text:
\section{Introduction}

Jump Markov linear systems have received considerable attention due
to its applications in a wide variety of signal processing systems
and control systems \cite{2005mjls,book1,fd,train}. For
discrete-time jump Markov linear systems, the dynamics are
represented by a number of modes governed by a finite state Markov
chain and within each mode the continuous state is described by a
stochastic difference equation. Unfortunately, computing the optimal
state estimate of jump Markov linear systems requires exponential
complexity as time progresses. As a result, many suboptimal filters
have been proposed such as the generalized pseudo Bayesian
\cite{gpb}, the interacting multiple model (IMM) estimator
\cite{1988imm}, the particle filter
\cite{2000mmpf,2001ieeesp,2003immpf} and the array algorithms
\cite{2009ieeeacarray}. See the survey for more detailed discussions
on multiple model methods \cite{survey5}.

State estimation for jump Markov systems with unknown measurement
noise statistics has been investigated in recent years. In
\cite{2008rekfjmns}, a robust extended Kalman filter (EKF) has been
developed for jump Markov nonlinear systems with uncertain noise,
where the uncertainty of noise covariance matrix is limited by an
upper bound and the filter is derived by solving a nonlinear
programming problem with inequality constraints. In
\cite{1994taclmmse}, a linear minimum mean square error (LMMSE)
estimator has been proposed for jump Markov linear systems without
Gaussian assumptions on the noise and the estimator has been
extended to develop an optimal polynomial filter for stochastic
systems with switching measurements in \cite{2009germani}. In
\cite{2006ieeecs}, a minimax filter has been derived for stochastic
bimodal systems with unknown binary switching statistics. In
\cite{2010immhinf}, the $H_\infty$ filter has been combined with the
IMM approach, where the purpose of the $H_\infty$ filter is to
minimize the worst possible effects of the unknown noise to the
estimation errors. In addition, some weighting parameters should be
designed carefully to guarantee the existence and the performance of
the $H_\infty$ filter.

Recently, the random matrix approach has been used for state
estimation of stochastic systems with unknown measurement noise
covariance \cite{2009ieeetac,2011ieeetaes}. By using different
conjugate prior distributions for the unknown measurement noise
covariance, state estimation for jump Markov linear systems with
unknown measurement noise covariance has been addressed in the
framework of Bayesian estimation. In \cite{ourwork1}, by treating
the conjugate prior for the noise variance parameters as the
inverse-Gamma distribution, an IMM estimator has been developed for
jump Markov linear systems. However, a serious limitation in this
filter is that the noise covariance is restricted as a diagonal
matrix. This assumption is used due to the fact that each diagonal
element of the matrix can be modeled by an inverse-Gamma
distribution but not the matrix itself. In fact, a matrix can be
considered as multivariate random variable and the inverse-Wishart
distribution can be used as the conjugate prior for the covariance
matrix of a multivariate Gaussian distribution \cite{1999mvd}. By
using the inverse-Wishart distribution as the conjugate prior for
the measurement noise covariance, an IMM estimator has been proposed
in \cite{ourwork2}. Due to the presence of the estimation of random
matrices, one difficulty encountered in the IMM estimation is the
combination of a set of weighted inverse-Wishart distributions. In
\cite{ourwork2}, an inverse-Wishart distribution is used to
approximate a set of weighted inverse-Wishart distributions by
matching the first order moment and the mean squared estimation
errors. However, it is not clear whether it is effective to
approximate a set of weighted inverse-Wishart distributions by using
the moment matching method.

In this paper, we attempt to propose a novel IMM estimator for jump
Markov linear systems with unknown measurement noise covariance. By
modeling the unknown measurement covariance as a random matrix
according to an inverse-Wishart distribution, the state and the
random matrix are estimated jointly in the framework of Bayesian
estimation. Instead of using the moment matching approach to address
the combination of a set of weighted inverse-Wishart distributions,
this difficulty is overcome by minimizing the weighted
Kullback-Leibler divergence for inverse-Wishart distributions. It is
shown that a closed form solution can be derived for the
optimization problem and the resulting solution coincides with an
inverse-Wishart distribution. A simulation study of maneuvering
target tracking is provided to illustrate the effectiveness of the
proposed filter. Simulation results show that the proposed filter
performs better than the previous work using the moment matching
approach.

The rest of this paper is organized as follows. In section
\Rmnum{2}, the problem of state estimation for jump Markov linear
systems is formulated. In section \Rmnum{3}, the weighted
Kullback-Leibler divergence is introduced and it is applied in the
IMM approach to develop a novel estimator. A numerical example is
provided in section \Rmnum{4}, followed by conclusions in section
\Rmnum{5}.

\section{Problem formulation}

Consider the following jump Markov linear system
\begin{align}\label{eq1}
x_k&=F_{k-1}(r_k)x_{k-1}+G_{k-1}(r_k)w_{k-1}(r_k)\\
z_k&=H_kx_k+v_k
\end{align}
where $x_k\in\mathbb{R}^n$ and $z_k\in\mathbb{R}^m$ denote the state
and the measurement vectors, respectively. $r_k$ is a discrete
variable denoting the state of a Markov chain and taking values in
the set $\mathcal{M}\triangleq\{1,2,\cdots,M\}$ according to the
transition probability matrix $\Pi=[\pi_{ij}]_{M\times M}$ with
\begin{align}
\pi_{ij}&\triangleq\mathbb{P}\{r_{k}=j|r_{k-1}=i\}\\
&\sum_{n=1}^M\pi_{ij}=1,~i\in\mathcal{M}
\end{align}
The quantities $F_{k-1}(r_k)$, $G_{k-1}(r_k)$ and $H_k$ are known
matrices. Note that the measurement equation (2) does not evolve
with time according to the Markov state. This is a reasonable
condition since the measurement is generally insensitive to the
state of the model. The process noise $w_{k-1}(r_k)$ corresponding
to mode $r_k$ and the measurement noise $v_k$ are assumed to be
mutually uncorrelated zero-mean white Gaussian processes with
covariance matrices $Q_{k-1}(r_k)$ and $R_k$, respectively. The
measurement noise covariance $R_k$ is assumed to be unknown and it
is modeled as a random matrix with the conjugate prior of an
inverse-Wishart distribution \cite{1999mvd}.

The aim of this paper is to derive the estimates of the state $x_k$
and the random matrix $R_k$ in the framework of Bayesian estimation.
To this end, the IMM approach is adopted to derive the estimates
recursively. One cycle of the IMM estimator consists of four steps
including interacting of mode-conditioned estimates,
mode-conditioned filtering, mode probability update and fusion of
mode-conditioned estimates \cite{1988imm}. Specifically, at each
time step, the initial condition for the filter matched to a certain
model is derived by mixing the estimates of all filters at the
previous time step. This is followed by a regular filtering step,
performed in parallel for each model. Then, the mode probability is
updated by using the measurement and a combination of the updated
estimates of all filters yields the final estimates. For IMM
estimation with random matrices, one difficulty encountered is how
to combine a set of weighted inverse-Wishart distributions in the
interacting and fusion steps. Moreover, the combined probability
density function is expected to be an inverse-Wishart distribution
which facilitates to derive the Bayesian estimation recursion. This
is illustrated in the following formulation.

\emph{Problem Formulation}: Assume that the mode-conditioned
posterior density function at time step $k-1$ is approximated by a
product of Gaussian and inverse-Wishart (GIW) distributions
\begin{align}\label{eq4}
p(x_{k-1},R_{k-1}|r_{k-1}=i,Z_{k-1})=\mathcal{N}(x_{k-1};\hat{x}_{k-1}^i,P_{k-1}^i)
\mathcal{IW}_m(R_{k-1};\nu_{k-1}^i,\Sigma_{k-1}^i)
\end{align}
where $Z_{k-1}\triangleq\{z_1,\cdots,z_{k-1}\}$ is the cumulative
set of measurements up to time $k-1$. $\mathcal{N}(x;\hat{x},P)$
denotes the probability density function of Gaussian distribution
with mean $\hat{x}$ and covariance matrix $P$
\begin{align}
\mathcal{N}(x;\hat{x},P)=\frac{1}{(2\pi)^{n/2}|P|^{1/2}}\exp\Big{[}-\frac{1}{2}(x-\hat{x})^TP^{-1}(x-\hat{x})\Big{]}
\end{align}
The notation $\mathcal{IW}_m(R;\nu,\Sigma)$ represents the
probability density function of an inverse-Wishart distribution with
degree $\nu$ and scalar matrix $\Sigma$
\begin{align}\label{eq5}
\mathcal{IW}_m(R;\nu,\Sigma)=\frac{2^{\frac{-(\nu-m-1)m}{2}}|\Sigma|^{\frac{\nu-m-1}{2}}}{\Gamma_m(\frac{\nu-m-1}{2})|R|^{\frac{\nu}{2}}}
\exp\Big{[}-\frac{1}{2}\mathrm{Tr}(R^{-1}\Sigma)\Big{]}
\end{align}
with $\Gamma_m(\cdot)$ being the multivariate Gamma function and
$\mathrm{Tr}$ being the trace function of a matrix.

Assume that the mode probabilities are also derived at time step
$k-1$
\begin{align}
\mathbb{P}\{r_{k-1}=i|Z_{k-1}\}=\mu_{k-1}^i
\end{align}

The problem considered in this paper is to, given a set of
mode-conditioned posterior density functions (5) and mode
probabilities (8), obtain a solution to the mixed probability
density function is of the same function form as (5), i.e.,
\begin{align}
p(x_{k-1},R_{k-1}|r_k=j,Z_{k-1})=\mathcal{N}(x_{k-1};\hat{x}_{k-1}^{0j},P_{k-1}^{0j})
\mathcal{IW}_m(R_{k-1};\nu_{k-1}^{0j},\Sigma_{k-1}^{0j})
\end{align}
and the fusion of mode-conditioned posterior density function at
time $k$
\begin{align}
p(x_{k},R_{k}|Z_{k})=\mathcal{N}(x_{k};\hat{x}_{k},P_{k})
\mathcal{IW}_m(R_{k};\nu_{k},\Sigma_{k})
\end{align}

\section{Proposed estimator}

In this section, the Kullback-Leibler divergence is briefly
reviewed, based on which the weighted Kullback-Leibler divergence is
defined to derive an optimal probability density function for a set
of weighted inverse-Wishart distributions. Then, the proposed
approach is utilized to address the problem of combination of
inverse-Wishart distributions in the IMM estimator.

\subsection{Kullback-Leibler divergence}

Let
\begin{align}
\mathcal{P}\triangleq\{p(x):
\mathbb{R}^n\rightarrow\mathbb{R}~\mathrm{such
~that}~\int_{\mathbb{R}^n}p(x)dx=1~ \mathrm{and}~ p(x)\geq0,\forall
x\in\mathbb{R}^n\}
\end{align}
denotes the set of probability density functions over
$\mathbb{R}^n$.

From the information-theoretic point of view, the difference between
two probability density functions $p(x)$ and $q(x)$ in $\mathcal{P}$
can be measured by the following Kullback-Leibler divergence
\begin{align}
\mathrm{D}_{\mathrm{KL}}(p||q)=\int p(x)\log\frac{p(x)}{q(x)}dx
\end{align}

In Bayesian statistics, the Kullback-Leibler divergence can be used
as a measure of the information gain in moving from a prior
probability density function $q(x)$ to a posterior probability
density function $p(x)$. The Kullback-Leibler divergence satisfies
$\mathrm{D}_{\mathrm{KL}}(p||q)\geq0$ with equality if, and only if
$p(x)=q(x)$. However, it is not a symmetrical quantity, that is to
say $\mathrm{D}_{\mathrm{KL}}(p||q)\neq
\mathrm{D}_{\mathrm{KL}}(q||p)$ \cite{prml}. Thus, the
Kullback-Leibler divergence should not be taken as a distance
rigorously. Nevertheless, the Kullback-Leibler divergence has been
shown to be geometrically important and it can be evaluated
numerically. In addition, the Kullback-Leibler divergence can be
considered an example of the Ali-Silvey class of information
theoretic measures \cite{alisil}, and it quantities how close a
probability distribution is to a candidate. The Kullback-Leibler
divergence can be used to find a probability distribution that best
approximates the candidate in the sense of minimizing the
Kullback-Leibler divergence. To represent the difference between a
probability density function and a set of probability density
functions, we adopt the following definition of the weighted
Kullback-Leibler divergence \cite{2014kld}.

\begin{definition}
Given $N$ probability density functions $p_i(x)\in\mathcal{P}$, and
relative weights $\lambda_i$ satisfying
\begin{align}
\lambda_i\geq0,~~~\sum_{i=1}^N\lambda_i=1
\end{align}
their weighted Kullback-Leibler divergence is defined as follows
\begin{align}
\bar{p}(x)=\arg\inf_{p\in\mathcal{P}}\sum_{i=1}^N\lambda_i\mathrm{D}_{\mathrm{KL}}(p||p_i)
\end{align}
\end{definition}

It can be seen that the weighted Kullback-Leibler divergence
$\bar{p}(x)$ is the one that minimizes the sum of the information
gains from the initial probability density functions. Thus, it is
coherent with the Principle of Minimum Discrimination Information
(PMDI) according to which the probability density function best
represents the current state of knowledge is the one which produces
an information gain as small as possible \cite{it}. It has been
shown that the above weighted Kullback-Leibler divergence can be
derived explicitly as follows.
\begin{lemma}(\cite{2014kld})
The weighted Kullback-Leibler divergence defined in (14) turns out
to be
\begin{align}
\bar{p}(x)=\frac{\prod_{i=1}^N[p^i(x)]^{\lambda_i}}{\int\prod_{i=1}^N[p^i(x)]^{\lambda_i}dx}
\end{align}
\end{lemma}

By applying the weighted Kullback-Leibler divergence to the
inverse-Wishart distributions, we can obtain a closed form solution
to (15), as shown in the following theorem.

\begin{theorem}
Given $N$ inverse-Wishart probability density functions
$\mathcal{IW}_m(X;a_i;A_i)$ and weights $\lambda_i$ satisfying (13),
their weighted Kullback-Leibler divergence in (15) takes the form
\begin{align}
\bar{p}(X)=\mathcal{IW}_m(X;\bar{a};\bar{A})
\end{align}
where
\begin{align}
\bar{a}&=\sum_{i=1}^N\lambda_i a_i\\
\bar{A}&=\sum_{i=1}^N\lambda_i A_i
\end{align}
\end{theorem}

\textbf{Proof}. From the definition of the inverse-Wishart
distribution (7), we have
\begin{align}
\prod_{i=1}^N[\mathcal{IW}_m(X;a_i;A_i)]^{\lambda_i}&\propto\prod_{i=1}^N|X|^{-\frac{\lambda_i
a_i}{2}}
\exp\Big{[}-\frac{1}{2}\mathrm{Tr}(\lambda_i X^{-1}A_i)\Big{]}\nonumber\\
&\propto|X|^{-\frac{\sum_{i=1}^N\lambda_i a_i}{2}}
\exp\Big{[}-\frac{1}{2}\mathrm{Tr}(X^{-1}\sum_{i=1}^N\lambda_i A_i)\Big{]}\nonumber\\
&\propto\mathcal{IW}_m(X;
\sum_{i=1}^N\lambda_ia_i;\sum_{i=1}^N\lambda_iA_i)
\end{align}

Notice that the denominator of (15) is a constant, hence
\begin{align}
\bar{p}(X)=c\mathcal{IW}_m(X; \bar{a};\bar{A})
\end{align}
where $c$ is a normalizing constant, $\bar{a}$ and $\bar{A}$ are
given by (17)-(18), respectively.

Since $\bar{p}(X)$ is a probability density function, we have
\begin{align}
\int \bar{p}(X)dX=c\int\mathcal{IW}_m(X; \bar{a};\bar{A})dX=c=1
\end{align}
the result is proved. $\blacksquare$

\begin{remark}
Theorem 1 states that the weighted Kullback-Leibler divergence
provides an optimal probability density function to a set of
weighted inverse-Wishart distributions. Moreover, the resulting
solution coincides with an inverse-Wishart distribution, where the
parameters $\bar{a}$ and $\bar{A}$ can be simply obtained by the
algebraic average. This strategy can be applied in the IMM
estimation with random matrices by treating mode-conditioned
posterior density functions and mode probabilities as a set of
weighted inverse-Wishart distributions.
\end{remark}

\subsection{IMM estimator by Kullback-Leibler divergence}

As the IMM approach has been well studied in the previous work
\cite{1988imm,ourwork1,ourwork2}, we present one cycle of recursion
in the following steps.

\emph{Step 1}. Interacting of mode-conditioned estimates

Since the state $x_{k-1}$ and the random matrix $R_{k-1}$ are
independent, the posterior density function (5) can be rewritten as
\begin{align}\label{eq4}
p(x_{k-1}|r_{k-1}=i,Z_{k-1})&=\mathcal{N}(x_{k-1};\hat{x}_{k-1}^i,P_{k-1}^i)\\
p(R_{k-1}|r_{k-1}=i,Z_{k-1})&=
\mathcal{IW}(R_{k-1};\nu_{k-1}^i,\Sigma_{k-1}^i)
\end{align}

The mixed posterior density function for the state $x_{k-1}$ is
given by
\begin{align}\label{eq6}
p(x_{k-1}|r_k=j,Z_{k-1}) &=\sum_{i=1}^Mp(x_{k-1}|r_{k-1}=i,Z_{k-1})
\mathbb{P}\{r_{k-1}=i|r_k=j\}\nonumber\\
&=\sum_{i=1}^M\mu_{k-1}^{i|j}\mathcal{N}(x_{k-1};\hat{x}_{k-1}^i,P_{k-1}^i)\nonumber\\
&\approx\mathcal{N}(x_{k-1};\hat{x}_{k-1}^{0j},P_{k-1}^{0j})
\end{align}
where the moment matching method is used to approximate the Gaussian
mixture terms
\begin{align}\label{eq7}
\mu^{i|j}_{k-1}&=\frac{\pi_{ij}\mu^{i}_{k-1}}{\sum_{l=1}^M\pi_{lj}\mu^l_{k-1}}\\
\hat{x}^{0j}_{k-1}&=\sum_{i=1}^M\mu^{i|j}_{k-1}\hat{x}^{i}_{k-1}\\
P^{0j}_{k-1} &=\sum_{i=1}^M\mu^{i|j}_{k-1}\big{[}P^{i}_{k-1}+
(\hat{x}^{i}_{k-1}-\hat{x}^{0j}_{k-1})
(\hat{x}^{i}_{k-1}-\hat{x}^{0j}_{k-1})^T\big{]}
\end{align}

The mixed posterior density function for the random matrix $R_{k-1}$
is derived by
\begin{align}
p(R_{k-1}|r_{k}=j,Z_{k-1})=\arg\inf_{p}\sum_{i=1}^M\mu_{k-1}^{i|j}\mathrm{D}_{\mathrm{KL}}(p||p(R_{k-1}|r_{k-1}=i,Z_{k-1}))
\end{align}

By using Theorem 1, the weighted Kullback-Leibler divergence is
given by
\begin{align}\label{eq8}
p(R_{k-1}|r_k=j,Z_{k-1})=\mathcal{IW}_m(R_{k-1};\nu_{k-1}^{0j},\Sigma_{k-1}^{0j})
\end{align}
where
\begin{align}
\nu_{k-1}^{0j}&=\sum_{i=1}^M\mu_{k-1}^{i|j}\nu_{k-1}^i\\
\Sigma_{k-1}^{0j}&=\sum_{i=1}^M\mu_{k-1}^{i|j}\Sigma _{k-1}^i
\end{align}

\emph{Step 2}. Mode-conditioned filtering

As in \cite{ourwork2}, taking the mixing estimates as inputs of
filters, the mode-conditioned posterior density function at time $k$
can be obtained by using variational Bayesian approximation
\begin{align}\label{eq42}
p(x_k|r_k=j,Z_k)&\approx\mathcal{N}(x_k;\hat{x}_{k}^j,P_{k}^j)\\
p(R_k|r_k=j,Z_k)&\approx\mathcal{IW}_m(R_{k};\nu_{k}^j,\Sigma_k^j)
\end{align}

\emph{Step 3}. Update of mode probabilities

As in \cite{ourwork2}, the mode probabilities are updated by
\begin{align}\label{eq48}
\mu_k^j=
\frac{\Lambda_k^j\sum_{l=1}^M\pi_{lj}\mu^l_{k-1}}{\sum_{i=1}^M\sum_{l=1}^M\pi_{li}\mu^l_{k-1}\Lambda_k^i}
\end{align}
where $\Lambda_k^i$ the likelihood function.

\emph{Step 4}. Fusion of mode-conditioned estimates

The overall posterior density function for the state $x_{k}$ is
given by
\begin{align}\label{eq6}
p(x_{k}|Z_{k})
&=\sum_{j=1}^Mp(x_{k}|r_{k}=j,Z_{k})\mathbb{P}\{r_k=j|Z_k\}\nonumber\\
&=\sum_{j=1}^M\mu_{k}^{j}\mathcal{N}(x_{k};\hat{x}_{k}^j,P_{k}^j)\nonumber\\
&\approx\mathcal{N}(x_{k};\hat{x}_{k},P_{k})
\end{align}
where the moment matching method is used to approximate the Gaussian
mixture terms
\begin{align}\label{eq7}
\hat{x}_{k}&=\sum_{j=1}^M\mu^{j}_{k}\hat{x}^{j}_{k}\\
P_{k} &=\sum_{j=1}^M\mu^{j}_{k}\big{[}P^{j}_{k}+
(\hat{x}^{j}_{k}-\hat{x}_{k}) (\hat{x}^{j}_{k}-\hat{x}_{k})^T\big{]}
\end{align}

The overall posterior density function for the random matrix $R_{k}$
is derived by
\begin{align}
p(R_{k}|Z_{k})=\arg\inf_{p}\sum_{j=1}^M\mu_{k}^{j}\mathrm{D}_{\mathrm{KL}}(p||p(R_{k}|r_{k}=j,Z_{k}))
\end{align}

By using Theorem 1, the weighted Kullback-Leibler divergence is
given by
\begin{align}\label{eq8}
p(R_{k}|Z_{k})=\mathcal{IW}_m(R_{k};\nu_{k},\Sigma_{k})
\end{align}
where
\begin{align}
\nu_{k}&=\sum_{j=1}^M\mu_{k}^{j}\nu_{k}^j\\
\Sigma_{k}&=\sum_{j=1}^M\mu_{k}^{j}\Sigma _{k}^j
\end{align}

Notice that the overall estimate of the random matrix is the
expectation of the inverse-Wishart distribution (39)
\begin{align}
\hat{R}_k=\frac{\Sigma_{k}}{\nu_{k}-2m-2}
\end{align}

\begin{remark}
In the proposed filter, two different strategies are utilized to
fuse the mode-conditioned estimates of the state and the random
matrix. Specifically, for the probability density function of the
state $x_k$, the moment matching approach is used to approximate a
set of weighted Gaussian distributions, which is widely used in the
IMM estimation. For the probability density function of the random
matrix $R_k$, the weighted Kullback-Leibler divergence is adopted to
approximate a set of weighted inverse-Wishart distributions. The
weighted Kullback-Leibler divergence is adopted because only the
first order moment can be matched for a set of weighted
inverse-Wishart distributions by using the moment matching approach.
Moreover, the weighted Kullback-Leibler divergence provides a closed
form solution with an inverse-Wishart distribution.
\end{remark}

\begin{remark}
The difference between the proposed filter and the previous version
in \cite{ourwork2} is that the sum of weighted inverse-Wishart
distributions in \emph{Step 1} and \emph{Step 4} is approximated by
using the weighted Kullback-Leibler divergence instead of moment
matching method. Specifically, an inverse-Wishart distribution is
used to approximate the sum of weighted inverse-Wishart
distributions in \cite{ourwork2}, where the first order moment and
the mean-squared estimation error are matched to determine the
parameters of the inverse-Wishart distribution, e.g., the overall
estimate of the random matrix in \cite{ourwork2} is given by
\begin{align}
\hat{R}_k=\sum_{j=1}^N\frac{\mu_{k}^j\Sigma_k^j}{\nu_k^j-2m-2}
\end{align}
It can be seen that the overall estimates of the random matrix in
(42) and (43) are not matched in general. However, they are matched
if $\nu_k^i=\nu_k^j$ for all $i,j=1,2,\cdots,M$.
\end{remark}

\section{Numerical Example}

In this section, we compare the performance of the proposed filter
with the previous work via a two-dimensional (2-D) maneuvering
target tracking example. In order to produce a fair comparison, the
tracking parameters in \cite{ourwork2} are adopted. To be specific,
the target dynamics is described by the following coordinated turn
model
\begin{align}\label{eq52}
x_k=\begin{bmatrix} 1&\frac{\sin(\omega
T)}{\omega}&0&-\frac{1-\cos(\omega T)}{\omega}\\
0&\cos(\omega T)&0&-\sin(\omega T)\\
0&\frac{1-\cos(\omega T)}{\omega}&1&\frac{\sin(\omega T)}{\omega}\\
0&\sin(\omega T)&0&\cos(\omega T)
\end{bmatrix}x_{k-1}+w_{k-1}
\end{align}
where $x_k=(p_{x,k}~v_{x,k}~p_{y,k}~v_{y,k})^T$ denotes the target
state. $\omega$ denotes the coordinated turn rate and $T=1$ is the
sampling time period. The process noise $w_{k-1}$ is zero-mean white
Gaussian with covariance matrix
\begin{align}\label{eq53}
Q=qI_{2\times2}\otimes\begin{bmatrix}
T^4/4&T^3/2\\
T^3/2&T^2\\
\end{bmatrix}
\end{align}
where $q=0.09$ is the level of power spectral density and $\otimes$
denotes the Kronecker product.

Three models corresponding to different turn rates are used in the
simulations, i.e., $-4^{\circ}/\mathrm{s}$, $0^{\circ}/\mathrm{s}$
and $4^{\circ}/\mathrm{s}$. The switching between three models is
governed by a first order time-homogeneous Markov chain with known
transition probabilities $\pi_{ii}=0.8$ ($i=1,2,3$) and
$\pi_{ij}=0.1$ ($i\neq j$). It is assumed that only the target
positions are measured and the measurement noise is zero-mean white
Gaussian with unknown covariance matrix
\begin{align}
R_k=\begin{bmatrix}r&r/20\\r/20&r\end{bmatrix}
\end{align}
where $r$ is the level of power spectral density .

To evaluate the performance of the proposed filter, the IMM-KF with
known measurement noise covariance matrix is considered as the
baseline algorithm. For simplicity of notation, the proposed filter
with weighted Kullback-Leibler divergence is shortly denoted by
IMM-KL and the IMM estimation with moment matching approach is
shortly denoted by IMM-MM \cite{ourwork2}. Simulation results are
derived from 1000 Monte Carlo runs, where the root mean square error
(RMSE) in position and the estimation error of the random matrix
$R_k$ defined in \cite{2012vbrtttsp} are used.

The initial inverse-Wishart distribution for $R_{0}$ is chosen as
$\nu_0^i=20$ and $\Sigma_0^i=\mathrm{diag}\{50,50\}$ $(i=1,2,3)$.
The number of fixed iteration steps in the variational Bayesian
update is taken to be $N_c=2$ to derive mode-conditioned estimates.
The level of the measurement noise density is taken to be $r=200$.
The RMSE in position are shown in Fig.1. The simulation results show
that the IMM-KL outperforms the IMM-MM and the IMM-KL converges
faster than the IMM-MM at the beginning of the simulation intervals.
Especially, the IMM-KL generates almost identical results with the
IMM-KF as time progresses. The IMM-KF performs better than the
IMM-MM and IMM-KL at the beginning of the simulation intervals. This
is due to the fact that there is a large gap between the initial
prior distributions and the truth for covariance matrix. The
estimation errors with respect to the random matrix $R_k$ are shown
in Fig.2. It can be seen that the IMM-KL achieves higher accuracy
than the IMM-MM.

To further evaluate the performance of the proposed filter with
respect to different levels $r$, the averaged RMSE in position and
the averaged estimation errors with respect to the covariance matrix
are presented in Fig.3 and Fig.4, respectively. It can be observed
that the performance of the proposed IMM-KL is comparable to that of
the IMM-KF with known covariance matrix. The IMM-KL outperforms the
IMM-MM with respect to the estimation of covariance matrix.

\begin{figure}
\centering
\includegraphics[width=0.7\hsize]{fig1.eps}
\caption{RMSE in position versus time.} \label{fig1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\hsize]{fig2.eps}
\caption{Estimation error of the covariance matrix versus time.}
\label{fig2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\hsize]{fig3.eps}
\caption{Averaged RMSE in position versus different levels $r$.}
\label{fig3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\hsize]{fig4.eps}
\caption{Estimation error of the covariance matrix versus different
levels $r$.} \label{fig4}
\end{figure}

\section{Conclusion}

In this paper, we proposed a novel IMM estimation approach with
random matrices. Instead of using the moment matching method to
address of the combination of a set of weighted inverse-Wishart
distributions in the IMM estimation, the weighted Kullback-Leibler
divergence is applied and a closed form solution can be derived.
Simulation results show that the proposed filter outperforms the
previous work using the moment matching method. The proposed
approach can be expected to be used for maneuvering extended targets
tracking where the target extent is modeled via a random matrix
\cite{2008wkaes,2011etttsp,2014taekg,2012lan1,2012lan2,2014tsplan}.

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text:
\section{Introduction}\label{section 1}
Over the last decades, the problem of the choice of inputs so as to obtain maximal information from the experiments has been given a great deal of interest in the engineering literature (see \textit{e.g.} \cite{Levadi, Aoki, NG77, NG84, Mehra74a, Mehra74b,Goodwin2007}) as well as  in the statistical literature (see \textit{e.g.} \cite{Wald1943,Kiefer,Whittele1973,Brouste2012}). Many of these works mentioned above focused on experiments design for identification of directly observed dynamic system parameters. The classical approach for experiment design consists of a two-step procedure: maximize the Fisher information under energy constraint of the input and find an adaptive estimation procedure. In this field, there are many approaches such as sequential design, Bayesian design, mini-max and robust design (see \textit{e.g.} \cite{Levadi,Lopez1975} and the references therein). The referred  works not only considered the identification of directly observed dynamic systems such as \cite{Mehra74a} but also the partially observed systems \cite{Mehra74b}.

For the experiment design of the autoregressive with exogenous terms (ARX) model, many works focused on the system driven by the white noises (see \textit{e.g} \cite{NG84,Ljung}) since the independence of white noises is a very good property for the estimation procedure. However, there was little work on the experiment design in the case of dependent noises, such as the fractional Gaussian noise.

In this paper, we consider the ARX(1) model with the fractional Gaussian noise. Using the technique of the Kalman filtering (see \textit{e.g.} \cite{BK,BCK}) and following the idea of maximum likelihood estimation (MLE) for the AR(1) with regular dependent Gaussian noise, we study the asymptotic design problem of the input of the ARX(1) model and give an efficient estimator of the unknown signal drift parameter. For these purposes, we will separate the desired problem in two subproblems. On the one hand, we obtain the optimal input by maximizing the Fisher information under certain constraints on the energy of the input. Using Laplace transform computations, we can see that the asymptotical optimal input does not depend on the unknown parameter and so the MLE is a good candidate to reach efficient estimation. On the other hand, we investigate the asymptotic properties of the MLE. The consistency,
asymptotic normality and convergence of the moments of the MLE are proposed.

The organization of this paper is as follows. In Section \ref{section 2}, we state the problem of experimental design and asymptotic results of this paper. Section 3 provides some simulation examples to show the performance of the proposed MLE.  All the proofs are collected in the Appendix.

\section{Statement of the problems and asymptotic behaviors}\label{section 2}
We consider the model
\begin{equation}\label{model}
X_n=\vartheta X_{n-1}+u(n)+\xi_n,\,0< \vartheta<1, X_0=0\,,
\end{equation}
where $u(n)$ denotes the deterministic function of $n$ and $\xi=(\xi_n,\,n\in \mathbb{Z})$ is the centred regular stationary Gaussian noise with
\begin{equation}\label{spectral density}
\int_{-\pi}^{\pi}|\log f_{\xi}(\lambda)|d\lambda<\infty\,,
\end{equation}
where $f_{\xi}(\lambda)$ is the spectral density of $\xi$.

We suppose that the covariance $c=c(m,n),\, m,n\geq 1$ is
$$
\mathbf{E}\xi_m\xi_n=c(m,n)=\rho(|n-m|),\, \rho(0)=1\,,
$$
which is positive defined. In fact, the condition \eqref{spectral density} can be deduced by $\rho \sim cn^{-\alpha}$ with $\alpha>0$.

\begin{rem}
There are many Gaussian process satisfying the condition \eqref{spectral density}, such as three well-known noises: MA(1), AR(1) and fractional Gaussian noise (fGn). Let us mention that, in the case of fGn, the covariance function only depends on the Hurst parameter $H$. As we can see later, one can estimate $\vartheta$ and $H$ simultaneously.
\end{rem}

Now, we consider the problem of estimating the unknown parameter $\vartheta$ with the observation data
 $X^{(N)}=\left(X_n,\, n=1,\cdots, N\right)$.  Let $L(\vartheta, X^{(N)})$ be the likelihood function for $\vartheta$. Then the Fisher information can be written as
 $$
 \mathcal{I}_N(\vartheta,u)=-\mathbf{E}_{\vartheta}\frac{\partial^2}{\partial \vartheta^2}\ln L(\vartheta,X^{(N)}).
 $$

In fact, the function $u(n)$ is a deterministic function which is contained in the control space $\mathcal{U}_N$ of real (or complex function):
$$
\mathcal{U}_N=\left\{u(n)\big| \frac{1}{N}\sum_{n=1}^{N}\left\|\frac{\sum\limits_{m=1}^nk(n,m)u(m)}{\sigma_{n+1}}  \right\|^2\leq 1  \right\}\,,
$$
where the function $k(n,m)$ and $\sigma_n$ are defined in \eqref{eq:sigma} and \eqref{eq:k}.

Moreover, we define
$$
\mathcal{J}_N(\vartheta)=\sup_{u\in \mathcal{U}_N}\mathcal{I}_N(\vartheta,u)\,.
$$

Our main goal is to find an adapted estimator $\bar{\vartheta}_N$ of the parameter $\vartheta$, which is asymptotically efficient  in the sense that, for any compact $\mathbb{K}\in (0,1)=\{\vartheta|0<\vartheta<1\}$,
\begin{equation}\label{efficiency}
\sup_{\vartheta \in \mathbb{K}}\mathcal{J}_N(\vartheta)\mathbf{E}_{\vartheta}(\bar{\vartheta}_N-\vartheta)^2=1+o(1),
\end{equation}
as $N\rightarrow \infty$.

In what follows, we state the following results.

\begin{thm}\label{optimal input}
The asymptotical optimal input in the class of control $\mathcal{U}_T$ is $u_{opt}(n)=\sum\limits_{m=1}^nK(n,m)\sigma_{m+1}$. Moreover,
$$
\lim_{N\rightarrow \infty}\frac{\mathcal{J}_N(\vartheta)}{N}=\mathcal{I}(\vartheta)\,,
$$
where $\mathcal{I}(\vartheta)=\frac{1}{1-\vartheta^2}+\frac{1}{(1-\vartheta)^2}$.
\end{thm}

\begin{rem}\label{rem1}
For the case of $-1<\vartheta<0$, the optimal input will be $u_{opt}(n)=(-1)^n\sum\limits_{m=1}^nK(n,m)\sigma_{m+1}$ or $u_{opt}(n)=(-1)^{n+1}\sum\limits_{m=1}^nK(n,m)\sigma_{m+1}$ and $\mathcal{I}(\vartheta)=\frac{1}{1-\vartheta^2}+\frac{1}{(1+\vartheta)^2}$. It is worth emphasizing that the proof of this case is the same of $\vartheta>0$.
\end{rem}

Since the optimal input does not depend on the unknown parameter $\vartheta$, we can consider $\bar{\vartheta}$ as the MLE $\hat{\vartheta}_N$. The following theorem states that $\hat{\vartheta}_N$ will reach the efficiency of \eqref{efficiency}.

\begin{thm}\label{asymptotical normal}
With the optimal input $u_{opt}(n)$ defined in Theorem \ref{optimal input}, for $0<\vartheta<1$, the MLE $\hat{\vartheta}_N$ has the following properties:
\begin{itemize}
\item $\hat{\vartheta}_N$ is strong consistency, that is $\hat{\vartheta}_N\overset{a.s.}{\rightarrow }\vartheta$ as $N\rightarrow\infty$.

\item $\hat{\vartheta}_N$ is uniformly consistent on compact $\mathbb{K}\subset \mathbb{R}$, \textit{i.e.} for any $\nu>0$
$$
\lim_{N\rightarrow \infty}\sup_{\vartheta \in \mathbb{K}}\mathbf{P}_{\vartheta}^N\left\{\left|\hat{\vartheta}_N-\vartheta\right|>\nu\right\}=0
$$

\item $\hat{\vartheta}_N$ is uniformly on compacts asymptotically normal, i.e., as $N\rightarrow\infty$,
$$
\lim_{N\rightarrow \infty}\sup_{\vartheta\in \mathbb{K}}\left|\mathbf{E}_{\vartheta}f\left(\sqrt{N}\left(\hat{\vartheta}_N-\vartheta\right)-\mathbf{E}f(\xi)\right)\right|=0,\, \forall f\in \mathcal{C}_b\,,
$$
where $\xi$ is a zero mean Gaussian random variable with variance $\mathcal{I}^{-1}(\vartheta)$ defined in Theorem \ref{optimal input}.
Moreover, we have the uniform on $\vartheta\in \mathbb{K}$ convergence of the moments: for any $q>0$,
$$
\lim_{N\rightarrow \infty}\sup_{\vartheta\in \mathbb{K}}\left|\mathbf{E}_{\vartheta}\left|\sqrt{N}\left(\hat{\vartheta}_N-\vartheta\right)\right|^q -\mathbf{E}|\xi|^q                   \right|.
$$
\end{itemize}
\end{thm}
\begin{rem}
For $\vartheta<0$, we have the same asymptotic results and the same Fisher information $\mathcal{I}(\vartheta)$, which is defined in Remark \ref{rem1}.
\end{rem}

\section{Simulation study}\label{simulation}
In this section, we study the finite sample properties of the proposed MLE $\hat{\vartheta}_N$. We conduct Monte Carlo studies for different values of $\vartheta$ to numerically investigate the efficiency of the MLE. Since the simulation of the noise (AR(1), MA(1), fGn) has been studied in \cite{BCK} with $u(n)=0$, in this paper, we only present the results of the fGn with $H=0.6$. In fact, the main obstacle of the Monte Carlo simulation is the difficulty of simulating fGn, in contrast to white noises. In the literature, there are some methods to deal with the problem of simulating fGn (see \cite{coeurjolly2000simulation}). In this paper, we simulate fGn by the Wood-Chan method, which simulates fGn by extracting the square root of the following autocovariance matrix:
\begin{eqnarray*}
\Sigma = \frac{h^{2H}}{2}\left(
\begin{array}{cccc}
\Sigma_{0}  & \Sigma_{1}  &\ldots &  \Sigma_{m-1}  \\
\Sigma_{1}  & \Sigma_{0}  &\ldots &  \Sigma_{m-2}  \\
\ldots      &\ldots       &\ldots & \ldots         \\
\Sigma_{m-1}& \Sigma_{m-2}&\ldots &  \Sigma_{0}  \\
\end{array}
\right)\,,
\end{eqnarray*}
\noindent where
\begin{equation*}
\Sigma_{j}=\left\{\begin{array}{ll}\vspace{2mm}
\gamma(j)  \quad\quad\quad\quad\quad\quad\quad   0\leq j\leq m/2\\
\gamma(m-j)  \quad\quad\quad\quad\quad           m/2 \leq j\leq m-1
\end{array}
\right.\,,
\end{equation*}
and
\begin{align*}
\gamma(i-j)=&\frac{h^{2H}}{2}\left( \mid i-j-1 \mid^{2H} + \mid i-j+1 \mid^{2H}-2 \mid i-j \mid^{2H} \right) \,, \;\; i,j=0,1,\ldots,N-1 \,.
\end{align*}
By construction, the matrix $\Sigma$ is symmetric and circulant. For
algorithmic rapidity, $m\geq 2(N-1)$. Note that $\Sigma$ can be
decomposed as $\Sigma=R\Lambda R^{\ast}$ where $\Lambda$ is the
diagonal matrix of eigenvectors of $\Sigma$ and $R$ is a unitary
matrix which is given by
\begin{equation*}
R_{j,k}=\frac{1}{\sqrt{m}}\exp\left(-2 i \pi \frac{jk}{m}\right)
\quad j, k=0,1,\ldots,m-1  \,, \quad (i=\sqrt{-1})\;.
\end{equation*}

Moreover, $R^{\ast}$ denotes the adjoint of $R$, i.e., $R^{\ast}_{j,k}$ is the complex conjugate of $R_{k,j}$. If $W=R\Lambda^{1/2} R^{\ast}Z$ $\left(Z\sim\mathcal {N}\left(0,I_{m}\right) \right)$ where $Z$ is a $m$-dimensional vector of standard normal distribution, then $W\sim \mathcal {N}\left(0,\Sigma\right)$. As a consequence,  the simulation of fGn can be done in following steps:

\begin{description}
 \item[{\bf(\romannumeral1)}] Constructing the circulative matrix $\Sigma$ from the autocovariance function $\gamma$.
 \item[{\bf(\romannumeral2)}] Calculating the eigenvalues of $\Sigma$ by Fast Fourier Transform
\begin{equation}
\lambda_{k}=\sum_{j=0}^{m-1}\Sigma_{j}\exp\left(-2 i \pi
\frac{jk}{m}\right)\,,\quad k=0,1,\ldots,m-1\;. \label{4.3}
\end{equation}
 \item[{\bf(\romannumeral3)}] Simulating for $F = RZ$ with
\begin{equation*}
F_{j}=\left\{\begin{array}{ll}\vspace{2mm}
U_{0}  \quad\quad \quad\quad\quad\quad\quad\quad\quad                j=0\,,\\
\frac{1}{\sqrt{2}} \left(U_{j}+iV_{j}\right)  \quad\quad\quad\quad   0<j<m/2\,,\\
V_{0} \quad\quad \quad\quad\quad\quad\quad\quad\quad                 j=m/2\,,\\
\frac{1}{\sqrt{2}} \left(U_{N-j}-iV_{N-j}\right)  \quad\;\;
m/2<j<m\,,
\end{array}
\right.
\end{equation*}
\noindent where $U_{j}$ and $V_{j}$ are independent standard normal
random variables.
 \item[{\bf(\romannumeral4)}] Reconstructing of the fGn $\xi$ by Fast Fourier Transform
\begin{equation}\label{fGn}
\xi_{k}=\frac{1}{\sqrt{m}}\sum_{j=0}^{m-1}\sqrt{\lambda_{j}}F_{j}\exp\left(-2
i \pi \frac{jk}{m}\right)\,,\quad k=0,1,\ldots,m-1  \,.
\end{equation}
\noindent with $\lambda_{j}$ defined by \eqref{4.3}.
\end{description}

Once we obtain fGn, we can simulate $X_{n}$ using \eqref{model} and calculate the MLE $\hat{\vartheta}_N$. Setting $N=2500$, we carry out a simulation study proposed above. Then, we obtain the MLE $\hat{\vartheta}_N$ by using some generating datasets. For each case, replications involving $l=5000$ samples are simulated from the true model. To evidence the asymptotic laws of $\hat{\vartheta}_N$, we can now investigate the asymptotic distributions of the following statistics:
\begin{equation}\label{function phi}
\Phi(N,\vartheta,X)=\sqrt{N}\left(\hat{\vartheta}_{N}-\theta\right)\,.
\end{equation}

The results are presented in the following Figures and the following Table, which reports the variance of $\Phi(N,\vartheta,X)$ with different values of $\vartheta$.

\begin{center}
\resizebox{160mm}{80mm}{\includegraphics{1.eps}}\\
Fig.1. Histogram of the statistic $\Phi(N,\vartheta,X)$ with $\vartheta=-0.7$.
\end{center}

\begin{center}
\resizebox{160mm}{80mm}{\includegraphics{2.eps}}\\
Fig.2. Histogram of the statistic $\Phi(N,\vartheta,X)$ with $\vartheta=-0.4$.
\end{center}

\begin{center}
\resizebox{160mm}{80mm}{\includegraphics{3.eps}}\\
Fig.3. Histogram of the statistic $\Phi(N,\vartheta,X)$ with $\vartheta=0.4$.
\end{center}

\begin{center}
\resizebox{160mm}{80mm}{\includegraphics{4.eps}}\\
Fig.4. Histogram of the statistic $\Phi(N,\vartheta,X)$ with $\vartheta=0.7$.
\end{center}

\begin{center}
\begin{tabular}{ccccccccccccccc}
\multicolumn{4}{c}{Table 1.  The variance of $\Phi(N,\vartheta,X)$ with different values of $\vartheta$}\\
\toprule%
Vales of $\vartheta$ &-0.7&-0.4&0.4&0.7 \\
\hline
Theoretical variance of $\Phi(N,\vartheta,X)$&0.0765&0.2520&0.2520&0.0765\\
Empirical variance of $\Phi(N,\vartheta,X)$& 0.0763&0.2492&0.2489&0.0756 \\
\bottomrule
\end{tabular}
\end{center}

The histogram indicates that the normal approximation of the distribution of the statistic $\Phi(N,\vartheta,X)$ is reasonable even when sampling size $N$ is not so large. From Table 1, we can see that the empirical variance of $\Phi(N,\vartheta,X)$ are close to their theoretical variance, which confirms our theoretical result: the convergence of the distribution of $\Phi(N,\vartheta,X)$ is fast.

\section{Appendix}
The appendix provides the proofs of Theorem \ref{optimal input} and Theorem \ref{asymptotical normal}. For the sake of convenience, we first introduce some well-known results.

\subsection{Stationary Gaussian sequences}
We define
\begin{equation}\label{eq:sigma}
\sigma_1\epsilon_1=\xi_1,\,\,\, \sigma_n\epsilon_n=\xi_n-\mathbf{E}(\xi_n|\xi_1,\xi_2,\,\cdots,\, \xi_{n-1}),\, n\geq 2\,,
\end{equation}
where $\epsilon_n\sim\mathcal{N}(0,1),\, n\geq 1$ are independent. It follows from the Theorem of Normal Correlation (\cite{Liptser}, Theorem 13.1) that there exists a deterministic kernel denoted by $k(n,m),\, n\geq m,\, n\geq 1$ such that
\begin{equation}\label{eq:k}
\sigma_n\epsilon_n=\sum_{m=1}^n k(n,m)\xi_m,\,\,\, k(n,n)=1\,.
\end{equation}

For $n\geq 1$, let $\beta_{n-1}$ be the partial correlation coefficient
\begin{equation}\label{eq:k and beta}
-k(n,1)=\beta_{n-1},\, n\geq 1\,.
\end{equation}

It is worth mentioning that $\sigma_n\beta_n$ is nothing but the correlation between $\xi_0$ and $\epsilon_n$ namely $\sigma_n\beta_n=\mathbf{E}\xi_0\epsilon_n$ (see equation \eqref{eq: k rho sigma}). Then following the idea of \cite{Durbin}, we have the following relationship
\begin{equation}\label{eq:sigma and beta}
\sigma_n^2=\prod_{m=1}^{n-1}(1-\beta_m^2),\, n\geq 2,\, \sigma_1=1\,.
\end{equation}
\begin{equation}\label{eq: k rho sigma}
\sum_{m=1}^n k(n,m)\rho(m)=\beta_n\sigma_n^2\,.
\end{equation}
\begin{equation}
k(n+1,n+1-m)=k(n,n-m)-\beta_nk(n,m)\,.
\end{equation}

On the other hand, there exists an inverse deterministic kernel $K=(K(n,m),n\geq 1, n\geq m)$ such that
\begin{equation}\label{eq:K}
\xi_n=\sum_{m=1}^nK(n,m)\sigma_m\epsilon_m.
\end{equation}

\begin{rem}
From this replacement, we have
$$
\xi_n=\sum_{k=0}^{\infty}a_k\epsilon_{n-k}\,.
$$

Let us mentioned that we always assume $a_0\neq 0$, if not, we can renumber $a_k$. When the filter generated by $\xi_n,\xi_{n-1},\, \cdots,\, \xi_{-\infty}$ and $\epsilon_n,\, \cdots,\, \epsilon_{-\infty}$ are the same, we have $\lim\limits_{n\rightarrow \infty}=a_0^2$, which means $\sum\limits_{n=1}^{\infty}\beta_n<\infty$ or $\beta \rightarrow 0$ when $n\rightarrow \infty$.
\end{rem}

\subsection{Model Transformation}
Let us define the process $Z=(Z_n,\, n\geq 1)$ such that
\begin{equation}\label{eq:Z}
Z_n=\sum_{m=1}^nk(n,m) X_m,\, n\geq 1\,,
\end{equation}
where $k(n,m)$ is the kernel defined in \eqref{eq:k}.

Similar to \eqref{eq:K}, we have
$$
X_n=\sum_{m=1}^n K(n,m)Z_m\,.
$$

It is worth mentioning that the process $Z$ has the same filtration of $X$. In the following parts, let the observation be $(Z_1,\, Z_2,\,\cdots,\, Z_N)$. Actually, it was shown in \cite{BCK} the process $Z$ can be considered as the first component of a $2$-dimensional AR(1) process $\zeta=\zeta_n,\, n\geq 1$, which is defined by:
$$
\zeta_n=\left(\begin{array}{c}Z_n \\ \sum\limits_{r=1}^{n-1}\beta_rZ_r    \end{array}\right)\,.
$$

It is not hard to obtain that $\zeta_n$ is a $2$-dimensional Markov process which satisfies the following equation:
\begin{equation}\label{eq:zeta}
\zeta_n=A_{n-1}\zeta_{n-1}+bv(n)+b\sigma_n\epsilon_n,\, n\geq 1,\, \zeta_0=\mathbf{0}_{2\times 1}\,,
\end{equation}
with
\begin{equation}\label{eq:An and b}
A_n=\left(\begin{array}{cc}\vartheta & \vartheta\beta_n \\\beta_n & 1\end{array}\right),\, b=\left(\begin{array}{c}1 \\0\end{array}\right)
\end{equation}
and $\epsilon_n\sim\mathcal{N}(0,1)$ are independent. Here the function $v(n)$ will be in the space of
$$
\mathcal{V}_N=\left\{v\big| \frac{1}{N}\sum_{n=1}^N\left\|\frac{v(n)}{\sigma_{n+1}}  \right\|^2\leq 1   \right\}\,.
$$

\subsection{Fisher Information}
As we have interpreted, the observation will be the first component of the process $\zeta=(\zeta_n,\, n\geq 1)$. Now from the equation \eqref{eq:zeta}, it is easy to write the Likelihood function $L(\vartheta,X^{(N)})$, which depends on the function $v(n)$:
\begin{equation}\label{likelihood function}
L(\vartheta,X^{(N)})=\prod_{n=1}^N\frac{1}{\sqrt{2\pi\sigma_n^2}}
\exp\left(-\frac{1}{2}\sum_{n=1}^N
\left(\frac{b^*\left(\zeta_n-A_{n-1}^{\vartheta}\zeta_{n-1}\right)}
{\sigma_n}\right)^2\right).
\end{equation}

Consequently, the Fisher Information $\mathcal{I}_N(\vartheta, v)$ can be written as
\begin{equation}\label{Fisher Information 1}
\mathcal{I}_N(\vartheta,v)=-\mathbf{E}_{\vartheta}\frac{\partial^2}{\partial \vartheta^2}\ln L(\vartheta, X^{(N)})=\mathbf{E}_{\vartheta}\sum_{n=1}^{N-1}\left(\frac{a_{n}^*\zeta_{n}}{\sigma_{n+1}}\right)^2,
\end{equation}
where $a_n=\left(\begin{array}{c}1 \\\beta_n\end{array}\right)$.

In what follows, we present the proof of our main results.
\subsection{Proof of Theorem \ref{optimal input} }
To prove the theorem \ref{optimal input}, we separate the Fisher Information of \eqref{Fisher Information 1} into two parts:
\begin{eqnarray*}
\mathcal{I}_N(\vartheta,v) &=& \mathbf{E}_{\vartheta}\sum_{n=1}^{N-1} \left(\frac{a_{n}^*\zeta_{n}-a_{n}^*\mathbf{E}_{\vartheta}\zeta_{n}+a_{n}^*\mathbf{E}_{\vartheta}\zeta_{n}}{\sigma_{n+1}}\right)^2 \\
&=&\mathbf{E}_{\vartheta}\sum_{n=1}^{N-1} \left(\frac{a_{n}^*(\zeta_{n}-\mathbf{E}_{\vartheta}\zeta_{n})}{\sigma_{n+1}}   \right)^2+\sum_{n=1}^{N-1}\left(\frac{a_{n}^*\mathbf{E}_{\vartheta}\zeta_{n}}{\sigma_{n+1}}    \right)^2\\
&=&\mathbf{E}_{\vartheta}\sum_{n=1}^{N-1} \left(\frac{a_{n}^*\mathcal{P}^{\vartheta}_n}{\sigma_{n+1}}   \right)^2+\sum_{n=1}^{N-1}\left(\frac{a_{n}^*\mathbf{E}_{\vartheta}\zeta_{n}}{\sigma_{n+1}}    \right)^2\\
&=&\mathcal{I}_{1,N}(\vartheta)+\mathcal{I}_{2,N}(\vartheta,v).
\end{eqnarray*}
where $\mathcal{P}_n^{\vartheta}$ satisfies the following equation:
$$
\mathcal{P}^{\vartheta}_n=A_{n-1}^{\vartheta}\mathcal{P}^{\vartheta}_{n-1}+b\sigma_n\epsilon_n,\, \mathcal{P}^{\vartheta}_0=\mathbf{0}_{2\times 1}.
$$
Obviously, $\mathcal{I}_{1}(\vartheta)$ does not depends on $v(n)$. Thus, and as presented in \cite{BCK}, we have
$$
\lim_{N\rightarrow \infty}\mathbf{E}_{\vartheta}\exp\left(-\frac{1}{2N}\sum_{n=1}^{N-1}\frac{(a_n^*\mathcal{P}^{\vartheta}_n)^2}{\sigma_{n+1}^2} \right)=\exp\left(-\frac{1}{2}\mathcal{I}_1(\vartheta)\right)
$$
and $\mathcal{I}_1(\vartheta)=\frac{1}{1-\vartheta^2}$.

A standard calculation yields
\begin{equation}\label{I1}
\lim_{N\rightarrow \infty}\frac{\mathcal{I}_{1,N}(\vartheta)}{N}=\mathcal{I}_1(\vartheta)=\frac{1}{1-\vartheta^2}.
\end{equation}

To compute $\mathcal{I}_{2,N}(\vartheta)$, let $s(n)=\frac{\mathbf{E}_{\vartheta}\zeta_n}{\sigma_{n+1}}$. Then, we can see that $s(n)$ satisfies the following equation:
\begin{equation}\label{eq:s}
s(n)=A_{n-1}s(n-1)\frac{\sigma_n}{\sigma_{n+1}}+bf(n)\,,
\end{equation}
where $f(n)=\frac{v(n)}{\sigma_{n+1}}$ and it is bounded.

Note that $\beta_n\rightarrow 0$ and
$\frac{\sigma_n}{\sigma_{n+1}}\rightarrow 1$, we assume that for $n=1,2,\cdots$, $\frac{\sigma_n}{\sigma_{n+1}}\leq (1+\varepsilon)$ and $\beta_n\leq \varepsilon$ for the sufficiently small positive constant $\varepsilon$ and $(1+\varepsilon)\vartheta<1$. Consequently, we can state the following result.

\begin{lem}Let $Y=(Y_n,\, n\geq 1)$ be the 2-dimension equation, which  satisfies the following equation:
$$
Y_n=\left(\begin{array}{cc}\vartheta & 0 \\0 & 1\end{array}\right)Y_{n-1}+bf(n),\, Y(0)=y_0\,.
$$

Then, we have
$$
\lim_{N \rightarrow \infty}\frac{1}{N}\sum_{n=1}^N(a_n^*s(n))^2=\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{n=1}^N (b^*Y_n)^2\,.
$$

\begin{proof}
For the sake of notational simplicity, we introduce a 2-dimensional equation $Y=(Y'_n,\, n\geq 1)$, which satisfies the following equation:
$$
Y'_n=\left(\begin{array}{cc}\vartheta & 0 \\0 & 1\end{array}\right)Y_{n-1}\frac{\sigma_n}{\sigma_{n+1}}+bf(n),\, Y'_0=y'_0\,.
$$

In this situation, we have three comparison. First, we compare $b^*s(n)-b^*Y'_n$. A standard calculation implies
$$
s(n)-Y'_n=\left(\begin{array}{cc}0 & \vartheta \beta_{n-1} \\\beta_{n-1} & 0\end{array}\right)(s(n-1)-Y'_{n-1})\frac{\sigma_n}{\sigma_{n+1}},
$$
which implies $b^*(s(n)-Y'_n)\rightarrow 0$.

Now, we compare $b^*Y_n-b^*Y'_n$. A simple calculation shows that
$$
b^*(Y'_n-Y_n)=\vartheta(b^*(Y'_n-Y_n))+\vartheta\left(\frac{\sigma_n}{\sigma_{n+1}}-1\right)b^*Y'_{n-1}\,,
$$
which implies $b^*(Y'_n-Y_n)\rightarrow 0$ since $f(n)$ is bounded and $\frac{\sigma_n}{\sigma_{n+1}}-1\rightarrow 0$.

Finally, as $\beta_n\rightarrow 0$ and the component of $s(n)$ is bounded, we can easily obtain $a_n^*s(n)-b^*s(n)\rightarrow 0$, which achieves the proof.
\end{proof}
\end{lem}

Now, we define $\alpha(n)=b^*Y_n$. Then $\alpha(n)=\vartheta \alpha(n-1)+f(n)$, where  $f(n)$ is in the space of $\mathcal{F}_N=\left\{f(n)\big|\frac{1}{N}\sum\limits_{n=1}^Nf^2(n)\leq 1\right\}$. Since the initial value $\alpha(0)$ will not change our result, we assume $\alpha(0)=0$ without loss of generality.

Let $\mathcal{J}_{2,N}=\sup\limits_{v\in \mathcal{V}_N}\mathcal{I}_{2,N}(\vartheta,v)$. Then, it is clear that
\begin{equation}\label{J2N}
\lim_{N\rightarrow\infty}   \frac{\mathcal{J}_{2,N}(\vartheta)}{N-1}=\lim_{N\rightarrow \infty}\frac{1}{N}\sup\limits_{f\in \mathcal{F}_N}\sum_{n=1}^N\alpha^2(n).
\end{equation}
Now to prove theorem \ref{optimal input}, we only need the following lemma.

\begin{lem}
As $\mathcal{J}_{2,N}(\vartheta)$ presented in \eqref{J2N}, we have
\begin{equation}\label{J2}
\lim_{N\rightarrow \infty}\frac{\mathcal{J}_{2,N}(\vartheta)}{N-1}=\mathcal{I}_2(\vartheta)\,,
\end{equation}
where $\mathcal{I}_2(\vartheta)=\frac{1}{(1-\vartheta)^2}$.
\end{lem}
\begin{proof}
First of all, taking $f(n)=1$, it is easy to get the lower bound
$$
\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{n=1}^N\alpha^2(n)\geq \frac{1}{(1-\vartheta)^2}.
$$

Moreover, a simple calculation shows that
$$
\alpha(n)=\varphi(n)\sum\limits_{i=1}^{n}\varphi^{-1}(i)f(i),\, n\geq 1,\, \alpha(0)=0\,,
 $$
where
$$
\varphi(n)=\vartheta \varphi(n-1), \, \varphi(0)=1\,.
$$

Obviously, we can rewrite $\frac{1}{N}\sum_{n=1}^N\alpha^2(n)$ as
$$
\frac{1}{N}\sum_{n=1}^N\alpha^2(n)=\sum_{n=1}^{N}\left(\varphi(n)\sum_{i=1}^n\varphi^{-1}(i)\frac{f(i)}{\sqrt{N}}\right)\left(\varphi(n)\sum_{i=1}^n\varphi^{-1}(i)\frac{f(i)}{\sqrt{N}}\right).
$$
or
$$
\frac{1}{N}\sum_{n=1}^N\alpha^2(n)=\sum_{i=1}^{N}\sum_{j=1}^{N}F_{N}(i,j)\frac{f(i)}{\sqrt{N}}\frac{f(j)}{\sqrt{N}},
$$
where
$$
F_{N}(i,j)=\sum_{\ell=i\bigvee j}^{N}\left(\varphi(\ell)\varphi^{-1}(i)\right)\left(\varphi(\ell)\varphi^{-1}(j)\right)\,.
$$

Let $\phi_n=\varphi^{-1}(n)\sum\limits_{\ell=n}^{N}\varphi_{\ell}\epsilon_{\ell}$ with $\epsilon_{\ell}\sim\mathcal{N}(0,1)$ are independent. Then, we have
$$
F_{N}(i,j)=\mathbf{E}(\phi_i\phi_j)
$$
and
$$
\phi_{n-1}=\vartheta \phi_n+\epsilon_{n-1},\, \phi_{N}=0.
$$

Let us mentioned that $F_{N}(i,j)$ is a compact symmetric operator for fixed $N$. We should estimate the spectral gap (the first eigenvalue $\nu_1(N)$) of the operator. The estimation of the spectral gap is based on the Laplace transform of $\sum\limits_{i=1}^{N}\phi_i^2$, which is written as
$$
L_{N}(a)=\mathbf{E}_{\vartheta}\exp \left(-\frac{a}{2}\sum_{i=1}^{N}\phi_i^2\right)\,,
$$
for sufficiently small negative $a<0$.

On one hand, when $a>-\frac{2}{\nu_1(N)}$, $\phi$ is a centred Gaussian process with covariance operator $F_{N}$. Using Mercer's theorem and Parseval's identity, $L_T(a)$ can be represented by
\begin{equation}\label{Laplace transform 1}
L_N(a)=\prod_{i \geq 1}(1+a\nu_i(N))^{-1/2}\,,
\end{equation}
where $\nu_i(N)$ is the sequence of positive eigenvalues of the covariance operator.

On the other hand, a straightforward algebraic calculation shows
\begin{equation}\label{Laplace transform 2}
L_N(a)=\left(\vartheta^{N-1}\Psi_{N}^1\right)^{-1/2}\,,
\end{equation}
where
$$
\Psi_{N}=\left(\begin{array}{cc}1 & 0\end{array}\right)\left(\begin{array}{cc}\frac{1}{\vartheta} & \frac{1}{\vartheta} \\\frac{a}{\vartheta} & \frac{a}{\vartheta}+\vartheta\end{array}\right)^{N-1}\left(\begin{array}{c}1 \\0\end{array}\right).
$$
For
$$
\Delta=\left(\frac{1+a}{\vartheta}+\vartheta \right)^2-4\geq 0\,,
$$
there exists two real eigenvalues $\lambda_2\geq 1,\, \lambda_1\leq 1$ of the matrix
$$
\left(\begin{array}{cc}\frac{1}{\vartheta} & \frac{1}{\vartheta} \\\frac{a}{\vartheta} & \frac{a}{\vartheta}+\vartheta\end{array}\right).
$$

Then, we can see that
$$
\Psi_N=\left(\frac{\lambda^{N-1}-\lambda^{N-1}}{\vartheta^2}-\frac{\lambda_2^{N-2}-\lambda_1^{N-2}}{\vartheta}\right)\frac{\vartheta}{\lambda_2-\lambda_1}\geq 0\,.
$$

That is to say for $\vartheta>0$ and for any $0>a\geq -(1-\vartheta)^2$, $L_N(a)\geq 0$. Thus, $\lim\limits_{N\rightarrow \infty}\nu_1(N)\leq \frac{1}{(1-\vartheta)^2}$ and we complete the proof.
\end{proof}

\begin{rem}
For $-1<\vartheta<0$, $\Delta\geq 0$ means $\frac{1+a}{\vartheta}+\vartheta\leq -2$ and $0>a>-(1+\vartheta)^2$. As a consequence, we have $\nu_1(N)\leq (1+\vartheta)^2$.
\end{rem}

\subsection{Proof of theorem \ref{asymptotical normal}}
Let $v_{opt}(n)=\sigma_{n+1}$ and $\zeta^o=(\zeta^o_n,\, n\geq 1)$ be the process $\zeta$ with the function $v_{opt(n)}$. Then, we have
$$
\zeta^o_n=A_{n-1}\zeta^o_{n-1}+bv_{opt}(n)+b\sigma_n\epsilon_n,\,\,\, \zeta_0=\mathbf{0}_{2\times 1}
$$

To estimate the parameter $\vartheta$ from the observations $\zeta_1, \zeta_2,\,\cdots,\, \zeta_N$, we can write the MLE of $\vartheta$ uing
\eqref{likelihood function}
\begin{equation}\label{eq:MLE}
\hat{\vartheta}_N=\left(\sum_{n=1}^N\left(\frac{a_n^*\zeta_n}{\sigma_{n+1}}\right)^2 \right)^{-1}\left(\sum_{n=1}^N\frac{a_n^*\zeta_nb^*\zeta_{n+1}}{\sigma_{n+1}^2}  \right)\,.
\end{equation}

A standard calculation yields
$$
\hat{\vartheta}_N-\vartheta=\frac{M_N}{\langle M\rangle_N}\,,
$$
where
$$
M_N=\sum_{n=1}^N\frac{a_n^*\zeta_n}{\sigma_{n+1}}\epsilon_{n+1},\,\,\,\, \langle M\rangle_N=\sum_{n+1}^N\left(\frac{a_n^*\zeta_n}{\sigma_{n+1}}\right)^2\,.
$$

The second and third conclusion about the asymptotically normality of theorem \ref{asymptotical normal} are crucially based on the asymptotical study of the Laplace transform
$$
\mathcal{L}_N^{\vartheta}\left(\frac{\mu}{N}\right)=\mathbf{E}_{\vartheta}\exp\left(-\frac{\mu}{2N}\langle M\rangle_N \right)\,,
$$
for $N\rightarrow \infty$.

First, we can rewrite $\mathcal{L}^{\vartheta}_N\left(\frac{\mu}{N}\right)$ by the following formula:
$$
\mathcal{L}^{\vartheta}_N\left(\frac{\mu}{N}\right)=\mathbf{E}_{\vartheta}\exp\left(-\frac{1}{2}\sum_{n=1}^N\zeta_n^*\mathcal{M}_n\zeta_n   \right)\,,
$$
where $\mathcal{M}_n=\frac{\mu}{N\sigma_{n+1}^2}a_na_n*$.

Inspired by the Appendix of Chapter 2 of \cite{Cthesis}, we have the following result.

\begin{lem}\label{formula Laplace}
For any $N$, the following equality holds:
$$
\mathcal{L}^{\vartheta}_N\left(\frac{\mu}{N}\right)=\prod_{n=1}^N[\det(\mathbf{Id}+\gamma(n,n)\mathcal{M}_n)]^{-1/2}\exp\left(-\frac{1}{2}\sum_{n=1}^Nz_n^*\mathcal{M}_n (\mathbf{Id}+\gamma(n,n)\mathcal{M}_n)^{-1}z_n         \right)\,,
$$
where $(\gamma(n,m),\, 1\leq m\leq n)$ is the unique solution of the equation
\begin{equation}\label{eq:gamma n m}
\gamma(n,m)=\left[\prod_{r=m+1}^n A_{r-1}(\mathbf{Id}+\gamma(r,r)\mathcal{M}_r)^{-1}\right]\gamma(m,m),
\end{equation}
 and the function $(\gamma(n,n,\,n\geq 1))$ is the solution of the Ricatti equation:
\begin{equation}\label{eq:gamma}
\gamma(n,n)=A_{n-1}(\mathbf{Id}+\gamma(n-1,n-1)\mathcal{M}_{n-1})^{-1}\gamma(n-1,n-1)A_{n-1}^*+\sigma_n^2bb^*.
\end{equation}

It is worth to emphasize that $(z_n,\, 1\leq n\leq N)$ is the unique solution of the equation
$$
z_n=m_n-\sum_{r=1}^{n-1}\gamma(n,r)[\mathbf{Id}+\gamma(r,r)\mathcal{M}_r]^{-1}\mathcal{M}_rz_r,\,z_0=m_0.
$$
where $m_n=\mathbf{E}\zeta^o_n$.
\end{lem}

With the explicit formula of the Laplace transform presented in Lemma \ref{formula Laplace}, we have its asymptotical property.

\begin{lem}\label{le:conv laplace}
Under the condition \eqref{spectral density}, for any $\mu\in \mathbb{R}$, we have
\begin{equation}\label{conv laplace}
\lim_{N\rightarrow \infty}\mathcal{L}^{\vartheta}_N\left(\frac{\mu}{N}\right)=\exp\left(-\frac{\mu}{2}\mathcal{I}(\vartheta)\right)
\end{equation}
where $\mathcal{I}(\vartheta)=\frac{1}{1-\vartheta^2}+\frac{1}{(1-\vartheta)^2}$.
\end{lem}

\begin{proof}
In \cite{BCK} we have stated that
\begin{equation}\label{part1}
\lim_{N\rightarrow \infty} \prod_{n=1}^N[\det(\mathbf{Id}+\gamma(n,n)\mathcal{M}_n)]^{-1/2}=\frac{\mu}{1-\vartheta^2}.
\end{equation}

Since the component of $\gamma(n,n)$ is bounded, we have
$$
\lim_{N\rightarrow \infty} \mathbf{Id}+\gamma(n,n)\mathcal{M}_n=\mathbf{Id}.
$$

On the other hand,
\begin{equation}\label{part2}
\sum_{n=1}^Nm_n^*\mathcal{M}_nm_n=\frac{\mu}{N}\sum_{n=1}^N\left(\frac{a_n^*\mathbf{E}\zeta^0_n}{\sigma_{n+1}}\right)^2\longrightarrow \frac{\mu}{(1-\vartheta)^2},\, N\rightarrow \infty.
\end{equation}
which has been presented in last part. At last, notice
$$
\sum_{r=1}^{n-1}\gamma(n,r)[\mathbf{Id}+\gamma(r,r)\mathcal{M}_r]^{-1}\mathcal{M}_rz_r=\sum_{r=1}^{n-1}\left[\prod_{\tau=r+1}^nA_{\tau-1}(\mathbf{Id}+\gamma(\tau,\tau)\mathcal{M}_{\tau})^{-1}   \right][\mathbf{Id}+\gamma(r,r)\mathcal{M}_r]\mathcal{M}_rz_r
$$
we have that
\begin{equation}\label{conv 0}
\lim_{N\rightarrow \infty}\sum_{r=1}^{n-1}\gamma(n,r)[\mathbf{Id}+\gamma(r,r)\mathcal{M}_r]^{-1}\mathcal{M}_rz_r=0.
\end{equation}
Combining \eqref{part1} and \eqref{part2} and \eqref{conv 0}, the Lemma  \ref{le:conv laplace} achieves.
\end{proof}

From this conclusion, it is immediate that
$$
\mathbf{P}_{\vartheta}\lim_{N\rightarrow \infty}\frac{1}{N}\langle M\rangle_N=\mathcal{I}(\vartheta)\,.
$$

Moreover, using the central limit theorem for martingale, we have
$$
\frac{1}{\sqrt{N}}M_N\Longrightarrow \mathcal{N}(0,\mathcal{I}(\vartheta)).
$$

Consequently, the asymptotical part of the theorem \ref{asymptotical normal} is obtained.

The strong consistent is immediate when we change $\frac{\mu}{N}$ with a positive proper constant $\mu$ in Lemma \ref{le:conv laplace} because the determinant part tends to $0$ as  presented in section 5.2 of \cite{BCK}and the extra part is bounded.


Women take twice as long to urinate in public toilets as men. It's not their fault: having to deal with more cumbersome clothes, menstruation and sometimes small children, they easily use up 90 seconds, while men are done in half the time. But architects usually design equal numbers of toilet stalls for each gender, when women should have double. That explains the queues.
Women take twice as long to urinate in public toilets as men.
In the USA, the "potty parity" movement, founded by lawyer John Banzhaf, campaigns to redress the imbalance. In the US Capitol, for example, where laws are made, congressmen caught short while voting have access to a washroom a few meters from the debating floor equipped with six stalls, four urinals, gilt mirrors, a fireplace and television. Congresswomen have to cross either a tourist-filled hall, or a long corridor past several secretaries, before reaching a facility only accessible with a code. Though a Women's Restroom Equity Bill was debated in 2010, it has yet to become law.


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text:
Now Reading Five rainy-day tips for entertaining kids…
Five rainy-day tips for entertaining kids in Tokyo
Posted on 16 May, 2013 | By Hester
Post Category - TravelTravel
On a recent family trip to Japan we were greeted with a frankly epic downpour that lasted for three days straight. Caught on the back foot, our temple-hopping plans were quickly shelved in favour of some indoor child-friendly distractions, keeping us out of our (typically tiny) Japanese hotel room and keeping everyone sane!
Here's our fave five from the trip:
1. Tokyo Fire Museum
Start here and combine with number 2 on this list for an easy morning of entertainment. This museum covers six floors and delves into the history behind Tokyo's fire service, making for an interesting read for parents. There's a dress-up corner where kids can play the hero, but the real draws here are, of course, the nee-naws! With genuine vintage fire trucks galore (some that you can climb into), and even a helicopter, there are plenty of photo ops to be had and bells to be rung to sound the alert. Heaven!
Where: Tokyo Fire Museum, Yotsuya 3-10, Shinjuku, Tokyo.
How much: Free!
Getting there: Jump on Tokyo Metro's Marunochi Line to Yotsuya Sanchome Station. Exit 2 takes you to the museum entrance.
2. Tokyo Toy Museum
It may come second on our list, but the Tokyo Toy Museum should be first on everyone's Tokyo sightseeing list – it's that good! Don't let the word 'museum' fool you as only a small number of displays here are off-limits for little fingers, making the majority of the three-storey space a totally hands-on experience. Packed to the rafters with gorgeous handmade wooden toys, musical instruments and games, the friendly museum staff are more than happy to get down on their hands and knees and show your little one just how everything works. This is low-fi play at its best with not a screen to be seen – and you absolutely have to see the 'sandbox' area comprising over 20,000 perfectly polished wooden pebbles. The separate toddler room is packed full of tactile wonders, making it a perfect hangout for the under-twos, and there's a whole game and puzzle zone for older kids. Many of the toys are available to buy from the in-house shop at the end of your visit, cutting down on potential home-time tantrums.
Where: Tokyo Toy Museum, Yotsuya 4-20, Shinjuku, Tokyo.
How much: ¥700 for adults, ¥500 for children aged 3 and over, or ¥1000 for a parent and child combined ticket.
Getting there: As with the Fire Museum, head to Yotsuya Sanchome Station and take Exit 2. Turn right at street level, then take the third street on the right and continue for around five minutes until you come to the museum on the right.
3. Toyota Megaweb (and the train ride there!)
Another one for the vehicle enthusiasts! Megaweb is a giant Toyota car theme park-cum-showroom, which, in itself, doesn't sound terribly interesting. But for a car-obsessed toddler, this was the perfect way to spend a rainy afternoon. Parents can enjoy playing on the futuristic 'personal mobility' pods that you can drive around a purpose-built track winding its way around the showroom, and older kids can experience high-speed racing in the driving simulators and virtual reality theatre.
Added bonus activity: We arrived at Megaweb via the metro line that runs along Tokyo's waterfront. Anyone who's travelled on London's Docklands Light Railway will be familiar with the setup – it's an automatically guided rail system that allows you to sit right at the very front, giving you the driver's-eye view of the tracks and stations ahead. This kept a little boy who usually has ants in his pants quiet and perfectly still for nearly 40 minutes, making this an essential bolt on to the visit!
Where: Toyota Megaweb, 1-3-12 Aomi, Koto-ku, Tokyo.
How to get there: Take the Yurikamome line to Aomi station and Megaweb is right in front of you as you alight.
4. Tokyo Tower and Aquarium
Another rainy day to kill, and this one just happened to be local to our hotel, making it a shoe-in for the list. The Tokyo Tower is a great point from which to orientate yourself and is a landmark in its own right, being somewhat reminiscent of another, (more famous) tower overlooking the Seine! The huge orange and white structure looms over Shiba Park and offers a birds'-eye view over the vast megalopolis that is the Japanese capital. Staff are extremely helpful and ensured that our stroller-toting party was bumped to the front of the queue for the high-speed lifts.
Once we'd tired of the view – which includes Mount Fuji on a clear day – and the (unnerving for mama) glass floor, we decided to investigate the small aquarium on the tower's ground floor. Although nowhere near as impressive as our own Ocean Park, this did offer some child-friendly distractions, including a small display of jellyfish that fascinated my son for quite some time, and an open seashore exhibit with hermit crabs scuttling around in psychedelically decorated shells.
Where: Tokyo Tower, 4-2-8 Shibakoen, Minato-ku, Tokyo.
How much: Tower admission: (main observatory) ¥820 per adult and ¥310/¥460 for children 4 and over.
Aquarium: ¥1000 per adult
How to get there: Take the Oedo line to Akabanebashi station, the Hibiya Line to Kamiyacho Station (Exit 1) or the Mita line to Onarimon Station (Exit A1). All are around a 5-minute walk away from the tower.
5. Tokyo Baby Café
Finally if, after all that traipsing around entertaining your little one, you're in need of some refreshment, head to the Tokyo Baby Café for a little family-friendly coffee break. This weird and wonderful establishment is fully stocked with toys and book, but the real draw is the quirky design. The interior plays around with perspective, using spaces in different ways for adults and children, e.g. tables become houses for kids, while a giant sofa becomes an entire play area.
Exclusively for the under-7's (and parents) and pregnant women, this place will buy you plenty of time to enjoy your latte and kids are well catered for too, with scrummy, healthy meal options.
Where: B1F, 4-5-12 Jingumae, ?Shibuya-ku, Tokyo.
How much: ¥500 per 30 minutes.
How to get there: Take the to metro Omotesando station, exit A2 (Chiyoda, Ginza and Hanzomon lines). The café is a 2-minute walk away.
Hester is a co-founder of Sassy Mama, and lives in tropical Singapore with her two young daughters. With an unquenchable wanderlust and a love for grim Scandinavian crime novels and anything salted caramel, she's an expert at both mama guilt and cake-eating as a bonding activity.
Posted 1 year ago | By Zeyna
Travelling With Kids: How to Avoid Toddler Tantrums While Flying


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– Cotton: Designed with 100% cotton top layer.
– Latex: Made with high quality latex with activated carbon.


text:
Thursday, Mar 09, 2017 08:40 AM
Bears release Cutler after 8 seasons
Quarterback Jay Cutler's polarizing eight-year tenure in Chicago officially ended Thursday when he was released by the Bears.
Acquired in a blockbuster trade with the Denver Broncos on April 2, 2009, Cutler set virtually every Bears career passing record but posted a mediocre 51-51 regular-season record and led the team to just one playoff berth.
*Jay Cutler compied a 51-51 record in eight seasons as Bears starting quarterback. *<br>
"I appreciate Jay's professionalism throughout this process and throughout my two years with him here in Chicago," Bears general manager Ryan Pace said in a statement. "I will always appreciate his toughness and respect his accomplishments on the field with the Bears. He leaves here holding nearly every passing record with this storied franchise and I wish him nothing but the best going forward."
Cutler's best season with the Bears may have been in 2015 when he posted a career-high 92.3 passer rating and reduced his interceptions from 18 in 2014 to 11.
But the veteran quarterback was limited to just five games in 2016, completing 81 of 137 passes for 1,059 yards with four touchdowns, five interceptions and a 78.1 passer rating. He missed five contests with a thumb injury he suffered in Week 2 and sat out the final six games with a shoulder injury.
"Jay was always one of the biggest competitors on our roster," said Bears coach John Fox said. "He battled every day to get better, both himself and his teammates. He was a team guy who would offer anything he could to help the Bears. Wherever his career may take him next, I wish him nothing but success."
Cutler played for six different offensive coordinators with the Bears: Ron Turner (2009), Mike Martz (2010-11), Mike Tice (2012), Aaron Kromer (2013-14), Adam Gase (2015) and Dowell Loggains (2016).
Cutler arrived in 2009 with high expectations, one year after being voted to the Pro Bowl with the Broncos. The Bears acquired him along with a fifth-round pick they spent on Pro Bowl return specialist Johnny Knox in exchange for quarterback Kyle Orton, first-round selections in 2009 and 2010 and a third-round choice in 2009.
Cutler struggled with interceptions in his first season in Chicago, throwing a career-high 26 picks. That included four interceptions in his Bears debut in a Monday night loss in Green Bay and later five picks in a Thursday night defeat to the 49ers.
Cutler rebounded in 2010, however, passing for 3,274 yards with 23 touchdowns and 15 interceptions in leading the Bears to the NFC North championship with an 11-5 record. In a 35-24 playoff win over the Seahawks, he passed for 274 yards with two TDs, no interceptions and a 111.3 rating.
The following week in the NFC Championship Game against the Packers, however, he connected on just 6 of 14 passes for 80 yards and one interception before exiting with a knee injury he sustained late in the first half.
Cutler led the Bears to five straight wins in 2011, improving their record to 7-3. But he suffered a season-ending thumb injury in a victory over the Chargers and was lost for the season. The Bears dropped their next five games without Cutler, falling out of playoff contention.
Cutler ranks first on the Bears' all-time list in career passer rating (85.2), completions (2,020), attempts (3,271), completion percentage (61.8), passing yards (23,443), passing yards per game (229.8), passing touchdowns (154) and 300-yard passing games (16).
"We are grateful to Jay for all he did as a Bear," said Bears chairman George H. McCaskey. "His ability, toughness, and intelligence were on daily display at Halas Hall and Soldier Field. He had an extraordinary impact off the field, doing things for people—especially kids—without expecting or wanting any recognition. I was and am a big fan of his. We wish Jay, Kristin and their three kids all the best."
Cutler released the following statement to the Chicago Sun-Times: "I would like to first thank the city of Chicago and its passionate fans for a memorable eight years. I grew up in Indiana rooting for the Bears as kid, so it was an honor to wear the Bears uniform and play quarterback at Soldier Field for my favorite childhood team.
"I would like to thank Virginia, George and the rest of the McCaskey family for their support over my tenure in Chicago as well as Jerry Angelo, Phil Emery and Ryan Pace. It was a great experience to have played with such a great list of teammates during my time as a Bear. Also, I would like to thank all of the great employees of the Bears from the locker room to the front office.
"Chicago is the city where I met my wonderful wife Kristin. I have been fortunate to have three beautiful children here. Our family will leave Chicago with great memories and relationships on and off the field. We look forward to the next chapter in our lives and wish the best to Bear fans everywhere."
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Urlacher happy to be in NFL 100 commercial
Bears Hall of Fame middle linebacker Brian Urlacher enjoyed being part of the NFL 100 commercial that aired during the Super Bowl and features many of the best players in NFL history.
Friends, teammates, opponents congratulate Urlacher via Twitter
Before, during and after Saturday's Pro Football Hall of Fame enshrinement ceremony, those that knew and respected the newest Bears hall-of-famer the most took to social media with words of congratulations.
Urlacher shows humility, gratitude in speech
In typical Brian Urlacher fashion, the former star middle linebacker spent most of his Hall of Fame induction speech Saturday night thanking family, friends and former coaches and teammates for helping him reach the pinnacle of the sport.
Wims hoping to build on impressive debut
Bears rookie receiver Javon Wims made the most of his expanded playing time Thursday night against the Ravens. But the seventh-round pick still has plenty of room for improvement.
Urlacher's career by the numbers
Hall of Fame-bound Brian Urlacher played all 13 of his illustrious NFL seasons with the Bears from 2000-12. Check out some numbers that defined his career.
NFL coaches, players share admiration for Urlacher
With Brian Urlacher set to be enshrined in the Pro Football Hall of Fame Saturday night, we take a look back at what several individuals who played and coached with and against the star middle linebacker had to say about Urlacher when he retired from the NFL after the 2012 season.
Bears knew Urlacher would be special from Day 1
The Bears were enticed by the athletic ability Brian Urlacher displayed at New Mexico and knew he'd be special when they selected him in the 2000 NFL Draft.
Miller thankful Bears re-signed him
Injured tight end Zach Miller expressed his gratitude to the Bears for re-signing him, calling the move "a testament to the entire organization and the McCaskey family."
Offseason program ends with minicamp
This week's minicamp will give quarterback Mitchell Trubisky a final chance to hone his skills and digest the new offense before the Bears take a six-week hiatus in advance of training camp.
How is Leonard Floyd progressing?
Senior writer Larry Mayer discusses Leonard Floyd's recovery from a knee injury, the undrafted free agent with the best chance of making the roster and if Tarik Cohen and Taylor Gabriel will be on the field together.
Burton takes pride in leadership ability
The leadership skills that new Bears tight end Trey Burton displays on a daily basis at Halas Hall stem from his days as a quarterback at Venice High School in Florida.
Bears announce preseason broadcast team
The Bears on Thursday announced that Adam Amin and Kyle Brandt will join former quarterback Jim Miller on their 2018 Fox 32 preseason broadcasts.


Good news for fellow journalists who wish to come to Belarus!
On November 25 Russian state newswire "RIA Novosti", quoting their own correspondent, ran urgent news that Ukrainian military "opened massive artillery fire, shelling residential areas of Donetsk".
Mykola Balaban, publisher of The Village Ukraine magazine, was arrested by Belarusian police in Minsk, where he was due to attend an international Media Management and IT forum. Police came to his hotel room at 5 in the morning, took him to a police department and put him in a cell. After immigration officers arrived, the journalist was told to hand in his belt and shoelaces, which usually means imprisonment.
There was a cliché in the Soviet Union calling Belarus, Russia and Ukraine "sisters", symbolising extraordinary closeness and sisterly love between the people of the three republics.
Come to Belarus for 5 days without a visa!
Good news again on the political liberalization front — Belarus introduces five days visa-free travel for citizens of EU, US, Japan and many other countries — in total 80 countries. The visa exemption applies only to those arriving by air and does not apply to those arriving to Belarus from Russia. The regulation is due to come into force in mid-February.
Belarusian customs have returned the gear confiscated from the Australian journalist Amos Roberts about a year ago. The teddy bear was valued at 10c in the confiscation list, although they used to sell at several hundred US dollars at the black market.
Is it worth for foreign journalists to work under the radar in Belarus?
To put it short – no. I have spent 15 days including Christmas in jail for not having the press accreditation when working for German TV, so I can tell you for sure - not worth it.
On March 15 a major snowstorm "Xavier" hit Belarus. The whole traffic in Minsk came to a halt, some metro stations couldn't handle the inflow of passengers and were closed town, people were stuck in snow in their cars both in Minsk and on highways, the only option for some was to walk back home from work through knee-deep snow.
In summer of 2011, series of mass protests happened in several big and smaller cities of Belarus. People, mainly educated computer-literate youth, came out on the streets to express their discontent with the existing economic and political situation in the country. Because of peaceful silent character protests were called "Silent Revolution" (or "Revolution through Social Networks" as people informed each other about the rallies with the use of social media). This article is a short description of the study aimed to analyze coverage of protests in Belarusian media. It offers observations about the difference in the nature of discourse on the Revolution through Social Networks and the voices (sources) presented in major Belarusian state-run and independent newspapers.
Several reasons, which are not however the purpose of this discussion, created a new public situation on the eve of the 2010 presidential elections in Belarus. One of the main outcomes of it was the restored "publicness" of an alternative political discourse.


Micah Sheveloff is a Boston-born Pianist singer/songwriter who has merged a classical music Foundation with a unique Feet For Popular Song. Sheveloff's career has included live and studio work with numerous artists, however it is the solo work beginning in 2011 that has defined his voice as an artist. Sheveloff has been a regular Performer at such NYC venues as The Bitter End and Cafe Vivaldi. Recently. Sheveloff has relocated to Saint Petersburg, FL where he continues onward as a singer/songwriter/Producer.


My lifelong passion for storytelling via still photography grew into a fascination and love of film - so much so, that I've left a career in cancer research and completed training as a Film Crew Technician. My heart is set on Production Office and Art Department roles as my own Call to Adventure beckons! Currently on the Overflow List for the IATSE Local 480 Film Mechanics Union (New Mexico). I'm looking forward to sharing with and learning from others in the fascinating field of film and TV production!


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The Maury County Historical Society
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Email: info@historicmaury.com
Historic Maury
June 2015, 51:2 Historic Maury
June 2016, 52, No 2 Historic Maury
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Home / Publications / Featured publication
"Historic Maury County – Places and People"
by Alice Algood (Editor) and Mildred Hartsfield (Illustrator)
The Maury County Historical Society was organized on 28 September 1905, and was active for a number of years, falling dormant, unfortunately, for nearly 50 years. In 1964, the original charter was found at the Athenaeum among local historian Frank H. Smith's papers. Mr. Smith had served as the Society's first secretary in 1905. Frank H. Smith was the son of Franklin Gillette and Sarah Davis Smith, founders of the Athenaeum. In 1965, a group of citizens who had long been interested in history met at the home of Jill Garrett and decided to reactivate the Society. Those present were Virginia Wood Alexander (Mrs. Charles C.), Alice Wright Algood (Mrs. Reuben C.), Dr. C. Y. Clark, Franklin Fulton, Jill Knight Garrett (Mrs. Ted), Marise Parish Lightfoot (Mrs. Jack), Evelyn Hodge Shackelford (Mrs. William), and Judge W. B. Turner.
One of this group's endeavors to rekinkle Maury Countians' interest in Maury County was a project to highlight the historic places and people of the county. Mildred Hartsfield's art had delighted people for many years, and the Society, having sponsored many historical books' and monographs' publication, recognized the unique significance of her renderings of Maury County's historical legacy and began to explore the possibility of featuring her work in a book. Hartsfield selected historic sites for inclusion from among her portfolio of works that spanned 30 years.
Fred Hawkins, Vice President for Publications at the time, began to collect data on these various sites, but his untimely death terminated his efforts. The Society asked Alice Wright Algood, longtime member of the Society, board member, a true Maury County native, and seventh generation Tennessean devoted to her faith, her family, her community and its history, and her nation, to assume the responsibility of editing the book. Her role in preservation throughout the nation, and specifically in Tennessee, placed her in the role of President of the Association of Preservation of Tennessee Antiquities (APTA), President of the Tennessee State Museum Foundation, a member of the Tennessee Historic Commission, and Chairman of the Tennessee Governor's Residence Foundation. She was appointed by President Ronald Reagan and confirmed by the Senate of the United States to a 10-year term on the National Museum Services Board. Named also to the book committee were Ryan E. Crumley, Ronnie Erwin, James "Rex" Hartsfield, Charles Hinson, and Montee Sneed.
The remarkable ability of Mildred Hartsfield, a self-taught artist, is displayed in every work that she does, whether pen and ink, watercolors, or oils. Mildred Hartsfield finds and depicts beauty in all venues of her art. Her work graces the corporate offices of General Motors in Michigan, the Maury County Courthouse, the Maury Regional Hospital, and countless homes in this area as well as in other states.
Mrs. Hartsfield and her family moved to Columbia in 1974, and established the Lemon Tree Gallery. She fell in love with the history of the many beautiful sites she found in Maury County and wished those places to be preserved. As she began to depict familiar scenes throughout the county, her work found an enthusiastic audience.
Many home owners engaged her to depict their homes. Her prints have been sold all over the country, and are proudly displayed in the homes of former Maury Countians who ached for a view of home.
Over 100 of Mildred Hartsfield's works of art are featured in this book!
Some of the famous county buildings have been lost over the years, but fortunately, Mildred recreated their beauty using old photographs. While very young, she drew her first home place, hoping that it would be there forever, but it, like so many others, is now gone. The wishes of the Maury County Historical Society, through the artistry of Mildred Hartsfield and publication of this book, were that, and still are that, these treasured sites will live forever in the eyes and hearts of everyone who is privileged to view Maury County, known as "The Dimple of the Universe."
To get your copy of Historic Maury County – Places and People click on Publications navigation tag below for current pricing and then click on Book Order Form navigation tag at the bottom of the Publications page. After entering the requested information on the Book Order Form, to print your order form for mailing, click on the Submit and Print button at the bottom of the form.
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Shipping & Returns : Music Bohemo - Recording Studio Equipment, Guitars, Basses, Tube Amplifiers, Analog Synthesizer, Microphones!
In-stock items usually ship within 24 hours of processing payment or the same day when possible. Special orders and backordered items will be drop-shipped from the manufacturer when possible to speed delivery.
We recycle everything and use boxes from our manufacturers when possible rather than boxes with our own branding. This may not be great for our image but it is good for the environment.
We email shipping updates and tracking info so you always know where your stuff is.
We want you to be happy with your gear and make music with the stuff you buy from us. If a product is defective, call us right away to get a RA# and return label and return it to us for a full refund or exchange. We will even pay shipping both ways as well as shipping on any replacement items.
If it's not defective but you just don't dig it, that's cool too. Call us to get a RA# and return label within 7 days of delivery and return it to us for a full refund or exchange less actual shipping cost both ways. Products must be received in exactly the same conditions we sent it to you and must be received no later that 15 days from the date of delivery. We are reasonable people who don't hide behind policy. If you have a problem just call us.
Don't be a jerk. If you misuse it, scratch it, dent it, modify it, adjust it, take parts from it or otherwise do anything we will have to clean, repair, replace or discount to sell, we will have to deduct these costs from any refund. All returns are subject to inspection, restocking fees and charges for damage, missing components, manuals, packing materials, signs of use, etc. The cost of damage, replacement parts, maintenance and the amount refund under such conditions is at our sole discretion.
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You are here: GWRRA University - Updated 7/16/2019 > About Us > FAQ
What is a District Training Event?
The event should be fun and social with education as it's theme. It will be a coordinated effort with the District and Chapters to include the Trainers, Educators and MEC's to meet the educational needs requested by the Members. It is suggested that they occur quarterly whenever possible.
How do I maintain my certification?
For University Trainers (UTREP, UTLEP, UTMEP) you must present two Seminars within two years. Once completed, submit form U.1 to certify.renew@gmail.com. For University Instructors, you must present two Seminars or Modules within two years, one of which will be observed by a University Instructor Trainer, who will submit a University Instructor Observation Form.
Who manages my University activity?
Until the new database is launched, your District Trainer will maintain a record of all the training modules and or seminars you have taught, based upon the Class Roster (U.3) you have provided. Until you are officially notified about the change, records of all classroom training attended are the responsibility of the member!
How do I get my training assignments?
Requests for Training Events will be coordinated through District University Coordinator. Requests for on-bike courses, and MFA, will be coordinated through the District Educator. Motorist Awareness is coordinated through the Motorist Awareness Program.
Generally, your training assignments will come from your District University Coordinator. Wing Ding assignments will be given by the University Team. You may also contact the District University Coordinator of a neighboring District and volunteer your training services. In that instance, your assignment will come from that District University Coordinator.
What is the GWRRA University?
GWRRA University is an all-encompassing education and training institution with two main purposes— 1) a vibrant, dynamic source from which well-trained leaders can continue the practice of education, coaching and mentoring; 2) to provide Members access to some of the finest safety, professional and personal, education and training in the world. This is the education and training they would have to pay thousands of dollars for somewhere else. The University is to become the keystone supporting the organization as it is today and developing leaders for the organization of the future.


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Home Biden Administration Biden Megadonor Scores $500 Million Federal Loan For Solar Company
Biden Megadonor Scores $500 Million Federal Loan For Solar Company
ARTICLE REPOSTED FROM THE WASHINGTON FREE BEACON
A solar energy company owned by a Biden megadonor received a $500 million government loan to build a manufacturing facility in India, the Biden administration announced this week, raising questions about whether the company's political clout played any role in the financing decision.
The U.S. International Development Finance Corporation granted the loan to First Solar, which is owned by billionaire Walmart heir Lukas Walton, to build a solar module plant in India. Walton contributed over $300,000 to President Joe Biden's campaign last year, and over $100,000 to the Democratic National Committee, according to campaign finance records.
The loan to First Solar is the "largest single debt financing transaction" issued by the DFC, the agency announced this week. The DFC said the investment in the India project will "promote DFC's commitment to diversifying supply chains," following demands from lawmakers that the agency avoid funding any solar projects connected to forced labor in China.
Ethics watchdogs said the loan raises questions about whether First Solar's political connections played a role in the DFC's decision. The federal financing agency, which was formerly known as the Overseas Private Investment Corporation, has faced criticism in the past for funding projects linked to political donors. The loan also comes nine years after the Obama administration came under fire for approving $3 billion in loan guarantees to the same company—funding that Republican lawmakers alleged the company wasn't qualified to receive.
"The United States International Development Finance Corporation, formerly known as OPIC, has a history of deals gone bad when mixing taxpayer dollars with politically connected entities like First Solar," said Tom Anderson, director of the Government Integrity Project at the National Legal and Policy Center. "This agency has a history of favoring entities backed by huge political contributors, like First Solar, by giving them less scrutiny while prioritizing politically connected projects above entities and individuals who are not politically active."
A DFC spokesperson, asking to speak on background, said the deal was carefully assessed and unrelated to politics but declined to say if the White House had any input in the decision.
"This deal is a milestone in the U.S. effort to drive alternative supply chains that do not utilize forced labor, and absolutely nothing to do with politics," the spokesperson told the Washington Free Beacon. "DFC-supported projects undergo a rigorous due diligence process and are assessed based on their creditworthiness, development outcomes, and foreign policy impact. The agency is governed by a public-private board of directors with bipartisan membership."
In 2010, OPIC expedited a $10 million loan to a Hillary Clinton donor while Clinton was serving as secretary of state. The donor was supposed to use the loan to fund a Haiti housing relief project, but instead pocketed the money. He was sentenced to 12 years in prison for the scam.
First Solar came under scrutiny in 2012, when the GOP-led House Oversight Committee investigated a series of Department of Energy loans issued to solar companies by the Obama administration. The investigation was prompted by a scandal involving another politically connected solar energy company, Solyndra, which had declared bankruptcy and defaulted on a $500 million federal loan.
A House investigation turned up internal DOE emails that indicated First Solar's technology was not considered "innovative" by the loan program's technical director, which was a requirement for receiving the financing.
"The information currently accessible to the public raises the question of whether these companies received approval to use federal lands and taxpayer monies based not on the best value for the American people but the political clout of the recipients," then-senator Jeff Sessions (R., Ala.), ranking member of the Senate Budget Committee, said at the time.
Although First Solar's chairman testified to Congress that the company was "financially strong," the business was forced to furlough half of its employees at its solar plant near Los Angeles weeks later.
Last year, First Solar paid $350 million to settle a shareholder lawsuit brought by two United Kingdom pension funds, which claimed the company inflated its stock price and misled investors about its finances.
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