wife's ex-lover officially charged with harassment | People
Like in Fatal Attraction, history seems to have repeated but in real life. Journalist Marco Bancaccia has been formally accused of harassing Colin Firth's wife, according to the British newspaper. The Times. Livia Firth and the Italian journalist had a relationship that lasted 11 months during a marriage separation that occurred between 2015 and 2016 and which they agreed to keep private.
Last year, Livia Firth denounced to the Italian police "the terrifying campaign of harassment" to which she was subjected by the journalist and an investigation was opened that forced the actor and his wife to explain under what circumstances this relationship had taken place extramarital and the reasons that had forced them to go to the police. During the investigation, "abusive messages and emails" were found on Brancaccia's phone despite the fact that he denied the facts.
Last Tuesday the lawyers of both parties appeared in a closed hearing that took place in Rome and in which the charges were formalized, although due to a strike the hearing was moved to next July, at which time the judge will decide whether whether or not you send the case to trial. Brancaccia faces a possible sentence of five years and in the word of the prosecutor María Monteleone "we have sufficient evidence, otherwise we would not have asked to go to trial."
Last March, Brancaccia, 55, denied the accusations that Firth, 57, and his wife, 48, have made against him and told the magazine People that he had barely had contact with Livia since their relationship ended. "The reality is that Livia and I have known each other since we were young and then we had an 11-month love relationship between 2015 and 2016," explained the journalist. "It ended in June 2016 and I did not call her again. I sent her two messages from WhatsApp, a text message birthday greeting and an email to Colin, which I really regret.
Italian journalist Marco Branccacia.Twitter
In this message, he told the British actor about the relationship he had with his wife. "I sent it in a moment of frustration, of sadness," Brancaccia has argued. "It was a terrible mistake, so much so that I even apologized. But going from that to being a stalker is not correct, "he explained.
Brancaccia also referred to telling Colin Firth the details of his relationship with his wife, after seeing the couple appear on television together. "It hurt a lot, so I wrote to him and told him everything. After a couple of days, he replied. He was polite. He wrote something like: 'You hurt me, but I understand that it hurts you too.' After that, I never came back to contact them, "he says.
The Italian journalist was also not shy about giving details about his 11-month affair. He said that he and Livia Firth used to travel so they could see each other "as many times as possible" and that during that period they visited Italy, Brazil, London and New York. "It was a love story," he said. And he explained that the end of their relationship occurred when he pressured Livia to divorce her husband. "When I asked her when she would tell Colin, she abruptly ended our relationship.
The Firths have recognized the adventure that Livia had with Marco Brancaccia during their separation. And they explained that when they resumed their marriage relationship, they faced for months a "terrifying campaign of harassment" by Branccacia, which is neatly documented. A situation that the Firths have never wanted to make public for obvious reasons. , much of which is documented. For obvious reasons, the Firths have never had any desire to go public with this matter. "
Firth and Livia were married in 1997 and have two children, Luca, 16, and Matteo, 15. Brancaccia reacted to the marriage lawsuit by presenting, in May 2017, a crime of libel against Livia Firth, on a false accusation against the police or judicial authority.
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Dali Ancient Town is about 13km away from Xiaguan in Dali City and is one of Dali's Three Ancients-ancient city, ancient pagoda and ancient steles. History of Dali can be dates back to 1382 during the Ming Dynasty and it has been listed among the top historic cities of China.
Unlike any other Chinese town, the traditional Bai ethnic folk houses give the town distinctive feel. The town's layout was uniform, with five main streets from south to north and eight main streets from east to west, while marketplaces were neatly arranged within the town, which has remained unchanged to this date. Besides the Bai houses, the houses with gray-green roof tiles, peculiar workshops, temples, schools and churches with antique flavor are scattered. Traditional marble artworks, like pencil vases, striped screen, and a variety of woven handicrafts of fine straw are laid chockablock on both sides of the street to be appreciated and purchased. Its grand city wall, traditional Bai houses and marvelous scenery have been attracted many visitors.
Dali is also famous for the many types of marble it produces, which are used primarily in construction and for decorative objects. In fact, Dali is so famous for the stone that the name of marble in Chinese is literally Dali Stone.
Dali's newly finished administrative district that houses the newly opened Dali International Convention Center is in Longshan District. The Dali government's urban planning keeps its old and new districts separate, so those coming here for the first time often feel they have entered a time tunnel.
The new city is known as Xiaguan and is south of the old city. The old city was built during Ming Dynasty emperor Hongwu's reign (1368–1398). Owing to the distance between them, the old city is still peaceful and quiet.
Dali Ancient City roads are still maintained the chessboard type grid structure since the Ming and Qing dynasty, known as the nine streets and 18 lanes. The north-south two gate towers have been restored. A street across the city from south to north, deep and quite street lane is criss-cross from west to east. The whole city is full of grey tiles dwellings with cobblestone wall.
Wenxian Building is located in 1 km away from the south gate of Dali Ancient City, which is known as the first gate of the ancient city, and is the landmark of Dali Ancient City. It was established in the Qing dynasty emperor Kangxi period. Horizontal inscribed board of the building was written "historic town with rich culture" by Piantu, the chief commander of Yunnan Province in Kangxi period fortieth year (AD 1701). Wenxian Building stretch across the road which leads to Dali Ancient City from the south, and on both sides of the road lined with willow trees which is quite picturesque. Wenxian Building is two floors of gable and hip roof type, it has typical Bai architectural features.
Wenxian Building has gone through damage and reparation for several times in the history, and Wenxian Building reconstruction work started on November 28, 1998, completed on April 29, 1999. After reconstruction, Wenxian Building is 60 meters wide from east to west, 30 meters long from north to south and the main building is 24 meters high with a building area of 1600 square meters.
Wuhua Building is state guesthouse of ancient Nanzhao king, also called to Five Flowers Building. Since Nanzhao, Wuhua Building was burned down for many times, and rebuilt many times, but it was smaller after rebuilding over and over. Today the area of Wuhua Building has formed a certain scale of calligraphy and painting market, operating varieties include calligraphy, Chinese paintings like landscape flowers and birds painting and figures painting, etc., and leather trade and oil painting sale is very active.
The east-west Huguo Road in the Dali Ancient City is known as the "foreigner street". One by one Chinese and western restaurants, cafes, tea houses and gift shops here, their signboard and advertising is written in English, attracting blond foreigners lingering here and looking for ancient archaic Chinese rhyme, gradually the street become a unique scenery.


Climate change could severely impact the world's coffee-producing nations and turn a cup of decent java into a luxury in the years to come.
Fall is always a good time to create new habits, and coffee chains know it. These days, they are desperately trying to find any excuse to get you to drink their java.
Sylvain Charlebois, Professor in Food Distribution and Policy, Dalhousie University. This article was originally published on The Conversation, and is republished here with permission. Read the original article.


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Foss's groundbreaking and distinctive science fiction art revolutionized paperback covers in the 1970s and 80s. Dramatically raising the bar for realism and invention, his trademark battle-weary spacecraft, dramatic alien landscapes and crumbling brutalist architecture irrevocably changed the aesthetic of science fiction art and cinema. Featuring work for books by Isaac Asimov, E. E. 'Doc' Smith, Arthur C. Clarke, A. E. Van Vogt and Philip K. Dick, and film design for Ridley Scott and Stanley Kubrick, this volume brings together many rare and classic images that have never been seen or reprinted before, making it the first comprehensive retrospective of Chris Foss's SF career.
"Chris Foss's name has become pre-eminent among SF artists… He is in love with the monstrous, with angular momentum, with inertia-free projectiles and irresistible objects. " — Brian Aldiss
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Rian Hughes is an award-winning graphic designer, illustrator, font designer and comics artist, noted for his work on 2000AD, and Dan Dare. His illustration work is highly distinctive, wearing its design influences on its sleeve. He is the editor and designer of several books including "Lifestyle Illustration of the 60s" and the recently released "Cult-ure".
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Zakir Husain Delhi College (Evening) invited applications for Assistant Professor Post. Eligible candidates may apply online on or before 23 June 2015.
Zakir Husain Delhi College (Evening) invited applications for recruitment to the post of Assistant Professor in various disciplines. Eligible candidates may apply online through the prescribed format on or before 23 June 2015.
Good academic record with minimum 55% marks or an equivalent grade of B in the 7 point scale with letter grades O, A, B, C, D, E & F at the Master degree level in the relevant subject from an Indian University or an equivalent degree from a foreign University.
Qualified the National Eligibility Test (NET) conducted by the UGC/CSIR or a similar test accredited by the UGC (state level eligibility test SLET/SET).
Selection Procedure: Selection of the candidates will be made on the basis of performance in Interview/exam conducted by the administration.
How to Apply: Candidates are required to apply online through College Website http://zakirhusainpgeve.in on or before 23 June 2015.


We at the Golden State Pooled Trust respect that you have choices and that it can be overwhelming to make important life decisions affecting the personal and financial well-being of yourself or your loved one. It's important that you find independent counsel from an attorney who is familiar with special needs trusts (see here for a list of attorneys) to help you determine your options and select the best path forward.
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