text:
\section{Introduction}

Incomplete data are very common in statistics: when the mechanism leading to incomplete data (m.l.i.d.) is fixed a relatively simple likelihood can be written in general. Often the m.l.i.d. can not be considered as fixed and the question arises whether it can still be ignored.
Rubin (1976) introduced the concept of ignorability for the simplest case in which the observation is a sequence of random variables and some of them are missing; he gave conditions
under which inference based on the assumption of fixed m.l.i.d. was valid, even in the case when in fact it
was not fixed. He established a typology of cases of missing data and showed in particular that the m.l.i.d. was
ignorable for likelihood inference in the case of missing at random (MAR) observations.

 In the framework of survival analysis the most frequent cases of incomplete data is right-censoring (Kaplan and Meier, 1958; Cox, 1972) and  interval censoring (Peto, 1973). Conditions under which the conventional likelihood for right-censored
survival data was valid have been studied (Lagakos, 1979; Kalbfleisch and Prentice, 1980). Andersen et al.
(1993) studied the concept of independent censoring in the counting process framework. Heitjan and Rubin
(1991) also studied some less conventional incomplete data cases which they called ``coarsening''. This topic
was also studied by Jacobsen and Keiding (1995), Gill et al. (1997) and Nielsen (2000). The problematic of non-ignorable m.l.i.d.
has prompted the development of joint models, in which the m.l.i.d. was included, for instance in a model proposed by Diggle and Kenward (1994); see Thi\'ebaut et al. (2005) for a recent example.

The aim of this paper is to study ignorability in the context of stochastic processes: these processes may be
counting processes but also continuous state-space processes, such as diffusion processes. For giving a rigorous
treatment of that topic, we will need to rely on basic probability tools. First we will speak in terms of
likelihood ratio which is defined as a Radon-Nikodym derivative: this enables to manipulate likelihood ratios
for the observation of stochastic processes (for a review see Barndorff-Nielsen and Sorensen, 1994). Local
equalities of $\sigma$-fields and of random variables will play an important role in the very definition of
ignorability and in the proofs and we recall a probability result described in Kallenberg (2001):
\begin{Lemma}\label{Kallenberg}
Let the  $\sigma$-fields ${{\cal F}}$, ${\cal G} \subset {\cal A}$ and
functions $\xi,\eta \in L^1$ be such that $A \cap {{\cal F}} =A \cap
{\cal G} $ and $\xi=\eta$ a.s. on some set $A \in {{\cal F}}\cap {\cal
G}$. Then ${\rm E}[\xi |{{\cal F}}]={\rm E}[\eta |{\cal G}]$ a.s. on $A$.
\end{Lemma}
 Also, results on the likelihood of point
processes due to Jacod (1975) will play a key role in several
proofs.

 We begin in section 2 by recalling the general definition of
the likelihood ratio and of marginal and conditional likelihoods; a set of useful formulae is given. In section
3 a coarsening model for stochastic processes is given. Then in section 4 we present a new formulation of the
incomplete observation problem based on $\sigma$-fields and we give a definition of ignorability. In section 5
we give general conditions of ignorability and in section 6 a dynamical condition which is more interpretable
and usable than the general ones in some contexts. Finally section 7 illustrates the use of the theory in
survival models, longitudinal data and joint models, and section 8 is a short conclusion.

\section{Full, marginal and conditional likelihood}
Consider a measurable space $(\Omega, {\cal F})$ and a family of
measures $\{P_{\theta}\}_{\theta \in \Theta}$ absolutely
continuous relatively to a dominant measure $P_{\theta_0}$. For
${{\cal X}}$ a sub-$\sigma$-field of ${{\cal F}}$ the likelihood ratio on ${{\cal X}}$
is defined by:
$${\cal L}_{\cal X}^{\theta/\theta_0}=\frac{dP_{\theta}}{
dP_{\theta_0}}_{| {\cal X}}\hspace{5mm} {\rm a.s.} $$ where
$\frac{dP_{\theta}}{ dP_{\theta_0}}_{|{\cal X}}$ is the
Radon-Nikodym derivative of $P_{\theta}$ relatively to
$P_{\theta_0}$. Recall that $\frac{dP_{\theta}}{ dP_{\theta_0}}_{|
{\cal X}}$ is the ${\cal X}$-measurable random variable such that
$  P_{\theta}(F)=\int_F \frac{dP_{\theta}}{ dP_{\theta_0}}_{|
{\cal X}}dP_{\theta_0}, F\in {\cal X}$. If ${{\cal X}} \subset {{\cal Y}} \subset
{{\cal F}} $ we have the fundamental formula (Williams, 1991):
$${\cal L}^{\theta/\theta_0}_{{{\cal X}}}={\rm E}_{\theta_0} [{\cal
L}^{\theta/\theta_0}_{\cal Y}|{{\cal X}}] \hspace{5mm} {\rm a.s.}.$$
Note that because conditional expectations and likelihood ratios
are defined a.s., all the equalities involving them are to be
understood as a.s., even if this is not specified for sake of
notational simplicity.

When ${{\cal F}}$ is generated by two random elements $X$ and $Y$ and denote by ${{\cal X}}$ and ${{\cal Y}}$ the $\sigma$-fields they
generate respectively; thus ${{\cal F}}={{\cal X}} \vee {{\cal Y}}$; we will note ${\cal L}^{\theta/\theta_0}_{\cal F}= {\cal
L}^{\theta/\theta_0}_{{{\cal X}} \vee {{\cal Y}}}={\cal L}^{\theta/\theta_0}_{{{\cal X}},{{\cal Y}}}$. The likelihoods
${\cal L}^{\theta/\theta_0}_{{{\cal X}}}$ and ${\cal L}^{\theta/\theta_0}_{{{\cal Y}}}$ are called marginal likelihoods, and are linked to
the full likelihood by the conditional expectations: ${\cal L}^{\theta/\theta_0}_{{{\cal X}}}={\rm E} [{\cal
L}^{\theta/\theta_0}_{{{\cal X}},{{\cal Y}}}|{{\cal X}}]$ and ${\cal L}^{\theta/\theta_0}_{{{\cal Y}}}={\rm E} [{\cal
L}^{\theta/\theta_0}_{{{\cal X}},{{\cal Y}}}|{{\cal Y}}]$, as derives from the fundamental formula.

Conditional likelihoods can also be defined (see Hoffmann-Jorgensen, 1994); for brevity we do not recall the
definition. The conditional likelihood ratio of $Y$ given ${{\cal X}}$ will be denoted ${\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}}$.
The following properties will be used in this paper:
\begin{eqnarray*}
{\rm i)}& {\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}} \mbox{ is }  {{\cal X}} \vee{{\cal Y}}-{\rm measurable}\\
{\rm ii)}& \forall A \in {{\cal Y}}, \;\; {\rm E}_{\theta} \left[I_{A} | {{\cal X}}\right]=
{\rm E}_{\theta_0} \left[I_{A}{\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}} | {{\cal X}}\right], a.s.\\
{\rm iii)}&  {\cal L}^{\theta/\theta_0}_{{{\cal Y}},{{\cal X}}}={\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}}{\cal L}^{\theta/\theta_0}_{{{\cal X}}} \hspace{5mm}
{\rm a.s.}\\
{\rm iv)} &  {{\cal Y}} \subset {{\cal X}} \Longrightarrow {\cal L}^{\theta/\theta_0}_{{{\cal Y}},{{\cal X}}}={\cal L}^{\theta/\theta_0}_{{{\cal X}}} \Longleftrightarrow} %\newcommand{\1bar}{\bf \bar 1
{\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}}=1 \Longleftrightarrow} %\newcommand{\1bar}{\bf \bar 1 {\cal L}^{\theta/\theta_0}_{{{\cal X}}|{{\cal Y}}}=
\frac{{\cal L}^{\theta/\theta_0}_{{{\cal X}}}}{{\cal L}^{\theta/\theta_0}_{_{{\cal Y}}}}\\
{\rm v)} &   {\cal L}^{\theta/\theta_0}_{{{\cal X}}_1,\ldots,{{\cal X}}_m}={\cal L}^{\theta/\theta_0}_{{{\cal X}}_1}\prod_{k=2}^m{\cal L}^{\theta/\theta_0}_{{{\cal X}}_k|{{\cal X}}_1,\ldots,{{\cal X}}_{k-1}}\\
{\rm vi)} & {{\cal X}} \vee {{\cal Y}}={{\cal X}} \vee {{\cal Z}} \Longrightarrow {\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}}={\cal L}^{\theta/\theta_0}_{{{\cal Z}}|{{\cal X}}}\\
\end{eqnarray*}
Note that ii) is the generalization of the main property of the likelihood ratio to conditional expectations;
thus ${\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}}$ deserves its name of conditional likelihood ratio; it also implies
${\rm E}_{\theta_0} \left[{\cal L}^{\theta/\theta_0}_{{{\cal Y}}|{{\cal X}}} | {{\cal X}}\right]=1$. Note also that under the assumption of a
family of equivalent measures, all the likelihoods are strictly positive a.s.

\section{A general coarsening model for processes}

We define a General Coarsening Model for Processes (GCMP). In many
real studies, in particular in epidemiology, we would have to
consider a sample of $n$ independent ``subjects'', to each of whom
a process $X^i=(X^i_t)=(X^i_t)_{t\ge 0}$ would be associated.
Since the likelihood would be the product of the individual
likelihoods, it is sufficient to consider only one process. We
first consider a process $X=(X_t)$ where $X_t$ takes values in
$\Re$, then we will extend the model to a multivariate process.
The main objective is to describe observation schemes for
processes in continuous time $t$, but $t$ may also be discrete so
that the results can be applied to finite collections of random
variables. We shall consider a response indicator process
$R=(R_t)$ taking value $1$ at $t$ if $X_t$ is observed and $0$
otherwise; this a generalization of the response indicator
variable introduced by Rubin (1976). This unifies different
concepts of censoring and observation of longitudinal data. For
instance particular cases are:\\ i) right-censored survival data:
case where $X$ is a $0-1$ counting process and $R_t=1$ if $t\le
C$, $0$ otherwise, where $C$ is a censoring variable;
\\ ii) left-censored survival data:  case where $X$ is a $0-1$ counting
process and $R_t=1$ if $t\ge C$, $0$ otherwise;
\\ iii) interval-censored survival data ($X$ is a counting
process) or repeated measurements ($X$ has a continuous state
space): case where  and $R_t=1$ if $t\in \{V_1, V_2, \ldots,
V_m\}$, $0$ otherwise.

\begin{figure}[!ht]
\begin{center}
\input{right-censoring.pstex_t}
\caption{right-censoring} \label{right-censoring}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\input{non-standard.pstex_t}
\caption{Continuous monitoring followed by discrete-time visits} \label{non-standard}
\end{center}
\end{figure}

Note that $C$ in cases i) and ii), and $V_j, j=1,\ldots, m$ in iii) are
random variables. Cases i) and ii) illustrate a situation where $R$ is either right-
or left-continuous at each jump time and correspond to
observation in continuous time on some windows (see Figure 1);
case iii) corresponds to observations in discrete time. In the
latter case $R_t=1$ only on a finite or denumerable set of the
half line $[0,+\infty[$. The GCMP allows to represent a large
number of non-standard observation schemes. For instance the
subjects can be observed on windows separated by periods where no
observation are taken.  In most applications the process $X$ will
be either observed continuously on some periods or  observed only
at discrete time points, but the GCMP can represent a mixing of the
two types of observation for the same process: for instance a
subject could be observed in continuous time when he is at
hospital and at discrete times when he has left hospital (see
Figure 2). Also the process $X$ is not necessarily a 0-1 counting
process but may be for instance a more general counting process,
allowing recurrent events, or a process with continuous
state-space like a diffusion process.

That the GCMP includes censoring or coarsening models for random variables is obvious from the fact that to each
random variable $Y$ we can associate a counting process $X_t=1_{\{Y \ge t\}}$. We can define a
coarsening model for a random variable $Y$  by partitions ${\cal P}$ defined by intervals $A_j$ and
indicator $\nu_j$ which take values $1$ or $0$: if $\nu_j=1$, $Y$ is exactly observed on $\{Y\in A_j\}$, if $\nu^i_j=0$ it is
only observed that $Y$ falls in $A^i_j$. This is  equivalent to the GCMP for $X_t$ defined by
$R_t=\sum\nu_j1_{A_j}(t)$. In order to construct a random mechanism we can take a random partition, for instance determined by a set of random variables.

The model can be extended to the case of multivariate processes, as may be required by the observation of
several processes on the same ``subject''. So, we may consider that $X_t$ takes values in $\Re ^d, d \ge 1$ and
we may also consider a multidimmensional response process $(R_t)$.

Still another extension is to allow that if $R_t=1$, $X_t$ is not observed completely, but according to a
fixed m.l.i.d.: we shall refer to that case as ``vertical coarsening''. This would apply to left-censored
observation of a biological marker due to a detection limit (such as HIV-RNA), as exemplified in section 7.2.

\noindent{\bf {\em Remark.}} The $(R_t)$ process is a generalization of the response indicator variables of Rubin (1976). It is different from the filtering process proposed in Andersen et al. (1993) (section III.4): for instance neither left-censoring nor discrete observation times can be treated with Andersen et al. filtering process. See also Arjas, Haara and Norros (1992) and Arjas and Haara (1992) for a detailed treatment of the filtering problem in the framework of marked point processes. The $(R_t)$ process can also be considered as a special case of the auxiliary random variable $G$ used by Heitjan and Rubin (1991) or Jacobsen and Keiding (1995) if we interpret this random variable as a random element in a Skorohod space.

\section{Ignorability}
\subsection{The sigma-field representation for incomplete data}
A model for the random element $X$ is a family of measures
$\{P_{\theta}\}_{\theta \in \Theta}$ on a $\sigma$-field ${{\cal X}}$
generated by $X$ (for us, $X$ will be a stochastic process: $X_t$ takes
values in $\Re^d$ while the path of $X$ is an element of a Skorohod space). We will assume that the measures in the family
are equivalent and take $P_{\theta_0}$ as the reference measure.
If we observe ${{\cal X}}$ we will use the likelihood ${\cal L}^{\theta
/\theta_0}_{{{\cal X}}}$ for inference about $\theta$. We will represent
the observed events by a $\sigma$-field ${{{\cal O}}}$. A general
definition of incomplete data is: ${{\cal X}} \not\subset {{\cal O}}$. A simple
case of incomplete data is when only a sub-$\sigma$-field of ${{\cal X}}$
has been observed: ${{\cal O}} \subset {{\cal X}}$.  A particular case occurs
when the m.l.i.d. can be represented by a GCMP with $R$ a
deterministic function. We shall denote by $\{R=r\}$  the event
$\{R_t=r_t, t\ge 0\}$ where $r_t$ is a particular path (an element
of the Skorohod space for instance). If $R$ is deterministic there
is a value (a path) $r$ such that $\{R=r\}=\Omega$; for instance
$R_t=1$ for  $t<c$ and $R_t=0$ for $t\ge c$, where $c$ is fixed.
In such a case we have  ${{\cal O}}={{\cal X}}^r=\sigma(X_t, t\ge 0 :
r_t=1)=\sigma(r_tX_t,t\ge 0)$. In that case we have ${\cal
L}_{{{\cal O}}}={\cal L}_{{{\cal X}}^r}$ which is in general relatively easy to
compute. For instance in the right-censored case above, if $X$ is a counting process, Jacod's formula (Jacod, 1975) can be used to obtain the likelihood; in the more particular case where $X$ is a $0-1$ counting process with only one jump time $T$, the likelihood takes the form $\frac{\alpha_{\theta}^{\delta}(T)\exp[-\int_0^c \alpha_{\theta}(u) du ]}{\alpha_0^{\delta}(T)\exp[-\int_0^c \alpha_0(u) du ]}$, where $\delta=1_{\{T\le c\}}$ and $\alpha_{\theta}(.)$ (resp. $\alpha_0(.)$) are the risk functions under $P_{\theta}$ (resp. $P_0$).
Feigin (1976) has given the likelihood for a diffusion process observed in continuous time.
If a process is observed at fixed times $v_1, \ldots, v_m$ the likelihood can be computed as the likelihood for observation of the vector of random variables $X_{v_1},\ldots, X_{v_m}$; if $X$ is a $0-1$ counting process this case has been denoted ``interval censoring'' and the likelihood is simple to compute (see Peto, 1973; Alioum and Commenges, 1996); if $X$ is a gaussian process, the vector  $X_{v_1},\ldots, X_{v_m}$ has a normal distribution which makes the likelihood easy to compute.

If $R$ is not fixed, the above definition of ${{\cal O}}$ is meaningless; we must include $R$ in the description of the
problem. We shall consider a larger $\sigma$-field ${{\cal F}}={{\cal X}} \vee {{\cal R}}$, where ${{\cal R}}$ is the $\sigma$-field generated by
$R$ for right- or left-continuous processes that is: ${{\cal R}}=\sigma(R_t, t\ge 0)$; if $R$ takes value $1$ at only a
finite (or denumerable) number of times (corresponding to the discrete observation case) we can take ${{\cal R}}$ as
generated by the counting process counting the number of observation times.
We consider that $R$ is observed (see Remark 2) so that a representation of ${{\cal O}}$ is ${{\cal O}}=\sigma(R_t X_t, R_t, t
\ge 0)$. In section 6 which develops a
dynamical approach to the problem, we shall define adequate filtrations; for instance if $R$ is c\`adl\`ag (right-continuous with left-hand limits) the observed filtration $({{\cal O}}_t)$ will be the family of $\sigma$-fields ${{\cal O}}_t=\sigma(R_uX_u,R_u, 0\le u \le t)$. We have of course ${{\cal O}} \subset {{\cal F}}$; generally we have  ${{\cal X}} \not\subset {{\cal O}}$ (incomplete data) and ${{\cal O}}
\ne {{\cal X}}^r $ (the observation is not a predetermined subset of values of $X$).

\noindent{\bf {\em Remark 1.}}  We might think that we could define an interesting $\sigma$-field by
$\sigma(R_t X_t, t \ge 0)$ which could take the role of the notation $x_{obs}$ used in most of the literature in
missing data (for instance Kenward and Molenberghs, 1998); however if $X_t \ne 0$ for all $t$, the latter
$\sigma$-field is equal to ${{\cal O}}$. When $R$ is random it is not possible to disentangle the observed part of $X$
from $R$; only the realized value of $R$ effects a partition between  $x_{obs}$ and $x_{mis}$. This is in fact
the meaning of ${{\cal X}}^r$ which is the observed part of ${{\cal X}}$ when $R=r$.

\noindent{\bf {\em Remark 2.}} It is natural to say that $R$ is observed: for each $t$ we know whether we observe $X_t$ or not. There is
however an important case where this natural assumption does not hold: in survival analysis, $X$ is a $0-1$
counting process, so after a jump has been observed there is no need for observation anymore; so we may ignore
whether we would have observed the process if it had been necessary. The simplest way to get out of this problem
is to put $R_t=1$ if $X_t$ is known. More generally assume that there is an absorbing state $a$ such that if
$X_t=a$, $X_{t+u}=a$ for $u>0$; define the ${{\cal O}}_t$-stopping time $T=\inf \{t: X_t=a \mbox{ and } R_t=1\}$. By convention put
$R_{T+u}=1, u>0$. This part of the law of $R$ is anyway unidentifiable. In the remaining of the paper we will
consider that $R$ is observed.

\subsection{Model and notations}
From now on, a  model for the random element $(X,R)$ is a family
of measures $\{P_{(\theta, \psi)}\}_{(\theta, \psi) \in \Theta
\times \Psi}$  on a mesurable space $(\Omega, {{\cal F}})$. $X$ (resp.
$R$) takes values in a mesurable space $(\Xi, \xi)$ (resp.
$(\Gamma, \rho)$). For us $X$ and $R$ will be  d-dimensional
c\`{a}dl\`{a}g stochastic processes, so $(\Xi, \xi)$ and  $(\Gamma, \rho)$ are Skorohod
spaces endowed with their Borel $\sigma$-fields.
 The parameter spaces $\Theta$
and $\Psi$ need not be finite dimensional. We will assume
that the measures in the family are equivalent and take
$P_{(\theta_0,\psi_0)}$ as the reference measure. $P_{\theta}$ is
the restriction of $P_{(\theta, \psi)}$ to ${{\cal X}}$: that is, the marginal probability of $X$ does not depend on $\psi$. The additional
parameter $\psi$ will be considered as a nuisance parameter. We
assume implicitly a "Non-Informativeness" assumption in the
coarsening mechanism, which is :

\begin{equation}\label{noninfo}
  P_{(\theta_1, \psi)}(A|{{\cal X}})=P_{(\theta_2, \psi)}(A|{{\cal X}})\mbox{, a.s., } A
\in {{\cal R}} \mbox{ for all } \theta_1,  \theta_2, \psi
\end{equation}

In words, the conditional probability of $R$ given $X$ does not depend
on $\theta$.
 This has an important consequence in terms of likelihood
ratio. The latter assumption can also be written as: ${\rm E}_{(\theta_1,
\psi)}(1_A|{{\cal X}})={\rm E}_{(\theta_2, \psi)}(1_A|{{\cal X}})$, $A \in {{\cal R}}$,
which remembering property ii) of the conditional likelihood is
equivalent to ${\rm E}_{(\theta_0, \psi_0)}(1_A{\cal
L}^{(\theta_1,\psi )/ (\theta_0,\psi_0)}_{{{\cal R}}|{\cal X}}|{{\cal X}})={\rm
E}_{(\theta_0, \psi_0)}(1_A{\cal L}^{(\theta_2,\psi) /
(\theta_0,\psi_0)}_{{{\cal R}}|{\cal X}}|{{\cal X}})$. This in turn implies that
${\cal L}^{(\theta_1,\psi) /( \theta_0,\psi_0)}_{{{\cal R}}|{\cal
X}}={\cal L}^{(\theta_2,\psi) /( \theta_0,\psi_0)}_{{{\cal R}}|{\cal X}}$
and we will denote this common value ${\cal L}^{\psi /
\psi_0}_{{{\cal R}}|{\cal X}}$. Moreover it can be proved (and is
intuitive) that ${\cal L}^{\psi_0 / \psi_0}_{{{\cal R}}|{\cal X}}=1$,
a.s..

From now on, we fix our notation for ${{\cal O}}$ and ${{\cal X}}^r$ by
${{\cal O}}=\sigma(R_t X_t, R_t, t \ge 0)$ and  for $r=(r_t)$ a
deterministic path of $R$,  ${{\cal X}}^r=\sigma(X_t, t\ge 0 :
r_t=1)=\sigma(r_tX_t,t\ge 0)$. Note that (\ref{noninfo})  remains
true even if $A \in {{\cal O}}$. The inferential likelihood is ${\cal
L}^{(\theta,\psi) /(\theta_0,\psi_0) }_{{{\cal O}}}$ (which is
${{\cal O}}$-measurable and thus can be computed from the observations)
and the fundamental property yields:
$${\cal L}^{(\theta,\psi) /(\theta_0,\psi_0) }_{{{\cal O}}}={\rm E}_{(\theta_0,\psi_0)} [{\cal L}^{(\theta,\psi) /(\theta_0,\psi_0) }_{\cal F}|{{\cal O}}],$$
where ${\cal L}^{\theta,\psi /\theta_0,\psi_0 }_{\cal F}$ is the full likelihood.

\subsection{Definition of ignorability}

If the m.l.i.d. (represented by $R$) is random it may still be tempting to ignore it, treating it as fixed, and
use for inference  ${\cal L}_{{{\cal X}}^r}$ which is relatively easy to compute and does not depend on $\psi$.

\begin{Definition}
In the GCMP, the likelihood ratio ignoring the m.l.i.d. is the likelihood ratio ${\cal L}^{\theta /
\theta_0}_{{{\cal X}}^r}$ obtained under the assumption that $R$ is fixed at its observed value $r$.
\end{Definition}

When the fixed m.l.i.d. assumption does not hold, the question arises to know in which cases (if ever)
${\cal L}^{\theta / \theta_0}_{{{\cal X}}^r}$ leads to the same inference about $\theta$ as ${\cal L}^{\theta,\psi_0/
\theta_0,\psi_0}_{{{\cal O}}}$, if the true value of $\psi$ was known to be $\psi_0$ or ${\cal L}^{\theta,\psi/
\theta_0,\psi_0}_{{{\cal O}}}$ if $\psi$ had to be estimated.

For defining ignorability we face the problem  that both
Radon-Nikodym derivatives and conditional expectations are defined
almost surely; for some results we must restrict the theoretical
framework to measures giving a non null probability to a
denumerable set of trajectories of $R$.

{\bf Assumption A1} There exists a denumerable set $(r_1, r_2, \ldots)$ such that $P_{\theta}(R=r_i)>0$ and $\sum_i
P_{\theta}(R=r_i)=1$, for all $\theta$.

 This is a theoretical limitation but this has no impact on application since in practice
the times are always rounded. Some of the results below will need this restriction, other will not.

\begin{Definition}[Ignorability]
The m.l.i.d. will be called {\bf ignorable on {\em r} } if
${\cal L}^{(\theta,\psi_0)/ (\theta_0,\psi_0)}_{{{\cal O}}} = {\cal L}^{\theta /
\theta_0}_{{{\cal X}}^r}$ \hspace{3mm} {a.s.} on $\{R=r\}$, for all
$\theta $, whatever $\psi_0$. It will be called {\bf ignorable} if
assumption A1 holds and
 the m.l.i.d. is ignorable on $r$ for all values of $r$.
\end{Definition}

\noindent {\bf {\em Remark.}}
 If the m.l.i.d. is  ignorable it is then obvious that ${\cal L}^{\theta
/ \theta_0}_{{{\cal X}}^r}$ and ${\cal L}^{(\theta,\psi_0 )/ (\theta_0,\psi_0)}_{{{\cal O}}}$ lead to the same inference about
$\theta$.

\subsection{Extension to vertical coarsening}
This approach can readily be extended to fixed vertical coarsening, that is the case where, when $R_t=1$, $X_t$
is incompletely observed according to a fixed m.l.i.d. (see section 3). Let ${{\cal O}}$ (resp. ${{\cal O}}'$) the observed
$\sigma$-field for the completely vertically observed mechanism (resp. the incompletely vertically observed
mechanism). Since the vertical m.l.i.d. is fixed, ${{\cal O}}' \subset {{\cal O}}$. On $\{R=r\}$ we have ${{\cal O}}={{\cal X}}^r$ and
${{\cal O}}'={{\cal X}}'^r$ (where ${{\cal X}}'^r$ is the coarse observation of ${{\cal X}}^r$) with ${{\cal X}}'^r \subset {{\cal X}}^r$. If ignorability
holds for the completely vertically observed couple $(X, R)$  we have on $\{R=r\} $, ${\cal L}^{\theta,\psi_0/
\theta_0,\psi_0}_{{{\cal O}}}={\cal L}^{\theta / \theta_0}_{{{\cal X}}^r}$. By Kallenberg Lemma we have, on $\{R=r\}$, ${\rm
E}({\cal L}^{\theta,\psi_0/ \theta_0,\psi_0}_{{{\cal O}}}|{{\cal O}}')={\rm E}({\cal L}^{\theta / \theta_0}_{{{\cal X}} ^r}|{{\cal X}}'^r)$, from which
we deduce that on $\{R=r\}$, ${\cal L}^{\theta,\psi_0/ \theta_0,\psi_0}_{{{\cal O}}'}={\cal L}^{\theta / \theta_0}_{{{\cal X}}'^r}$; that
is, ignorability also holds for the fixed vertically coarsened pattern.

\section{Static conditions of ignorability for the GCMP}

We give a first fact which does not seem to have been noted
previously in a general context.

\noindent {\bf{\em Fact.}} Ignorability on $\{R_t=1, t\ge
0\}$ always holds.

\noindent {\bf{\em Proof.}} On the
event $\{R_t=1, t\ge 0\}$, $X$ and $R$ are observed so that ${{\cal F}} = {{\cal O}}$;
thus from Lemma \ref{Kallenberg} we have
  ${\cal L}^{(\theta,\psi) /(\theta_0,\psi_0)
}_{{{\cal O}}}={\rm E}_{(\theta_0,\psi_0)}({\cal L}^{(\theta,\psi) /(\theta_0,\psi_0) }_{{{\cal F}}}|{{\cal O}})={\rm E}_{(\theta_0,\psi_0)}({\cal L}_{{{\cal F}}}|{{\cal F}})={\cal L}^{(\theta,\psi)
/(\theta_0,\psi_0) }_{{{\cal F}}}= {\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}} {\cal L}^{\theta/\theta_0}_{{\cal X}}$ (the last equality comes from property iii) of the likelihood); for $\psi=\psi_0$ we retrieve the likelihood ${\cal L}^{\theta/\theta_0}_{{\cal X}}$ for the complete
observation of $X$ and ignorability holds on $\{R_t=1,t\ge 0\}$.\hfill \ding{113}

We shall now study ``static'' conditions of ignorability, in contrast with the ``dynamic'' conditions of the next section.
Gill \& al (1997) have introduced two conditions of ignorability: CAR(REL) (Relative Coarsening At Random) and CAR(ABS) (Absoluter Coarsening At Random); these conditions were further studied by Nielsen (2000). We give two conditions of ignorability in
the GCMP framework. The first one is an adaption
of CAR(REL); the second one is stronger
but original and doesn't imply CAR(ABS).

\begin{Definition}[CAR(GCMP)]
We will say that CAR(GCMP) holds for the couple $(X,R)$ if ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}$
is ${{\cal O}}$-measurable for all $(\theta, \psi)$.
\end{Definition}

With our notations the condition CAR(REL) is "there exists a version of ${\cal L}^{\psi/\psi_0}_{{\cal O}|{\cal X}}(o,x)$ such that the
mapping $x \rightarrow {\cal L}^{\psi/\psi_0}_{{\cal O}|{\cal X}}(o,x)$ is constant for all $x$
compatible with $o$", where $o$ is an elementary event of ${{\cal O}}$, i.e. $o=(r,rx)$.
First note that with our assumptions, $x$ is compatible with $o=(r,y)$ if $rx=y$ and
 second note that ${\cal L}_{{{\cal R}}|{{\cal X}}}(r ; x)={\cal L}_{{{\cal O}}|{{\cal X}}}(o ; x)$ if
 $o=(r,rx)$ (by property vi) and because ${{\cal R}}\vee {{\cal X}}={{\cal O}} \vee {{\cal X}}$.
So in the GCMP setting the condition CAR(REL) becomes
"there exists a version of ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}(o,x)$ such that :
for all  $r$ and $ (x,x') $
verifying $rx=rx'$ then  ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}(r,x)= {\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}(r,x')"$.

\begin{Theorem}
CAR(GCMP) is equivalent to CAR(REL) in the GCMP setting.
\end{Theorem}

\noindent {\bf {\em Proof.}} If CAR(REL) is true then for all $(r,x)$, ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}(r ;
x)={\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}(r ; rx)$ (by taking $x'=rx$ in the above formula) and then
${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}$ is a function
 of $(R,RX)$ and so ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}$ is ${{\cal O}}$-measurable that is CAR(GCMP).
 Conversely, if ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}$ is ${{\cal O}}$-measurable, there is a version of it
 which is constant on the atoms of ${{\cal O}}$ i.e. on the set of the
 form $O_{r,y}=\{(r,x) \mbox{ such that } x \mbox{ verifies } rx=y
 \}$ and we have CAR(REL).\hfill \ding{113}

The next theorem shows that CAR(GCMP) implies a factorization of
the likelihood in two parts : one which depends on $\psi$ and the
second one on $\theta$; this may be called ``weak ignorability''.

\begin{Theorem}[Factorization]
If the couple $(R,X)$ satisfies CAR(GCMP) then we have
${\cal L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal O}}}={\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}
{\rm E}_{(\theta_0,\psi_0)}({\cal L}^{\theta/\theta_0}_{{\cal X}}|{{\cal O}})$ and ${\rm E}_{(\theta_0,\psi_0)}({\cal L}^{\theta/\theta_0}_{{\cal X}}|{{\cal O}})$ does not depend on $\psi_0$.
\end{Theorem}

\noindent {\bf {\em Proof.}} A proof can be obtained using the previous theorem
and the fact that CAR(REL) has been proved to imply a similar
factorization theorem. However a direct proof is quite simple in
our formalism. From the decomposition formula we have
${\cal L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal F}}}={\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}} {\cal L}^{\theta/\theta_0}_{{\cal X}}$ and we
can pull ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}$ out of the conditional expectation using CAR(GCMP)
thus obtaining ${\cal L}^{(\theta, \psi) /(\theta_0,
\psi_0)}_{{{\cal O}}}={\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}} {\rm E}_{(\theta_0,\psi_0)}({\cal L}^{\theta/\theta_0}_{{\cal X}}|{{\cal O}})$. It only remains to prove that
the last term does not depend on $\psi_0$. We have
\begin{eqnarray*}{\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal O}}}&=&{\cal L}^{(\theta,\psi)/(\theta_0,\psi_1)}_{{{\cal O}}}{\cal L}^{(\theta_0,\psi_1)/(\theta_0,\psi_0)}_{{{\cal O}}}\\
&=& {\cal L}^{\psi/\psi_1}_{{\cal R}|{\cal X}} {\rm E}_{(\theta_0,\psi_1)}[{\cal L}^{\theta/\theta_0}_{{\cal X}}|{{\cal O}}]{\cal L}^{\psi_1/\psi_0}_{{\cal R}|{\cal X}} {\rm E}_{(\theta_0,\psi_0)}[{\cal L}^{\theta_0/\theta_0}_{{\cal X}}|{{\cal O}}]\\
&=& {\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}} {\rm E}_{(\theta_0,\psi_1)}[{\cal L}^{\theta/\theta_0}_{{\cal X}}|{{\cal O}}]
\end{eqnarray*} \hfill \ding{113}

As a result, we can use ${\rm E}_{(\theta_0,\psi_0)}({\cal L}^{\theta/\theta_0}_{{\cal X}}|{{\cal O}})$ for inference on $\theta$, getting rid of the nuisance parameter $\psi$.
The following condition allows obtaining a more precise result locally.

\begin{Definition}[CAR(GCMP)-loc]
We will call CAR(GCMP)-loc on $r$ the condition:  ${\cal
L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal R}}|{\cal X}}={\cal
L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal R}}|{\cal X}^r}$ a.s. on
$\{R=r\}$.

\end{Definition}

\noindent{\bf {\em Remark.}} The above definition has a meaning only for $r$ such that $P(\{R=r\})>0$.

 The GCMP model verifies the condition  CAR(ABS) defined by
Nielsen (2000) if $\forall (\theta, \psi)$, $P_{(\theta, \psi)}$
is CAR(ABS) i.e. the following condition is true :
\begin{eqnarray}
\mbox{for } P_{\theta} \mbox{ a.e. $x$, $x'$, for every }A \in {{\cal O}}, & &
\nonumber \\
 P_{ \psi}(A \cap D_x \cap D_{x'}|X=x)& = & P_{ \psi}(A \cap
D_x \cap D_{x'}|X=x')\label{carabs}
 \end{eqnarray}
  where $D_x=\{(r,y)\in (\Xi,\Gamma); ry=rx\}\in {{\cal O}}$. Note moreover that as
  pointed by Gill \& all (1997), if $P_{(\theta_0, \psi_0)}$ is
  CAR(ABS) and the model is CAR(REL), then the model is CAR(ABS).

\begin{Theorem} If the GCMP is CAR(ABS), then CAR(GCMP)-loc holds on all $r$.
\end{Theorem}

\noindent {\bf {\em Proof.}} Assume the model is CAR(ABS) and remark that $\{R=r\}
\cap D_x=\{R=r\} \cap D_{rx}$ and that $\forall (\theta, \psi)$,
$P_{(\theta, \psi)}[{D_x}|X=x]=1$, then $\forall (\theta, \psi)$
for $P_{\theta}$ a.e. $x$
 and every ${{\cal O}}$-measurable $Z$, we get
\begin{eqnarray*}
{\rm E}_{(\theta,\psi)}[1_{\{R=r\}}Z|X=x]& = & {\rm E}_{(\theta,\psi)}[1_{\{R=r\}}Z 1_{D_x}|X=x]\\
& = & {\rm E}_{(\theta,\psi)}[1_{\{R=r\}}Z |X=rx]
\end{eqnarray*}
From this equality, we deduce that $\forall (\theta, \psi)$ for
 every $Z$, ${{\cal O}}$-measurable, ${\rm E}_{(\theta,\psi)}[1_{\{R=r\}}Z|{{\cal X}}] = {\rm E}_{(\theta,\psi)}[1_{\{R=r\}}Z
|{{\cal X}}^r]$. It follows that for every $A \in {{\cal R}}$:
${\rm E}_{(\theta_0,\psi_0)}[1_{\{R=r\}}1_A {\cal L}^{(\theta, \psi) /(\theta_0,
\psi_0)}_{{{\cal R}}|{\cal X}}|{{\cal X}}] = {\rm E}_{(\theta,\psi)}[1_{\{R=r\}}1_A {\cal
L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal R}}|{\cal X}^r}|{{\cal X}}]$ which
implies CAR(GCMP)-loc. \hfill \ding{113}

\begin{Theorem}\label{loc}
 CAR(GCMP)-loc on
$r$ implies ignorability on $r$.
\end{Theorem}

\noindent {\bf {\em Proof.}} By the iterated decomposition formula (property v), we have ${\cal L}^{(\theta,
\psi) /(\theta_0, \psi_0)}_{{{\cal X}},{{\cal R}}}={\cal L}_{{{\cal X}},{{\cal X}}^r,{{\cal R}}}={\cal L}^{(\theta, \psi) /(\theta_0,
\psi_0)}_{{{\cal X}}|{{\cal X}}^r,{{\cal R}}}{\cal L}_{{{\cal R}}|{{\cal X}}^r}{\cal L}_{{{\cal X}}^r}$. We have ${\cal L}^{(\theta, \psi) /(\theta_0,
\psi_0)}_{{{\cal O}}}={\rm E}[{\cal L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal X}},{{\cal R}}}|{{\cal O}}]={\rm E}[{\cal L}^{(\theta,
\psi) /(\theta_0, \psi_0)}_{{{\cal X}}|{{\cal X}}^r,{{\cal R}}}{\cal L}_{{{\cal R}}|{{\cal X}}^r}{\cal L}_{{{\cal X}}^r}|{{\cal O}}]$. On $\{R=r\}$, ${{\cal O}}= {{\cal X}}^r \vee
{{\cal R}}$ and by  Lemma \ref{Kallenberg} we have:
$${\cal L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal O}}}=
{\cal L}^{\theta/\theta_0}_{{{\cal X}}^r}{\cal L}^{(\theta, \psi)
/(\theta_0, \psi_0)}_{{{\cal R}}|{{\cal X}}^r}{\rm E}_{(\theta_0, \psi_0)} [{\cal
L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal X}}|{{\cal X}}^r,{{\cal R}}}|{{\cal X}}^r\vee
{{\cal R}}]={\cal L}^{\theta/\theta_0}_{{{\cal X}}^r}{\cal L}^{(\theta, \psi)
/(\theta_0, \psi_0)}_{{{\cal R}}|{{\cal X}}^r}.$$ If CAR(GCMP)-loc holds we have
on $\{R=r\}$, ${\cal L}^{(\theta, \psi) /(\theta_0,
\psi_0)}_{{{\cal R}}|{\cal X}^r}={\cal L}^{(\theta, \psi) /(\theta_0,
\psi_0)}_{{{\cal R}}|{\cal X}}$; we have ${\cal L}^{(\theta, \psi)
/(\theta_0, \psi_0)}_{{{\cal R}}|{\cal X}}={\cal L}^{\psi
/\psi_0}_{{{\cal R}}|{\cal X}}$. Thus with both conditions we have ${\cal
L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal O}}}={\cal
L}^{\theta/\theta_0}_{{{\cal X}}^r}{\cal L}^{\psi/\psi_0}_{{{\cal R}}|{{\cal X}}},$ a.s.
on ${\{R=r\}}$
 and this implies ignorability.\hfill \ding{113}

\section{Dynamical conditions of ignorability}
\subsection{$R$ representable by a counting process}
 Assume that we can represent $(R_t)$ by a marked point process $(N_t)$;
 if the component $R_h$ of $R$ takes value $1$ at isolated points $V_{h,1}, V_{h,2}, \ldots$
 (discrete-time observations), then its associated counting
 process $N^d_{ht}$ is  defined by
 $N^d_{h,t}=\sum_{j=1}^{\infty}1_{\{V_{h,j}\leq t\}}$. If the component
 $R_{h,t}$ is c\`{a}dl\`{a}g, it can be written as $R_{h,t}=\sum_{k=0}^{\infty}1_{[V_{h,2k},V_{h,2k+1}[}$
 and its associate counting process $N^c_{h,t}=\sum_{j=0}^{\infty}1_{\{V_{h,j}\leq t\}}$.
 If the component $R_h$ of $R$ is a mixing of the two kinds of
  observation,
  (discrete- and continuous-time) then at points of discontinuities we have
  to add a mark which indicates if the jump affects the discrete
  part of $R_h$ represented by $N^d_{h,t}$ or the c\`{a}dl\`{a}g one
  represented by $N^c_{h,t}$. In all cases $R$ can be represented by
  a marked point process $(N_t)$.
Denote by $({{\cal N}}_t)$ the self-exciting filtration of $(N_t)$.  We
define the filtration $({{\cal O}}_t)$ as the family of $\sigma$-fields
${{\cal O}}_t=\sigma(N_u,R_uX_u, 0\le u \le t)$. Let us call
$\Lambda^{{{\cal O}},N}=(\Lambda^{{{\cal O}},N}_t)$  and
$\Lambda^{{{\cal F}}^*,N}=(\Lambda^{{{\cal F}}^*,N}_t)$ the compensators of $N$
in the filtration $({{\cal O}}_t)$ and $({{\cal F}}^*_t)$ respectively and for
probability $P_{\theta,\psi}$, where $({{\cal F}}^*_t)$ is the family of
$\sigma$-fields ${{\cal F}}^*_t= {{\cal X}} \vee {{\cal O}}_t, t\ge 0$. The compensators
generally depend on $(\theta, \psi)$ but we omit this for
notational simplicity; the compensators for $P_{\theta_0,\psi_0}$
will be denoted $\Lambda_0$. In the following we will assume that
there exist a fixed time $\tau$ such that $R_{\tau+u}=R_{\tau}$
and  that there is no explosion of the process on $[0,\tau]$ (i.e.
$\sum_{t \leq \tau}\Delta R_t <\infty$) (thus
$\Lambda^{N}_{\tau+u}=\Lambda^{N}_{\tau}, u\ge 0$, for any
filtration).

\begin{Definition}[CAR(DYN)]

We will denote CAR(DYN) the condition:

$\rm{ \hspace{10mm}} \forall \;(\theta, \psi)\;\;  $ we have under
$P_{(\theta, \psi)}: \;\; (\Lambda^{{{\cal O}},N}_t) =
(\Lambda^{{{\cal F}}^*,N}_t)$, (up to indistinguishability).
\end{Definition}

\noindent{\bf {\em Remark.}} This is an absolute condition in the sense of Gill et al. (1997) and Nielsen (2000) since the CAR(DYN) criterion is defined for each probability separately while the definition of CAR(GCMP) bears on Radon-Nikodym derivatives.

\begin{Theorem}
CAR(DYN) implies CAR(GCMP).
\end{Theorem}

\noindent {\bf {\em Proof.}} Let us write the likelihood for $N$ and $X$. The
filtration $({{\cal F}}^*_t)$ is the self-generated filtration for $N$
when ${{\cal F}}^*_0={{\cal X}}$. Thus we have using Jacod's formula $
{\cal L}_{{{\cal F}}^*}={\cal L}^{\theta/\theta_0}_{{\cal X}}\phi(\Lambda^{{{\cal F}}^*,N}_u,\Lambda^{{{\cal F}}^*,N}_{0u},
N_u, u\ge 0)$, where $$\phi(\Lambda, \Lambda_0,
N)=\phi(\Lambda_u,\Lambda_{0u}, N_u , 0\le u)=\prod_{t \ge 0}\prod
_h \Biggl(\frac{d  \Lambda _h}{d\Lambda_{0 h}}_t\Biggr)^{\Delta
N_{ht}} \frac {\prod_{t \ge 0: \Delta N_{.t }\ne 1} (1-d  \Lambda
_{.t})}{\prod_{t \ge0: \Delta N_{.t} \ne 1} (1-d  \Lambda_{0
.t})},$$ where $N_.=\sum _h N_h$ and $\Lambda_.= \sum _h
\Lambda_h$. Identifying with the decomposition ${\cal L}_{{{\cal F}}^*}={\cal L}^{\theta/\theta_0}_{{\cal X}}
{\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}$ we find that ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}=\phi(\Lambda^{{{\cal F}}^*,N},
\Lambda^{{{\cal F}}^*,N}_0, N)$. With CAR(DYN) this is equal to $\phi
(\Lambda^{{{\cal O}},N}, \Lambda^{{{\cal O}},N}_0, N)$ which is
${{\cal O}}$-measurable, and thus CAR(GCMP) holds.\hfill \ding{113}

\begin{Theorem} \label{Dyn>loc}
CAR(DYN) implies CAR(GCMP)-loc for all $r$.
\end{Theorem}

\noindent {\bf {\em Proof.}} Let us define ${{\cal F}}^r_t={{\cal X}}^r \vee {{\cal N}}_t$ and
${{\cal F}}^r={{\cal X}}^r \vee {{\cal N}}$ then for all $t \in [0,t]$, ${{\cal F}}^r_t
\subseteq {{\cal F}}^*_t$. As seen in the previous proof, ${\cal
L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal R}}|{\cal
X}}=\phi(\Lambda^{{{\cal F}}^*,N}, \Lambda^{{{\cal F}}^*,N}_0, N)$ and ${\cal
L}^{(\theta, \psi) /(\theta_0, \psi_0)}_{{{\cal R}}|{\cal X}^r}=
\phi(\Lambda^{{{\cal F}}^r,N}, \Lambda^{{{\cal F}}^r,N}_0, N)$. So the proof
will be complete if $\forall t \in [0,t]$,
$\Lambda_t^{{{\cal F}}^r,N}1_{\{R=r\}}= \Lambda_t^{{{\cal F}}^*,N}1_{\{R=r\}}$.
We get
 \begin{eqnarray}
 \Lambda_t^{{{\cal F}}^r,N} 1_{\{R=r\}} & = & E[\Lambda_t^{{{\cal F}}^*,N} |{{\cal F}}^r_t]1_{\{R=r\}} \label{ino}\\
 & = & E[\Lambda_t^{{{\cal O}},N} 1_{\{R_{. \wedge t} =r_{. \wedge
t}\}}|{{\cal F}}^r_t] 1_{\{R=r\}} \label{dyn}\\
& = & \Lambda_t^{{{\cal O}},N}1_{\{R=r\}} \label{mesu}\end{eqnarray}

 (\ref{ino}) is due to the innovation theorem, (\ref{dyn}) is due
 to CAR(DYN) and the fact that
$\{R_{. \wedge t} =r_{. \wedge t}\} \subset \{R=r\}$ and $\{R_{.
\wedge t} =r_{. \wedge t}\}\in {{\cal O}}_t \cap {{\cal F}}^r_t$. At last the
local inclusion $\{R_{. \wedge t} =r_{. \wedge t}\}\cap {{\cal O}}_t
\subseteq \{R_{. \wedge t} =r_{. \wedge t}\}\cap {{\cal F}}^r_t$ implies
the ${{\cal F}}^r_t$-measurability of $\Lambda_t^{{{\cal O}},N} 1_{\{R_{. \wedge
t} =r_{. \wedge t}\}}$ and (\ref{mesu}). Using again CAR(DYN), we
have the desired equality. \hfill \ding{113}

\begin{Corollary} \label{Rprev}
If $R$ is ${{\cal O}}_t$-predictable, ignorability holds.
\end{Corollary}

\noindent {\bf {\em Proof.}} The associated counting process $N$ is itself ${{\cal F}}^*_t$-predictable. Thus the Doob-Meyer decomposition is $N_t=N_t+0$ in both $({{\cal O}}_t)$ and $({{\cal F}}^*_t)$. It follows that $\Lambda^{{{\cal F}}^*,N}=\Lambda^{{{\cal O}},N}=N$ which is CAR(DYN). By Theorem \ref{Dyn>loc} and \ref{loc} ignorability holds.\hfill \ding{113}

In the framework of counting processes, Andersen et al. (1993) have proposed a criterion of independent
right-censoring. We adapt their criterion  to right-continuous censoring processes and we restrict to the univariate case for simplicity. In that case $R_t=1_{\{t<C\}}$
where $C$ is a censoring variable and $N_t=1-R_t$. Let $X$ be a counting process and $\Lambda^{{{\cal X}},X}$ its
compensator in the self-generated filtration $({{\cal X}}_t)$. Let $({{\cal F}}_t)$ the filtration generated by both $X$ and $N$.
We have independent right-censoring if the compensator of $X$ is the same in the filtration including
information on the censoring that is : $\Lambda^{{{\cal F}},X}=\Lambda^{{{\cal X}},X}$.

\begin{Theorem}
Let $X$ be a counting process which admits a c\`ag (left-continuous) intensity $\lambda^{{{\cal X}},X}$ in the self-generated filtration $({{\cal X}}_t)$.
Consider a right-continuous right-censoring process of $X$ satisfying  CAR(DYN); then this is an  independent censoring. Inversely, independent censoring implies CAR(DYN).
\end{Theorem}

\noindent {\bf {\em Proof.}} The likelihood of the counting process $(X, N)$ can be written using Jacod's formula as
${\cal L}_{{{\cal X}},{{\cal N}}}=\phi(\Lambda^{{{\cal F}},X},\Lambda^{{{\cal F}},X}_0,X)\phi(\Lambda^{{{\cal F}},N}, \Lambda^{{{\cal F}},N}_0,N)$. As we have done above we can also write it:
${\cal L}_{{{\cal X}},{{\cal N}}}={\cal L}^{\theta/\theta_0}_{{\cal X}} \phi(\Lambda^{{{\cal F}}^*,N}, \Lambda^{{{\cal F}}^*,N}_0,N)$. Noting that ${\cal L}^{\theta/\theta_0}_{{\cal X}} = \phi(\Lambda^{{{\cal X}},X},\Lambda^{{{\cal X}},X}_0,X)$ and equating the two
representations we have:
$$\phi(\Lambda^{{{\cal F}},X}, \Lambda^{{{\cal F}},X}_0,X)\phi(\Lambda^{{{\cal F}},N}, \Lambda^{{{\cal F}},N}_0,N)=\phi(\Lambda^{{{\cal X}},X}, \Lambda^{{{\cal X}},X}_0, X) \phi(\Lambda^{{{\cal F}}^*,N} \Lambda^{{{\cal F}}^*,N}_0,N).$$
CAR(DYN) says that $\Lambda^{{{\cal F}}^*,N}=\Lambda^{{{\cal O}},N}$ and in this
right-censoring case we have $\Lambda^{{{\cal O}},N}=\Lambda^{{{\cal F}},N}$
(this is because ${{\cal O}}_t={{\cal F}}_t$ on $\{t\le C\}$ and
$\Lambda^{N}_{C+u}=\Lambda^{N}_{C}$ whatever the filtration). So
if CAR(DYN) holds, the above equation yields
$\phi(\Lambda^{{{\cal F}},X}, \Lambda^{{{\cal F}},X}_0,
X)=\phi(\Lambda^{{{\cal X}},X},\Lambda^{{{\cal X}},X}_0,X)$. This  must be true
almost surely, for all $(\theta, \psi)$, and moreover, we still
have this equality if we stop the observation at time $t$ or at a
$({{\cal X}}_t)$-stopping time $T$. All that we have to prove is that this
implies $\Lambda^{{{\cal F}},X}=\Lambda^{{{\cal X}},X}$, which is independent
censoring.

Let us begin with $X$ a $0-1$ counting process and denote its jump time $T$. If we stop observation at $t$, we have on $\{T>t\}$, $\frac{\exp \Lambda^{{{\cal F}},X}_t}{\exp \Lambda^{{{\cal F}},X}_{0t}}=\frac{\exp \Lambda^{{{\cal X}},X}_t}{\exp \Lambda^{{{\cal X}},X}_{0t}}$; because of left-continuity, we have also the equality of $t=T$ and because the intensity is equal to zero after $T$, the equality holds for all $t$ almost surely. Taking log and differentiating we obtain:
\begin{equation}  \lambda^{{{\cal F}},X}_t - \lambda^{{{\cal F}},X}_{0t}=\lambda^{{{\cal X}},X}_t - \lambda^{{{\cal X}},X}_{0t}.\label{differ}\end{equation}
The likelihood has a limit when  $t \rightarrow \infty$ and at the limit we have $\frac{\lambda^{{{\cal F}},X}_T\exp \Lambda^{{{\cal F}},X}_T}{\lambda^{{{\cal F}},X}_{0T}\exp \Lambda^{{{\cal F}},X}_{0T}}=\frac{\lambda^{{{\cal X}},X}_T\exp \Lambda^{{{\cal X}},X}_T}{\lambda^{{{\cal X}},X}_{0T}\exp \Lambda^{{{\cal X}},X}_{0T}}$ from which we successively deduce
$\frac{\lambda^{{{\cal F}},X}_T}{\lambda^{{{\cal F}},X}_{0T}}=\frac{\lambda^{{{\cal X}},X}_T}{\lambda^{{{\cal X}},X}_{0T}}$ and
$\frac{\lambda^{{{\cal F}},X}_t}{\lambda^{{{\cal F}},X}_{0t}}=\frac{\lambda^{{{\cal X}},X}_t}{\lambda^{{{\cal X}},X}_{0t}}$, almost surely for all $t$ on the support of the distribution of $T$. Combining this result with (\ref{differ}), we obtain $\lambda^{{{\cal F}},X}_t =\lambda^{{{\cal X}},X}_t $ $a.s.$, which for c\`ag processes implies indistinguishability of the intensities and of the cumulative intensities. If the process may have several jumps $T_1, T_2, \ldots$, we first prove by the same reasoning that we have equality of the intensities on  $\{t\le T_1\}$, then using this result and again the same reasoning we have equality on $]T_1, T_2]$ and so on.
All this reasoning is symmetrical so we can also prove that independent censoring implies CAR(DYN). \hfill \ding{113}

\subsection{Extension to left-continuous $R$}

In some situations it is natural to consider response indicator
processes which are left-continuous; case II right-censoring is an
example. In that case it is not possible to directly associate to
$R$ a counting process and hence to apply CAR(DYN). In order to
extend the application of CAR(DYN) to such processes, we will
consider them as limits of right-continuous processes. Let us
still consider the univariate case. Consider for instance the case
where $R_0=1$. $(R_t)$ may be left-continuous at jumps of odd
ranks: $V_{2j+1}, j\ge 0$; the process can be written
$R_t=\sum_{j\ge 0}1_{[V_{2j},V_{2j+1}]}$. Consider the sequence of
processes ${R^n_t}=(\sum_{j\ge 0}1_{[V_{2j}\le t <V^n_{2j+1}[})
\wedge 1$, defined by: $R^n_0=1$, $V^n_{2j+1}=V_{2j+1}+1/n$. The
limit of $(R^n_t)$ is $(R_t)$.
\begin{Theorem} \label{Rcag}
Consider a process $R=(R_t)$ which is right-continuous at upward
jumps and may left-continuous at downward jumps. Consider a
sequence of right-continuous processes $R^n=(R^n_t)$ constructed
as above; if each $(X,R^n)$ satisfies  CAR(GCMP)-loc on $r$
then ignorability holds for $(X,R)$ on $r$.
\end{Theorem}

\noindent {\bf {\em Proof.}} We note  ${{\cal O}}^n$, the observed $\sigma$-field
associated to ${R^n}$. If $(X,R^n)$ satisfies CAR(GCMP)-loc on
$r$ then (see the proof of Theorem (\ref{loc})), we have for
all $n$: on $\{R=r\}$,
${\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal O}}^n}=
{\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal R}}^n|{{\cal X}}}{\cal L}^{\theta/\theta_0}_{{{\cal X}}^{r^n}}$.
 ${{\cal O}}^n$ is
larger than ${{\cal O}}$: ${{\cal O}} \subset {{\cal O}}^n$ and it is clear that
${{\cal O}}^n$ is a decreasing sequence of $\sigma$-fields: ${{\cal O}}= \cap_n
{{\cal O}}^n= {{\cal O}}^{\infty}$. By the Downward Levy Theorem (Williams,
1990) we have: ${\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal O}}^n}={\rm
E}_{(\theta_0,\psi_0)}({\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal F}}}|{{\cal O}}^n)
\rightarrow {\rm
E}_{(\theta_0,\psi_0)}({\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal F}}}|{{\cal O}})=
{\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal O}}} $ $a.s.$. Using again
the Downward Levy Theorem we get ${\cal L}^{\theta/\theta_0}_{{{\cal X}}^{r^n}}
\rightarrow {\cal L}^{\theta/\theta_0}_{{{\cal X}}^{r}}$ a.s. Moreover note
that ${{\cal R}}^n={{\cal R}}$ because the process $R^n$ is deterministically
defined from $R$: this implies that
${\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal R}}^n|{{\cal X}}}={\cal L}^{\psi/\psi_0}_{{{\cal R}}|{{\cal X}}}$.
At the limit we have thus:
${\cal L}^{(\theta,\psi)/(\theta_0,\psi_0)}_{{{\cal O}}}=
{\cal L}^{\psi/\psi_0}_{{{\cal R}}|{{\cal X}}}{\cal L}^{\theta/\theta_0}_{{{\cal X}}^{r}}$ which
concludes the proof. A similar result could be obtained for upward
jumps. \hfill \ding{113}

 As an example consider the case of Type II right-censoring
where we have independent $0-1$ counting processes $(X^i_t)$, $i=1,\ldots,n$ and observation is stopped just
after observing the $d^{th}$ event. Thus the response indicator process $R$ is not independent on the
multivariate process $X$. In fact we have $R_t=1_{\{\bar X_{t}\le d\}}$, where $\bar X_t=\sum_i X^i_t$. This is
a case of a left-continuous process which has only one downward jump. Since $R$ is ${{\cal X}}$-measurable ${\cal L}^{\psi/\psi_0}_{{\cal R}|{\cal X}}=1$ (by
property iv) and is thus obviously ${{\cal O}}$-measurable which is CAR(GCMP). Consider now a slightly more
sophisticated model which we call randomized Type II censoring in which we may stop observation after each event
with a given probability depending of what have been observed. For instance let $(T_1, T_2,...,T_n)$ the times
of occurrence of the first, second,..., events, and let the probability of stopping observation just after $T_j$
(conditional on having observed $X$ until $T_j$) be $\frac{j-1}{j}$, $j=1,\ldots,n$; let $C$ be the jump time of
$R$ ($C=T_j$, for some $j$).
 We consider $R$ as the limit of the sequence of right-continuous processes $R^n$ such that $R^n_t=1_{\{t\ge C+1/n\}}$. We
can easily verify that these observation processes satisfy CAR(DYN) (because future values of $X$ are not used
for defining the probability of stopping observation), and thus CAR(GCMP) by Theorem 5; thus $R$ itself
satisfies CAR(GCMP) by Theorem \ref{Rcag}.

\section{Applications}

\subsection{Right-censoring of counting processes with time-dependent covariable}
We consider the modelling of independent counting processes $W^i=(W^i_t)$ with possibly time-dependent explanatory
variables $Z^i=(Z^i_t)$. We observe $X^i=(W^i,Z^i)$, $i=1,\ldots,n$ through a mechanism specified by $R=(R^{W^1},
\ldots,R^{W^n};R^{Z^1}, \ldots,R^{Z^n})$. The $(Z^i_t)$ are supposed to be completely observed so that
$R^{Z^i}_t=1$ for all $t$. We consider a family of probability measures $\{P^{\theta,\gamma,\psi}\}$,
where $\theta$ is the parameter of interest which parameterizes the dynamics of $(W^i_t)$ given the value of the
 explanatory variables, that is, the intensity of $(W^i_t)$ in the filtration $({{\cal X}}^i_t)$ (defined as the family of $\sigma$-fields ${{\cal X}}^i_t={{\cal W}}^i_t \vee {{\cal Z}}^i_t$) depends on
$\theta$ only. Let $\gamma$ parameterize the marginal law of $(Z^i_t)$ (this is possible for external
explanatory variables; if there is a causal effect of $W$ on $Z$ we must resort to joint modelling). Consider that assumption A1 holds; for right-censoring this means that the set of times at which observation may be stopped is denumerable; this is not a limitation in practice: for instance in an epidemiological cohort we may say that observation may be stopped each day at a fixed hour (we generally do not have a precision better than one day). If ignorability holds for $(X,R)$, we have on $\{R=r\}$,
 ${\cal L}^{\theta, \gamma / \theta_0, \gamma_0 }_{{{\cal X}}^r}=
{\cal L}^{\theta, \gamma, \psi_0 / \theta_0, \gamma_0, \psi_0}_{{{\cal O}}}$. But we have also ${\cal L}^{\theta, \gamma /
\theta_0, \gamma_0 }_{{{\cal X}}^r}={\cal L}^{\theta/ \theta_0}_{{{\cal W}}^r|Z^r}{\cal L}^{\gamma/ \gamma_0}_{Z^r}$, so that in terms of
inference about $\theta$ we only need to compute ${\cal L}^{\theta/ \theta_0}_{{{\cal W}}^r|Z^r}$. The conclusion is that although we can use the conditional likelihood of $W$ given $Z$ while we may consider the
ignorability condition for $(X,R)$, that is, the response indicator process may depend on both observed $W$ and
$Z$.

We give a particular example of artificial right censoring of $0-1$ counting processes which is compatible with
ignorability (assuming to simplify that there is no other source of random censoring). Consider a study where
$n$ subjects are potentially followed-up until a time $t^*$; it is assumed that the processes $(X^i,Z^i)$ are independent and identically distributed. Suppose that at a given time $t_1$ we make an analysis
of the data. Using for instance maximum likelihood estimators in a parametric model we can construct an
estimator $\hat \theta_1$ which by definition is ${{\cal O}}_{t_1}$-measurable. Let us suppose for simplicity that
$Z_t, t>t_1$ is known at $t_1$; it is then possible to compute an estimator
of the probability that subject $i$ experiences the event before the end of the study $P_{\hat
\theta}(W^i_{t^*}=1|{{\cal W}}_{t_1},{{\cal Z}}_{t_*})$, where ${{\cal W}}_t$ and ${{\cal Z}}_t$ denote $\sigma$-fields generated by the $n$ processes up to time $t$. For reducing the cost of the study without reducing too much its power,
we may decide to follow after $t_1$ only the subjects for whom $P_{\hat \theta}(W^i_{t^*}=1|{{\cal W}}^i_{t_1},{{\cal Z}}^i_{t^*})
\ge c$, for some chosen $c$. In the GCMP this means putting $R^{W^i}_t=0$ for $t>t_1$ for those subjects with an
estimated probability below $c$. It is clear that $R$ is ${{\cal O}}_t$-predictable which  from Corollary \ref{Rprev} implies ignorability on all $r$ because A1 holds. Note that ignorability is not dependent of the good specification of
the model used at $t_1$; of course the validity of the final analysis will depend on the good specification of
the model used for it.

\subsection{Longitudinal markers (continuous state-space processes)}
\subsubsection{Fixed observation times}
This is the classical set-up of ``repeated measurements'' or ``longitudinal data'' (in a narrow sense). We
consider independent continuous state-space processes $(X^i_t)$, $i=1,\ldots,n$; for simplicity we do not consider covariates. Here $(X^i_t)$ is planned to be observed at $v^i_j$, $j=1, \ldots,
m$ but there may be missing data. The response indicator processes for
$(X^i_t)$ can be written: $R_t^{X^i}=\sum_{j=1}^m 1_{\{t=v^i_j\}}Y_j^i$, where $Y_j^i$ are  binary variables. The jump at $v^i_j-$ of the
compensator of the counting process associated to $R^i$ in a filtration $({\cal G}_t)$ is $P(Y_j^i=1|{\cal G}_{v^i_j-})$. CAR(DYN) can thus be expressed as:
$P(Y_j^i=1|Y_1^i,\ldots, Y^i_{v_{j-1}^i},{{\cal W}})=P(Y_j^i=1|Y_1^i,\ldots, Y^i_{v_{j-1}^i},Y_1^iW_1^i,\ldots, Y^i_{v^i_{j-1}}W^i_{v^i_{j-1}})$, which implies (by taking conditional expectation) that
$$P(Y^i_j=1|Y^i_1,\ldots, Y^i_{v^i_{j-1}},{{\cal W}}^i_{v^i_{j}-})=P(Y^i_j=1|Y^i_1,\ldots, Y^i_{v^i_{j-1}},Y^i_1W^i_1,\ldots, Y^i_{v^i_{j-1}}W^i_{v^i_{j-1}});$$ this can intuitively be interpreted in saying that the
missing data mechanism may depend (only) on the observed $W^i$ and on the $Y^i$ up just before time $v^i_j$; this case could be treated with the conventional MAR concept.

\subsubsection{Random observation times}
We may consider determining observation times in order to reduce the costs of a study or to improve the
monitoring of patients. Consider the case of a study of the evolution of CD4 lymphocytes counts in HIV infected
patients; CD4 counts at time $t$ are  represented by $(X^i_t)$; the time of the next visit, $V^i_{j+1}$ may
depend on  CD4 count observed at $V^i_{j}$: $V^i_{j+1}=V^i_{j}+g(X^i_{V^i_{j}})$. For instance we could decide
to see a patient with a delay of three months if CD4 $\ge 500$, two months if $200 \le$ CD4 $<500$ and one month
if CD4 $< 200$. It is clear that CAR(DYN) would hold in this instance. On the contrary if the visit time was decided
based in part on clinical symptoms not included in the model, or if drop-out could be due to severe clinical
events or death (related to CD4 and not included in the model), CAR(DYN) would not hold.

\subsubsection{Vertically coarsened observations}
If we are interested in the evolution of HIV-RNA, we may consider the same issues as above ($(X^i_t)$ now
representing HIV-RNA), with the additional complexity of a (known) detection limit $\eta)$: this produces a left-censoring of
$X^i_t$:  if $R^i_t=1$ we observe $\max (X^i_t, \eta)$.
This is what we have called ``vertical coarsening''. Here the vertical m.l.i.d. is fixed so for studying
ignorability it is sufficient to study the case without vertical coarsening.

\subsection{Joint modeling}
One of the reasons for considering a joint model is precisely to remove the bias due to so called ``informative
censoring ``. Consider as in the example of 7.2.2 that we are interested in the evolution of CD4 lymphocytes
counts represented by $(W^i_t)$. Assume that subjects are lost from follow-up when they develop AIDS. There is
a rather strong relationship between CD4 lymphocytes counts and the risk of developing AIDS, so that the
intensity of the counting process describing drop-out depends on unobserved values of CD4 counts: CAR(DYN) does not
hold in a model which does not include AIDS.

Thus we may consider jointly modelling CD4 counts and AIDS and consider the process $X^i_t=(W^i_t,Y^i_t,Z^i_t)$,
where $W^i_t$ represents CD4 counts, $Y^i_t$ is a counting process which represents AIDS ans $Z^i_t$ is a
multivariate process of explanatory variables. We may allow as in 7.2.2 that the visit times depend on the
observed CD4 counts; we may also allow that the probability that the subject drop out after $V^i_j$ depends on
the AIDS status at this visit. The compensator of the associated counting process $N^i$ will be constant between
$V^i_j$ and $V^i_{j+1}$ and will make a jump at $V^i_{j+1}$ equal to one minus the probability of drop-out: if
the latter depends on the observed AIDS status $Y^i_{V^i_j}$ only, then CAR(DYN) holds; if it depends of unobserved
status $Y^i_t$ for $t\ne V_j$, then CAR(DYN) does not hold.

\section{Conclusion}
We have proposed a general coarsening model for processes (GCMP) and  developed a theory of ignorability in this framework. The theory applies to general stochastic processes having discrete or continuous state-space; in particular it applies to both counting processes and diffusion processes. The framework of repeated measurements can be represented as a continuous-time continuous state-space process observed at discrete times.
Our results hold even if the observed part of the process of interest $X$ has a null probability, which is the case in the examples of the previous section. We have given a factorization condition for the likelihood which allows to get rid of the nuisance parameter and can be called weak ignorability; we can define ignorability in a strong sense, that is equality between the correct likelihood and the likelihood ignoring the m.l.i.d.  on events $\{R=r\}$ of non-null probability. This restriction comes from the fact that likelihoods, as Radon-Nikodym derivatives, are not uniquely defined.

We have applied the results presented here to a multi-state model for dementia, institutionalization and death (Commenges and G\'egout-Petit, 2005). This model can be represented by a three-variate counting process to which we associate the three-variate response process. In this application the observed event of $X$ is generally of null probability although it is not a singleton; institutionalization can be observed either exactly or in an interval; we showed  that the m.l.i.d. could be ignorable.

\section*{References}
\setlength{\parindent}{0.0in}
Alioum A and Commenges, D. (1996) A proportional hazards  model for arbitrarily censored and truncated data. {\em Biometrics}, {\bf 52}, 95-107.\vspace{3mm}

Andersen, P. K., Borgan, \O., Gill, R. D.  and  Keiding, N. (1993) {\it
Statistical Models Based on Counting Processes}. New-York: Springer-Verlag.\vspace{3mm}

Arjas, E. and Haara, P. (1992) Observation scheme and likelihood. {\it Scandinavian Journal of Statistics}, {\bf 19}, 111-132.\vspace{3mm}

Arjas, E., Haara, P. and Norros, I. (1992) Filtering the histories of a partially observed marked point process. {\it Stochastic Processes and their Applications}, {\bf 40}, 225-250.\vspace{3mm}

Barndorff-Nielsen, O. E. and Sorensen, M. (1994) A review of some aspects of asymptotic likelihood theory for
stochastic processes. {\em Int. Statist. Rev.}, {\bf 62}, 133-165.\vspace{3mm}

Commenges, D. and G\'egout-Petit, A. (2005) Likelihood for generally coarsened observations from multi-state or counting process models. Submitted. \vspace{3mm}

Cox, D. R. (1972) Regression models and life tables. {\em Journal of the Royal Statistical Society, Series B },
{\bf 34}, 187-220.\vspace{3mm}

Diggle, P. J. and Kenward, M. G. (1994) Informative Drop-out in Longitudinal Data Analysis. {\em Applied
Statistics}, {\bf 43}, 49-93.\vspace{3mm}

Feigin, P. D. (1976) Maximum likelihood estimation for continuous-time stochastic processes. {\em Adv. Appl. Prob.}, {\bf 8}, 712-736.\vspace{3mm}

Gill, R. D., van der Laan, M. J. and Robins, J.M. (1997) Coarsening at random: characterizations, conjectures
and counter-examples, pp. 255-294 in: {\em State of the Art in Survival Analysis}, D.-Y. Lin and T.R. Fleming
(eds), Springer Lecture Notes in Statistics 123. \vspace{3mm}

Heitjan, D. F. and Rubin,  D. B. (1991) Ignorability and coarse data. {\em Annals of Statistics}, {\bf 19},
2244-2253. \vspace{3mm}

Hoffman-Jorgensen, J. (1994) Probability with View toward Statistics. New-York: Chapman \& Hall.\vspace{3mm}

Jacod, J. (1975) Multivariate point processes: predictable projection; Radon-Nikodym derivative, representation
of martingales. {\it Z. Wahrsheinlichkeitsth.}, {\bf 31}, 235-253.\vspace{3mm}

Jacod, J. and Shiryaev, A. N. (2003) {\em Limit Theorems for Stochastic Processes}. Springer,
New-York.\vspace{3mm}

Jacobsen, M. and Keiding, N. (1995) Coarsening at random in general sample spaces and random censoring in
continuous time. {\em Annals of Statistics}, {\bf 23}, 774-786.\vspace{3mm}

Kalbfleisch, J. D. and Prentice R. L. (1980) {\em The Statistical Analysis of Failure Time Data}, Wiley,
New-York.\vspace{3mm}

Kallenberg, O. (2001) {\em Foundations of modern probabilities}. Springer Verlag, New-York.\vspace{3mm}

Kaplan, E. L. and Meier, P. (1958)  Non-parametric estimation from incomplete observations, {\em Journal of the
American Statistical Association}, {\bf 80}, 457-481, 562-563.\vspace{3mm}

Kenward M. G. and Molenberghs G. (1998) Likelihood based frequentist inference when data are missing at random
{\em Statist. Sci.}, {\bf 13}, 236-247.\vspace{3mm}

Lagakos, S. (1979) General right-censoring and its impact on the analysis of survival data. {\em Biometrics},
{\bf 35}, 139-156.\vspace{3mm}

Nielsen, S.F. (2000) Relative coarsening at random. {\em Statistica Neerlandica}, {\bf 54}, 79-99.\vspace{3mm}

Peto, R. (1973) Experimental survival curves for interval-censored data {\em Applied Statistics}, {\bf  22},
86- 91.\vspace{3mm}

Rubin, D. B. (1976) Inference and missing data. {\em Biometrika}, {\bf 63}, 581-592.\vspace{3mm}

Thi\'ebaut, R. , Jacqmin-Gadda H. , Babiker A. and Commenges D. (2005) Joint Modelling of bivariate longitudinal data with informative drop-out and left-censoring, with application to the evolution of CD4+ cell count and HIV RNA viral load in response to treatment of HIV infection. {\em Statistics in  Medicine},  {\bf 24}, 65-82.\vspace{3mm}

Williams, D. (1991) {\em Probability with martingales}. Cambridge Univ. Press.

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Q: Best Rails 3 XML parser Which XML library, GEM would you recommend to use with Rails 3?

A: Nokogiri is widely used.

A: Try hpricot!
EDIT: I take that back, hpricot is primarily HTML. Nokogiri is probably the best option out there today. For a guide, check out this Engine Yard blog post.

redpajama_set_name: RedPajamaStackExchange


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I use the product on a medium level basis. It seems good so far.
It is used for personal fun stuff.
I use the ink for my daughters project.


text:
Home Sport Rugby Fijiana Xv's on track for 2022 Rugby World Cup
Fijiana Xv's on track for 2022 Rugby World Cup
Fijiana XV's have been pumping up preparation off the field leading up to the Rugby World Cup in New Zealand next year.
World Rugby made the decision to postpone the 2021 Rugby World Cup due to the uncertainty and the challenges of a global COVID-19 landscape.
Fijiana have not played a match together since they last played a year ago, the deferred of the RWC date has provided head coach Senirusi Seruvakula and his management teams the time to work closely with his squad in getting everything right.
Preparations has been well on track for the team but when Covid-19 surge the country around April, it brought everything to a halt.
However, the past 5 months of strict Covid restrictions has saw the Fijiana management team working around the clock to accommodate the players with the much-needed training programme to keep them on track.
Fijiana XV's captain Sereima Leweniqila explained that their management team have been very active in keeping the players engaged.
"Our coaches n SNC team have been very active from the beginning. We have our training chat groups where our progamme is posted on a daily basis and players are tasked to send in daily videos and results," said Leweniqila.
"It's easy to just message our management individually if we are facing any difficulties and they are very understanding, and at times we would have our zoom meetings with our different coaches just to get everyone on the same page," said Leweniqila.
LeweniqiIa emphasised how their journey to the RWC has been a roller coaster not just due to the current situation they're "facing but with the postponement of the world cup as well."
"We can't do much with what the country is going through, for each player it's a different story on the struggle we each face to try and keep the dream alive of playing in the world cup in these difficult times," said Leweniqila.
"Some of us have lost loved ones, contracted COVID-19, loss of job/ hours cut etc, then theres trying to keep fit and improvising with whatever we can find at home to use it as training equipment," she added.
"Not only the form of training that we had been undergoing we had found some to be new to us and also challenging, then we have a few stumbling blocks of smaller issues that we hit along the way," she added.
However, these did not deter them from pushing through, and with the Fijiana 7's success at the Olympics, it has definitely provided them with the urge to train hard, give it their best and aim for the top reward.
"Just an overall inspirational, motivational and just a proud moment for all women who play rugby, their win not only impacted the rugby fraternity but just women in General, winning an Olympic medal or just participating in the event is one of the highest forms of achievement in sport for any women," said Leweniqila.
"As for the Fijiana 15s i think it's given the players a sense of feeling and hope that anything is possible, we just have to work hard and work together," added Leweniqila.
SOURCE: FRU/PACNEWS
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redpajama_set_name: RedPajamaCommonCrawl


Spring is on its way; the quiet of winter is passing and the air is alive with hopeful bird sound. Black strings of Canada geese and white flashes of Tundra Swans make their way north. Song birds are getting excited and chatter noisily about. Resident Birder reports 207 gulls, 110 swan/goose species (too far away to id), 185 icterids, 1705 Canadian geese, and 143 tundra swans.
I'm hearing- Folks gathered in front of the fireplace conversing.
I'm wearing- A tweedy wool skirt and a red wool cardigan.
I'm thankful for- An excellent retreat day at our church this past weekend. It was a silent retreat for the women, while the men will have their turn next weekend.
This Tea Brack from King Arthur (aka Irish Freckle Bread) is becoming a favorite this Lent. Moist, dense and nutritious, it's made with lots of dried fruits which give it a pleasant sweetness. Mary Rose makes these, pretty much following the recipe. She makes two at a time, and the rounds keep nicely in cookie tins. Lacking currants she used a combination of raisins, dates, prunes and apricots for the fruits, and about 3/4 cup of honey instead of sugar for a double recipe. Very good with tea, add a bite of cheese and an orange and you have a little lunch.
Tomorrow we shall have Welsh Cakes for St. David's Day (but no leeks in our hats).
We reckoned that 'Washington pie' consisted of any good thing with cherries thus we enjoyed these almond scented cherry topped squares. I read aloud the Address of the Catholics of the United States to George Washington by the first American Bishop, John Carroll, along with Washington's reply in which he thanks the Catholics for their service (you can read both at that link, the address to Washington is at the bottom). More about Washington's relations with Catholics here and Bishop Carroll's eloquent eulogy for Washington here.
We remain on a bit of a George Washington trail, with all the old familiar picture books out in circulation: D'Aulaire's George Washington, George Washington A Picture Book Biography, When Washington Crossed the Delaware, A Birthday for General Washington, George Washington - A Discovery Book, Buttons for General Washington, The Story of Mount Vernon, Silver for General Washington, and the new old favorite, Meet George Washington.
Best of all, I am reading aloud The Life of George Washington for Children by E. Cecil from 1859. We have a very old and delicate copy that we acquired from my mother-in-law's collection. "This little book has been written with the hope of giving American children some knowledge of Washington's character. Great pains have been taken to make it accurate, and to avoid the use of long words; but it is not possible to render all the complications and responsibilities of Washington's public career perfectly plain to children. Enough, however, is accomplished if they can enter into the spirit of his life, and gain something of that loyalty to his memory which every American should feel."
It is very interesting, with lots of quotes and excerpts from letters and such making it seem authentic in its accuracy, and despite its claim to 'avoid long words' it is naturally written with far richer language than the sort of things produced for children nowadays. We are enjoying it greatly.
Living the Liturgical Year - We're almost halfway through Lent! Laetare Sunday is just around the corner, with Passiontide to follow.
while Josiah has starting turning dip pens.
I use clementine boxes to keep my seeds organized, one for flowers and one for vegetables. I glued in little pieces of slats to make sections to keep things neat and tidy; here in the veggie box I have seeds to start indoors, seeds to direct sow, and herbs.
and it was very, very quiet, with the little girls and their Pa hanging out in the biggest, coolest blanket fort ever.
Hope your week is off to a good start! We survived February, whoo hoo! Happy Leap Day!
PS You can read about the Bissextile Day, a sort of 'liturgical leap day', here.


CRME967 (875 x 710 x 635mm) 227 litre Mobile Truck with 4 x 100mm Nylon Castors (2 fixed, 2 swivel).
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text:
newschain
A better View of the News. Watch the latest breaking news, sport, celebrity and entertainment online. PPA Award finalist for the Diversity of Year Initiative 2020.
https://www.newschainonline.com
Nicky Devlin
Charles Dunne
Jim Goodwin
#Hearts Flynn#Motherwell
Sheffield United F.C.
Ryan Flynn urges St Mirren to make strong squad count in bid for top-six finish
By NewsChain Sport
Ryan Flynn believes St Mirren's squad is as strong as he has known it – but he warned they have to make it count.
Flynn joined the Buddies four years ago in January and has helped them rise from the bottom of the Scottish Championship to, currently, the top six of the Premiership.
Having reached two cup semi-finals last season, St Mirren's next target is to secure a top-six finish having missed out by a matter of minutes last term.
Speaking ahead of Saturday's trip to face Hearts, Flynn said: "It's really clear that the squad is really strong, especially this season, and we have to make it count.
"We have got people fighting for every position throughout the whole team. Hopefully we can do something good this season. It's good for competition for places but there is also such a good spirit through the team."
The strength of the squad was evident when former Motherwell defender Charles Dunne replaced the injured Conor McCarthy during last weekend's draw with Livingston.
"We are gutted that Conor is going to be missing for a period of time but Charles stepped in and was brilliant," Flynn said. "If you get that opportunity you want to grasp it."
Livingston captain Nicky Devlin's late equaliser made it seven draws from 14 matches for Saints in the league and manager Jim Goodwin warned they were in danger of being a "nearly team".
Flynn feels they cannot let the blow get to their heads and will get their rewards if they maintain their standards.
"You have got to make sure your performances stay up," the 33-year-old said. "You have got to put yourself in those positions, like at the weekend when we were 1-0 up. Obviously we let those points slip but we put ourselves in that position.
"We have to keep doing that. The last few games, the points we have got have not been a just reward for the performances we have been putting in.
"We will just keep plugging away, you don't throw in the towel, you keep working hard and I think over a course of a season you get your just rewards."
The game against Livi was Flynn's 100th appearance for the club.
"I have loved being here," the former Falkirk and Sheffield United player said. "When I first moved up it was a great group of lads so I settled in really quickly. It has just been one of those fits.
"The fans have always been with us through thick and thin, and obviously there have been some hard times when we have been scrapping at the bottom. That obviously helps as well."
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She worked in television production at MTV Latin America, Telemundo and Univision before becoming chief director of philanthropy and creative strategies at The Women's Fund Miami-Dade. The Frost is located at 10975 SW 17th St. on FIU's main campus. For information, call 305-348-2890 or visit www.thefrost.fiu.edu.
▪ Miami-based artist Pablo Cano is teaching a figure-sculpting workshop Sunday at the Bakehouse Art Complex in Wynwood. From 3 to 5 p.m., students will learn how to model the human head and body, as well as seated or reclined figures using 17th century techniques and handmade modeling tools. Space is limited, and registration is required. Tickets are $30 for members, $40 for non-members, and can be purchased at www.bacfl.org. The Bakehouse is located at 561 NW 32nd St., Miami. Call 305-576-2828 for more information.
▪ Anthony Liggins and Gallery 88 present Nicola Siso's photography in an exhibit titled "Cuba Is Calling." The show opens June 13 from 6 to 9 p.m. and will be on view through July 23. The gallery is located at 237 20th St., Miami Beach. Call 305-397-8098 or visit anthonyliggins.com.
▪ The "Let's Rock" exhibition at the Fahey/Klein Gallery in the Design District runs through Saturday. The collection of images includes work by some of the world's best photographers as they captured the spirit of legendary musicians such as John Lennon, Jim Morrison, Carly Simon, Joan Jett and Madonna. The gallery is located on the second floor of the Chrome Hearts Building, 4025 NE Second Ave., Miami. Call 786-953-7480 or visit faheykleingallery.com.
Christine Dolen and Galena Mosovich wrote this report. Send visual arts news to Jane Wooldridge at jwooldridge@miamiherald.com.
Ana Martinez has been named director of development for the Patricia & Phillip Frost Art Museum at Florida International University.
National Public Radio's Neda Ulaby will deliver a talk titled 'Arts Journalism in a Fractured Cultureverse' at Fort Lauderdale's ArtServe June 24.
Artist Pablo Cano will teach a figure-sculpting class at the Bakehouse Art Complex in Wynwood. Liam Crotty
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ANOTHER DAY, ANOTHER ADVENTURE
Walk The World
I Know This Place
Another day, another adventure.
Africa's most luxurious safari lodge is in a surprising place
Africa's most luxurious safari lodge is in an unexpected place
Rwanda is full of surprises – and this remarkable hotel is just one of them
Words by Ute Junker
Photos supplied
Pack your sense of wonder when you come to Rwanda. Tourism may be in its infancy here, but Rwanda delivers one of Africa's most memorable visitor experiences. The biggest draw is, of course, a close-up encounter with the famous mountain gorillas, but there is much else to marvel at, from the country's spectacular scenery to its surprising sense of peace and purpose.
Less than 30 years ago, Rwanda was convulsed by a bloody genocide. These days, peace has been restored, investment is flowing, and the country is learning to showcase its many attractions.
That includes its stunning landscapes, its verdant hills alternating with tranquil lakes. Apart from Volcanoes National Park, home to the famed gorillas, the country's loveliest national parks include Nyungwe Forest National Park, where you can track chimpanzees, and Akagera National Park, where giraffe, hippos and lions roam. Equally inviting is the magnificent Lake Kivu, one of Africa's Great Lakes.
The country's capital, Kigali, is worth a visit in its own right. The fascinating Kigali Genocide Memorial does a remarkable job of explaining Rwanda's darkest chapter – when one million Rwandans were murdered in 100 days – while other attractions include a surprising number of local art galleries, lively markets (check out the magnificent textiles) and East Africa's only whisky distillery.
Rwanda is also home to some of Africa's most luxurious lodges, including the marvellous One & Only Nyungwe House), For sheer wow factor, however, Wilderness Safari's remarkable Bisate Lodge is the one to beat.
Your first glimpse is enough to realise that this is not your average safari lodge. Perched on a jungle-covered mountain ridge, the accommodation consists of six pods, shaped like beehives and appearing to have been woven like baskets. Up close, the unusual design – inspired by Rwanda's traditional royal architecture – is more of a figure 8, with walls that curve up into the domed ceiling like space pods.
There's a king-size bed, two balconies, magnificent views onto the nearby peaks and a volcanic stone fireplace; perfect for curling up in front of during misty afternoons. The bathroom is huge, with a sexy black free-standing tub and a volcanic-stone shower.
Need more reasons to indulge in a spot of cocooning? The food is superb – the menus change daily – and the service is stellar. Come in from a trek and your muddy shoes will be whisked away, to be return later freshly cleaned; a laundry service does the same for your clothes. In-room massages are available, and whatever else you might need – from a cappuccino to help stoking your fire – is instantly delivered with a smile.
Luxury in the Cambodian wilderness
Bill Bensley's spectacular tented camp, Shinta Mani Wild, lets guests experience luxury in the heart of the Cambodian wilderness.
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UTE JUNKER
Luxury Travel and Lifestyle - Writer, editor, podcaster. Co-founder, tonicmag.com.au
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Dense arrays may lead to high-resolution printers
If lasers could be stacked close enough together in a laser printer such that each was responsible for placing a dot on the paper, it would eliminate the current need for laser beam-steering with a spinning polygonal mirror. This might lead to lower-cost printers and could certainly increase printing speed. Project leader Robert Thornton, Chris Chua, and David Treat at Xerox Palo Alto Research Center (PARC; Palo Alto, CA) have developed a very dense linear array of vertical-cavity surface-emitti
Yvonne Carts-Powell
If lasers could be stacked close enough together in a laser printer such that each was responsible for placing a dot on the paper, it would eliminate the current need for laser beam-steering with a spinning polygonal mirror. This might lead to lower-cost printers and could certainly increase printing speed. Project leader Robert Thornton, Chris Chua, and David Treat at Xerox Palo Alto Research Center (PARC; Palo Alto, CA) have developed a very dense linear array of vertical-cavity surface-emitting lasers (VCSELs). These could be used in laser printers, as well as optical signal-processing systems. For both applications, the more elements in an array, the better.
At the Conference on Lasers and Electro-Optics (CLEO; San Francisco, CA) in May, Chua presented paper CdThK6 on making independently addressable VCSEL arrays with only 3 µm from the center of one VCSEL to the center of the next. Current typical VCSEL arrays have center-to-center spacing of about 50-100 µm.
The high-density array from PARC is made possible by transparent contact technology and a planar laterally oxidized device architecture. CLEO session chair Kent Choquette of Sandia National Laboratories (Albuquerque, NM) says, "The technologies are old hat, but the ways the Xerox group is using them is pushing the envelope."
VCSEL design
"We are interested in replacing the polygon in a laser printer with an array of semiconductor lasers," explains Chua. If each dot corresponds to a laser, then a printer with 1200 dots-per-inch (dpi) resolution would require 1200 VCSELs per inch. "If you have a document 11 in. long, you need roughly 13,000 lasers," he adds. With a 3-µm pitch, "a device 4 cm long would contain more than 10,000 lasers." The advantage of having so many lasers in such a small area is that it allows for a reasonable manufacturing yield.
The devices must exhibit certain characteristics. Regular VCSEL arrays use opaque metallic top contacts that take up space between the emitting areas. For this device, however, the researchers used interdigitated transparent contacts made of indium tin oxide--the same material used for transparent contacts on active-matrix liquid-crystal displays (see figure on p. 20). As with these displays, this arrangement allows individual addressing of the VCSELs.
Heat, in particular, can be a problem; with the lasers so tightly packed, heat from one device can affect the surrounding devices. Individual lasers must, therefore, work at low operating currents and high power-conversion efficiency.
To achieve the high efficiency, the researchers use a variation of lateral oxidation, a technique developed in 1994 at the University of Texas (Dallas, TX). By selectively oxidizing an area around the active area of the VCSEL, the light is better confined. Lateral oxidation improves the efficiency of a VCSEL by orders of magnitude, says Chua. But the standard method is difficult to manufacture. It requires etching out a mesa (containing the VCSEL) to expose the layer to be oxidized. The etching reduces the planarity of the device, which makes establishing electrical contacts more difficult. This is why no commercially available VCSELs use this technique.
Instead, the researchers used photolithography and a dry-etch method to create a uniform grid of via holes into the epitaxially grown layers. Small circular patterns are made, then etched about 3.5 µm deep. The 2-µm-diameter holes allow access to the aluminum gallium arsenide (AlGaAs) layer, which is selectively oxidized. Each VCSEL is bordered by six via holes. The AlGaAs layer is oxidized outward from each hole until a 4-µm-diameter aperture is exposed at the center. This method maintains the planarity of the device.
The device was grown by metal-organic vapor-phase epitaxy on n-type GaAs substrate. The bottom mirror has 35 pairs of silicon-doped AlGaAs distributed Bragg reflectors. The resonant cavity is a one-wave-thick separate confinement heterostructure with four strained indium aluminum gallium arsenide (InAlGaAs) quantum wells. The oxidation layer of AlGaAs is grown above the active region, and 25 pairs of magnesium-doped distributed Bragg reflectors are deposited as the top mirror of the VCSEL.
The lasers operate at a wavelength of 813 nm with threshold currents of less than 200 µA and threshold voltages of less than 3.5 V. The arrays showed low interelement thermal crosstalk, which indicates that heating effects are within acceptable boundaries for the array. The array has electrical resistance of 3 MW between adjacent array elements. Output powers for a 16-element array ranged from 270 to 340 µW.
The researchers also looked at far-field interference patterns in the outputs of the lasers. A lack of interference effects in the optical far-field of the VCSEL beams indicates that the lasers do not have coherent optical crosstalk. Furthermore, the transverse mode of any one device is not affected by the state of its neighbors.
YVONNE CARTS-POWELL is a science writer based in Belmont, MA.
Optical micrograph of the linear VCSEL array shows each laser surrounded by six vias and covered by the end of a transparent electrical contact.


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Q: Is reinforcement learning the right approach for finding unknown functional relationship in this example? We have a machine learning problem as depicted in the following figure.
It has the following structure:

*

*Data consists out of float values $a_i$, $n_i$, $X_i$, $Y_i$ and an outcome $O_i$ (0 or 1).

*$X_i$ and $Y_i$ are decisive for $F_i$ and there is a functional relationship between $X_i$ and $Y_i$ and $F_i$ but we don't know the type of it and it might be very complex.

*What we know is how $F_i$ needs to be postprocessed to find $G_i$, i.e. we calculate $H_i=f(F_i,n_i)$ and compare $H_i$ against $a_i$ which gives $G_i=1$ if $H_i>a_i$ and 0 otherwise.

*What we want is the best agreement between all $G_i$ and $O_i$.
So to sum it up:
We are interested in a neural network (or similar) that allows us to predict $F_j$ for a new dataset $X_j$ and $Y_j$, but we can only train using the booleans $G_i$ and we do not know or have any access to any training value $F_i$.
We thought about reinforcement learning, but I only heard of applications in action taking depending on a state (e.g. pacman).
Can you tell me if reinforcement learning might work on this or can you direct me to any other field in machine learning that might help here?

A: I think if you use a neural network classification and train your model using back-propagation your model would find the best value for Fj. Therefore for your test set with given new Xj and Yj values your trained model will predict the Gj.
This way you don't need and Fj values since your network will choose Fj value to predict the outcome which is either 0 or 1.
Reinforcement Learning is used for maximising the reward from the environment, so you can use a normal neural network for predicting the Gj values
hope it helped