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\begin{document}
\centerline{\large \bf Conjoint Degradation Model of Disablement}
\centerline{\large \bf for  Survival and Longitudinal Data Measured with Errors}
\centerline{\large \bf in Presence of Covariates}

 \vskip 0.5cm
  \centerline{V.
Bagdonavi\v{c}ius$^1$ and M. Nikulin$^2$} \vskip 0.1cm \noindent $^{1}$
{\it \small \it Department of Mathematical statistics, University
of Vilnius, Vilnius, Lithuania}

\noindent $^{2}$ {\small \it EA 2961 "Mathematical Statistics and
Its Applications",  Victor Segalen University Bordeaux 2, France
\&  Laboratory of Statistical Methods, Steklov Mathematical
Institute, Saint Petersburg, Russia} \vskip 1cm



 \centerline{\bf Abstract} \vskip 0.5cm The paper
considers the semiparametric analysis of several new degradation
and failure time regression models  without and with time
depending covariates.  These joint models for survival and
longitudinal data measured with errors can be applied in studies
of  longevity, aging and degradation in survival analysis,
biostatistics, epidemiology, demography, biology and reliability.


\vskip 0.5cm
\par {\it Keywords}:  aging, censored data, consistency,  degradation,
degradation process, disability,failure, longevity, longitudinal
data, measurement error,  noise, nonparametric estimation, optimal
estimator,  path model, regression parameters, reliability,
semiparametric estimation, survival analysis, traumatic event,
Wiener process, Wulfsohn-Tsiatis model.
 \vskip 0.5cm

\par {\it AMS classification}:  62F10, 62J05, 62GO5, 62N05, 60M10,
62G30


\centerline{\bf 1. Introduction}

\vskip 0.5cm
 The purpose of this paper  is at first the construction of a statistical accelerated
 degradation
 model of disablement in the elderly to verify
    that a hierarchical relationship
 exists between the concepts of Activities Daily Living (ADL), Instrumental Activities  of Daily Living (IADL) and mobility and
 to use this model to study the evolution of disability.
    The concepts of ADL, IADL and mobility have been developed and widely used for the assessment of the consequences of disease.
    ADL include basic activities of personal care, IADL functions are concerned with a person's ability to perform activities in a
    given environment and according to sex-linked roles, and the definitions of mobility vary from basic function to more complex
    level of activities. The Rosow-Breslow (1966)  scale includes items which explore gross mobility. The disability is defined
    as a reduction or loss of functional capacity or activity resulting from an impairments.
    The PAQUID (Personnes Ag\'ees QUID)  study is an epidemiological study of functional and cerebral ageing conducted in
    75 randomly chosen parishes in two administartive areas "Gironde" and  "Dordogne" in the South-Western France since 1988/89.
    To be included at baseline subjects has to be aged 65 or more on the 1 January of 1998. 3777 persons gave their written
    consent to participate and were visited at home by specially trained psychologists  for the baseline interview and were followed-up
    3  and  5 years after in order to identify cases of {\it dementia}.
     The general descriptions of the PAQUID study one can see, for example, in Dartigues  and al. (1991),   Barberger-Gateau and al
      (1992, 2000). Following  Katz and al. (1970), Spector (1990) and Rosow and Breslow (1966) functional assessment variables used
      in the present study included:\\
      1. Five ADL items: bathing, dressing, going to toilet, transfer and feeding.\\
      2. Five IADL items for both sexes: ability to use telephone, shopping, mode of transportation, responsibility for own medication,
      ability  to handle finances.\\
      3. Three items from the Rosow scale; walk up and down to second floor, walk half a mile, and do heavy work around the house.\\

    For each of these three domains a subject was considered as "{\it dependent}" if he could not perform at least one activity of the
    domain without  a given level of help. Dependency for each domain was represented by a single item coded 0 (fully independent
    subject for the domain) versus 1.
        In Guttman's approach an unidimensional  scale is  in which the subject's responses to the items would place individuals in
        perfect order. According to this perfect scale the sum of correct responses to the items can be used to recreate all the responses
         of  a subject in a single digit. In this case a single disability scale  is obtained by summing up the responses to each of the
         three dichotomized disability items for all subjects fitting into the hierarchy:\\
          subjects without any dependency were coded 0,\\
         those only in th Rosow scale were coded 1,\\
          those dependent in Rosow and IADL scales not ADL scale were coded 2,\\
           and those dependent were coded 3.
           \par Zdorova-Cheminade (2003) studied  by simulation the considered model as
 statistical  degradation
     model of disablement in the elderly to verify
    that a hierarchical relationship
 exists between the concepts of Activities Daily Living, Instrumental Activities
   of Daily Living  and mobility and
 to use this model to study the evolution of disability.
 The cumulative disability scale was used to describe the degradation
             process in time. In longitudinal analysis an
            additional level was considered to the disability index to take death into
             account. It is evident that this approach
            can be used in many other medical studies where a degradation is observed,
             especially in oncology.

To construct a mathematical model we shall use the terminology
used in theory of reliability, (see,  Meeker and Escobar (1998),
Gavrilov and Gavrilova (2003), Bagdonavicius and Nikulin (2001)).
Functioning of a unit is characterized by its degradation process
and by the random moments of its potential failures. We call a
failure of a unit natural if the degradation attains some critical
level. Other failures are called traumatic. These can be related
with production defects, caused by mechanical damages or by
fatigue of components etc.
\par The intensities of the traumatic
failures depend on degradation. As a rule these intensities are
increasing functions of degradation values. Suppose that the
lifetime of a unit is determined by the
 degradation process
$Z(t)$ and the moment of its potential traumatic failure $T$. For example, $Z(t)$ may be
the value of tire wear at the moment when a tire has run $t$ km
(in this case "time" is the tire run), the size of fatigue crack,
the size of failure-causing conducting filament  of
chlorine-copper compound in a printed-circuit board, luminosity of
light emitting diode at the moment $t$ (Meeker and Escobar
(1998), Bagdonavi\v{c}ius and Nikulin (2002)), etc.

Denote by $T^0$ the moment of non-traumatic failure, i.e. the moment when the degradation attains some
critical value $z_0$. The moment of the
unit's failure is
$$
  \tau=T^0\wedge T.
$$
\vskip 0.5cm

\centerline{\bf 2. Modeling the degradation-failure time process}
\vskip 0.5cm
We suppose that the real degradation process is modeled by the general
path model (Meeker and Escobar (1998))
\beq
  Z_r(t)=g(t,A);
\eeq
here $A=(A_1,\dots,A_r)$ is a random vector with positive
components and the distribution function $F_A$, and $g$ is a
specified continuously differentiable  increasing in $t$ function. The typical
form of the degradation curves is
\beq
g(t,a)=e^{a_1}(1+t)^{a_2}.
\eeq
In the particular case of  linear degradation $a_2=0$.


Denote by $h$ the function inverse to $g$ with respect to the
first argument. Evidently, it is continuously differentiable and
increasing in $t$. Moreover,
\beq
  T^0=h(z_0,A).
\eeq
\noindent\par The observed degradation process $Z=Z(t), t\geq 0$ may be slightly different from the real degradation process.

Suppose that the values of the real degradation process are measured at time moments
$t_1,\dots,t_m$.

 \noindent{\bf Model 1} (Degradation with measurement errors). {\it The observed degradation values are
\beq
Z(t_j)=Z_r(t_j)\;U(t_j),
\eeq
where $e_j=\ln U(t_j)$ are i.i.d. random variables, $e_j\sim N(0,\sigma^2)$.}
\vskip 0.3cm
 \noindent {\bf Model 2} (Degradation with the noise). {\it The observed degradation process is
\beq
Z(t)=Z_r(t)\;U(t),
\eeq
where
\beq
V(t)=\ln U(t)=\sigma W(c(t)),
\eeq
$W$ is the standard Wiener process independent on $A$, and $c:[0,\infty\to [0,\infty)$, $c(0)=0$,
is a specified continuous and increasing time function, $c(0)=0$.}

For any $t>0$ the median of
the random variable $U(t)$is $1$.

The cumulative distribution function (c.d.f.) of the real degradation at the moment $t$ is
\beq
F_{r0}(z\mid t)=P(g(t,A)\leq z)=\intl_{g(t,a)\leq z}dF_A (a),
\eeq
the mean real degradation attained at the moment $t$ is
\beq
m_{r0}(t)=E g(t,A)=\intl_{\cal A} g(t,a) dF_A (a),
\eeq
where ${\cal A}$ is the set of possible values of the random vector $A$.

Note that the characteristics (7) and (8) have sense only in the ideal situation when the
probability of failure until the moment $t$ is zero. So we shall modify
them defining the c.d.f. and other characteristics of the degradation at any moment $t$
as conditional given survival up to this moment.

\noindent\par Let $T$ be the time to traumatic failure.

 \noindent{\bf Model 3.} {\it The  conditional survival of $T$ given the real degradation process has the form:
\beq
S_T(t\mid A)=P\{T>t\mid  g(s,A),0\leq s\leq t\}
  =\exp\Bigl\{-\int_0^{t}\lambda_0(s,\alpha)\lambda\bigl(g(s,A)\bigr)ds\Bigr\};
\eeq
here $\lambda$ is the unknown intensity function, $\lambda_0(s,\alpha)$ being from
a parametric family of hazard functions.}

Note
that the function $\lambda$ is defined on the set of degradation
values, not on the time scale.

The model states that the conditional hazard rate $\lambda_T(t\mid A)$ at the moment $t$ given
the degradation $g(s,A), 0\leq s\leq t$, has the form
\beq
\lambda_T(t\mid A)=\lambda_0(t,\alpha)\lambda(g(t,A)).
\eeq
The term  $\lambda(g(t,A))$ shows the influence of degradation on the hazard rate, the term
$\lambda_0(t,\alpha)$ shows the-influence of time on the hazard rate not explained by degradation. If,
for example, $\lambda_0(t,\alpha)=(1+t)^\alpha, e^{\alpha t}$, then  $\alpha=0$ corresponds
to the case when
the hazard rate at any moment $t$ is a function of the degradation level at this moment.

Wulfsohn and Tsiatis (1997) considered the so called joint model for survival and longitudinal data measured with error,
given by
$$
\lambda_T(t\mid A)=\lambda_0(t) e^{\beta (A_1+A_2t)}
$$
with bivariate normal distribution of of $(A_1,A_2)$. The difference: in our model the function
 $\lambda$, characterizing the influence of degradation on the hazard rate, is non-parametric, in the
 Wulfsohn-Tsiatis model this function is parametric. On the other hand, the baseline
  hazard rate $\lambda_0$ (it is proportional to the hazard rate which should be observed
  if the degradation would be absent) is parametric in our model and non-parametric
  in Wulfsohn-Tsiatis model. We consider the case when the distribution of $A$ is not
  specified.

Let us consider the model (9) and set
\beq
  \Lambda(z)=\int_0^z\lambda(y)dy.
\eeq
The conditional survival function (5) can be rewritten in the following form
\beq
S_T(t\mid A)
  =\exp\Bigl\{-\int_{g(0,A)}^{g(t,A)}\lambda_0(h(z,A),\alpha) h'(z,A)\,d\Lambda(z)\Bigr\},
\eeq
where $h$ is the function inverse to $g$ with respect to the
first argument.


\vskip 0.5cm
\centerline{\bf 3. Survival and degradation characteristics}
\vskip 0.5cm


The  survival function and the mean of
the time-to-falure $\tau$ are
\beq
  S_\tau(t)=P(\tau >t)
  =\intl_{{g(t,a)\leq z_0}}S_T(t\mid a)dF_A(a)
 ,
\eeq
and
\beq
  e_\tau=E(\tau)
  =\intl_{{\cal A}}\{h(z_0,a)S_T(h(z_0,a)\mid a)+
  $$
  $$
  \intl_{g(0,a)\wedge z_0}^{z_0} \lambda_0(h(z,a))
h(z,a)  h'(z,a)S_T(h(z,a)\mid a)d\Lambda(z)\}dF_A(a)
 ,
\eeq
respectively.

Set
\beq
\delta=\left\{\begin{array}{cc} 0,& \mbox{if}\quad \tau=T^{(0)},\\ 1,&
\mbox{if}\quad \tau=T.\\
\end{array} \right.
\eeq
Important reliability characteristics  are the
probability  of the traumatic failure in the interval $[0;t]$:
\beq
P^{(tr)}(t)=P\{\tau\le t,\ \delta=1\}=1-\intl_{\cal A}S_T(t\wedge h(z_0,a))\mid a)dF_A(a)
\eeq
and the
probability  of the non-traumatic failure in the interval $[0;t]$:
\beq
P^{(0)}(t)=\{\tau\le t,\ \delta=0\}=\intl_{g(t,a)\geq z_0}S_T(h(z_0,a))\mid a)dF_A(a).
\eeq
 Now we are able to modify the characteristics (7) and (8) to the case of the real situation.

The c.d.f. of the real degradation at the moment $t$ given the survival until this moment is:
\beq
F_{r}(z\mid t)=P(g(t,A)\leq z\mid \tau>t)=\frac1{S_\tau(t)}\intl_{g(t,a)\leq z}S_T(t\mid a)dF_A (a),
\eeq
if $z\leq z_0$, and $F_{r}(z\mid t)=1$, if $z>z_0$. The mean real degradation
attained at the moment $t$ given the survival until this moment is
\beq
m_{r}(t)=E (g(t,A)\mid \tau>t)=\frac1{S_\tau(t)}\intl_{\cal A} \{g(t,a)\wedge z_0-z_*\} S_T(t\mid a)dF_A (a),
\eeq
where $z_*$ is the minimal possible value of degradation.

For any $B\subset {\bf R},B_0\subset {\bf R}$, $t_1\leq s <t$ the probability that at the moment
$t$ the value of the real degradation will be in the set $B$ given that at the moment $s$ it is
in the set $B_0$ and given the survival until $t$, is
$$
P(g(t,A)\in B\mid g(s,A)\in B_0, T>t)
$$
\beq
=
{\intl_{g(t,a)\in B,g(s,a)\in B_0}S_T(t\mid a)d
F_A (a\mid t)}/{\intl_{g(s,a)\in B_0}S_T(t\mid a)dF_A (a\mid t)}.
\eeq
Denote by
$$
T^0(z)=h(z,A)
$$
the time needed to attain the level $z$ $(z\leq z_0)$ of the real degradation. The conditional survival function
and the conditional mean of
 $T^0(z)$ given the survival until  the degradation attains the level $z$ and given  survival until the
 moment $T^0(z)$ are
\beq
P(T^0(z)>t\mid T>T^0(z))=
\frac{\intl_{g(t,a)<z}S_T(h(z,a)\mid a)
dF_A (a)}{\intl_{{\cal A}}S_T(h(z,a)\mid a)
dF_A (a)},
\eeq
and
\beq
E(T^0(z)\mid T>T^0(z))=
\frac{\intl_{{\cal A}}h(z_0,a)S_T(h(z,a)\mid a)
dF_A (a)}{\intl_{{\cal A}}S_T(h(z,a)\mid a)
dF_A (a)},
\eeq
respectively.

  \vskip 0.5cm
\centerline{\bf 4. Modeling covariate effects on degradation and survival}
\vskip 0.5cm
Suppose units are observed under the covariate $x=x(t)=(x_1(t),\dots,x_m(t)),t\geq 0$, and
 the values of the real degradation process are measured at time moments
$t_1,\dots,t_m$.

In this case, following Bagdonavi\v{c}ius and  Nikulin (2001), the real degradation process is modeled by
\beq
Z_r(t\mid x)=g(\varphi(t,\beta,x),A),
\eeq
where
\beq
\varphi(t,\beta,x)=\int_0^{t}e^{\beta x^T(u)}du,
\eeq
is  the link function, $\beta=(\beta_1,\dots,\beta_m)$ is the vector of the  regression parameters.

If the covariate is constant over time then
\beq
\varphi(t,\beta,x)=e^{\beta x^T}t.
\eeq
\vskip 0.3cm
 \noindent {\bf Model 4} (Degradation with measurement errors). {\it The observed degradation values are
\beq
Z(t_j\mid x)=Z_r(t_j\mid x)\;U(t_j),
\eeq
where $e_j=\ln U(t_j)$ are i.i.d. random variables, $e_j\sim N(0,\sigma^2)$.}


\vskip 0.3cm
 \noindent {\bf Model 5} (Degradation with the noise). {\it The observed degradation process is
$$
Z(t\mid x)=Z_r(t\mid x)\;U(t),
$$
where
\beq
V(t)=\ln U(t)=\sigma W(c(t)),
\eeq
$W$ is the standard Wiener process independent on $A$, and $c:[0,\infty\to [0,\infty)$,
is a specified increasing and continuous time function, $c(0)=0$.}


The c.d.f. of the real degradation at the moment $t$ given the covariate $x(s),0\leq s \leq t$, is
\beq
F_{r0}(z\mid t,x)=\intl_{g(\varphi(t,\beta,x),a)\leq z}dF_A (a),
\eeq
the mean real degradation attained at the moment $t$ is
\beq
m_{r0}(t,x)=\intl_{\cal A} g(\varphi(t,\beta,x),a) dF_A (a).
\eeq
As in the case without covariates these characteristics have sense only in the ideal situation when the
probability of failure until the moment $t$ is zero.

Similarly as in (9), we model the survival distrubution:
\vskip 0.3cm
 \noindent{\bf Model 6} {\it The  probability of survival until the
moment $t$  given the real degradation process $Z_r(s\mid x),0\leq s\leq t$
is
$$
S_T(t\mid a,x)=P\{T>t\mid  A=a,x(s),0\leq s \leq t\}
  =\hskip 5cm
  $$
  \beq
\hskip 5cm  \exp\Bigl\{-\int_{0}^{t}\lambda_0(\varphi(s,\beta,x),\alpha)\lambda\bigl(g(\varphi(t,\beta,x),a)\bigr)ds\Bigr\},
\eeq
where the $\lambda$ is an unknown intensity function, $\lambda_0(s,\alpha)$ being from
a parametric family of hazard functions.}

The probability (30) can be rewritten in the following form
\beq
S_T(t\mid a,x)
  =\exp\Bigl\{-\int_{g(\varphi(0,\beta,x),a)}^{g(\varphi(t,\beta,x),a)}
 \lambda_0(h(z,a),\alpha)\psi'(h(z,a),\beta,x)h'(z,a)\,d\Lambda(z)\Bigr\},
\eeq
where $\psi(s,\beta,x)$ is the inverse function of $\varphi(t,\beta,x)$ with respect to the first argument,
$\psi'_1$ is the derivative of $\psi$ with respect to the first argument, and $\Lambda$ is defined by the formula (11).

 In the case of constant covariates
 $$
 \psi(t,\beta,x)=e^{-\beta x^T}t,\quad \psi'(t,\beta,x)=e^{-\beta x^T}.
 $$
The given model implies that the survival function and the mean of the random variable $\tau$ given the covariate $x$
are
\beq
  S_{\tau}(t\mid x)=P(\tau>t\mid x(s),0\leq s \leq t)
  =
  \intl_{g(\varphi(t,\beta,x),a)<z_0}S_T(t\mid a,x)dF_A(a).
  \eeq
 The
probability  of the traumatic failure  in the interval $[0;t]$under the covariate $x$ is:
\beq
P^{(tr)}(t\mid x)=P\{\tau\le t,\delta=1\mid x(s),0\leq s\leq t\}=\hskip 5cm
$$
$$
\hskip 5cm
1-\intl_{\cal A}S_T(t\wedge \psi (h(z_0,a),\beta,x))\mid a,x)dF_A(a)
\eeq
and the
probability  of non-traumatic failure in the interval $[0;t]$ under the covariate $x$ is
\beq
P^{(0)}(t\mid x)=\{\tau\le t,\delta=0\mid x(s),0\leq s \leq t\}=
\hskip 5cm
$$
$$
\hskip 5cm\intl_{g(\varphi (t,\beta,x)),a)\geq z_0}
S_T(\psi (h(z_0,a),\beta,x))\mid a,x)dF_A(a),
\eeq
where  $\delta$ is given by (15).

 The c.d.f. of the real degradation at the moment $t$ given the survival until this moment is
\beq
F_{r}(z\mid t,x)=P(Z_r(t\mid x)\leq z\mid  x(s),0\leq s \leq t,\tau>t)
$$
$$
=\frac1{S_\tau(t\mid x)}\intl_{g(\varphi(t,\beta,x),a)\leq z}S_T(t\mid a,x)dF_A (a),
\eeq
and the mean real degradation attained at the moment $t$ given the survival until this moment is
\beq
m_{r}(t,x)=E Z_r(t\mid x)=\frac1{S_T(t\mid x)}
\intl_{\cal A} (g(\varphi(t,\beta,x),a)\wedge z_0-z_*)S_T(t\mid a,x) dF_A (a).
\eeq
For any $B\subset {\bf R},B_0\subset {\bf R}$, the probability that at the moment
$t$ the value of the real degradation will be in the set $B$ given that at the moment $s$ it is
in the set $B_0$ and given $x(u),0\leq u\leq t$, is
$$
P(Z_r(t\mid x)\in B\mid Z_r(s\mid x)\in B_0, \tau>t, x(u),0\leq u\leq t)\hskip 5cm
$$
\beq
\hskip 6cm =
\frac{\intl_{g(\varphi(t,\beta,x),a)\in B,g(\varphi(s,\beta,x),a)\in B_0}S_T(t\mid a,x)
d
F_A (a)}{\intl_{g(\varphi(s,\beta,x),a)\in B_0}S_T(t\mid a,x)dF_A (a)}.
\eeq
Denote by
\beq
T^0(z,x)=\psi(h(z,A),\beta,x)
\eeq
the time needed to attain the level $z$ of the real degradation under the covariate $x$. The conditional survival
function and the conditional mean of
 $T^0(z,x)$ given the survival until  $T^0(z,x)$,  and given $x(u),0\leq u\leq t$, are
\beq
P(T^0(z,x)>t\mid \tau>T^0(z,x), x(u),0\leq u\leq t)=\hskip 5cm
$$
$$
\hskip 5cm
\frac{\intl_{g(\varphi(t,\beta,x),a)<z}S_T(\psi(h(z,a),\beta,x)\mid a,x)
dF_A (a)}{\intl_{\cal A}S_T(\psi(h(z,a),\beta,x)\mid a,x)
dF_A (a)},
\eeq
and
\beq
E(T^0(z)\mid \tau>T^0(z), x(u),0\leq u\leq t)=\hskip 5cm
$$
$$
\hskip 5cm \frac{\intl_{\cal A}\psi(h(z,a),\beta,x)
S_T(\psi(h(z,a),\beta,x)\mid a,x)dF_A (a)}{\intl_{\cal A}S_T(\psi(h(z,a),\beta,x)\mid a,x)
dF_A (a)},
\eeq
respectively.
\vskip 0.5cm
\centerline{\bf 5. Semiparametric estimation of degradation and survival characteristics:}
\centerline{\bf models without covariates}
\vskip 0.5cm


Assume that the c.d.f. $F_A$ and the intensities $\lambda$ are completely unknown.

Fix the degradation measurement moments $t_{i,1},\dots,t_{i,m_i}$ of the $i$th unit $(i=1\dots,n)$. If the failure time
 $\tau_i$ of this unit occurs in the interval $[t_{i,j_i},t_{i,j_i+1})$ $(j_i=1,\dots,m_i;t_{m_{i}+1}=\infty)$
 then
 the values $Z_{i1},\dots,Z_{i,j_i}$
of the degradation process $Z_i$ of the $i$th unit are observed at the time moments
$t_{i1},\dots,t_{i,j_i}$.   Set
$$
Y_{ij}=\ln Z_{ij},
\quad
Y_i=(Y_{i1},\dots,Y_{ij_i})^T.
$$
Then given $A_i=a_i$, $j_i$
$$
Y_i\sim N(\mu_i,\sigma^2\Sigma_i),
$$
where
$$
\mu_i=(\mu_{i1},\dots,\mu_{i,j_i})^T,\quad
\mu_{ij}=\mu_{ij}(a_i)=\ln g(t_{ij},a_i),\quad
 \Sigma_i=\mid\mid
s_{ikl}\mid\mid_{j_i\times j_i} ,
$$
\beq
 s_{ikl}=\Un_{\{k=l\}}\quad (\mbox{first model}),
 $$
 $$
 s_{ikl}=c_{ik}\wedge c_{il},\quad (\mbox{second model}),\quad c_{ij}=c(t_{ij}).
\eeq
Denote by $b_{ikl}$ the elements of the inverse matrix $\Sigma^{-1}_i$, and by
$N$ the number of units such that $j_i\geq r$ (usually $r=2$).

The predictors $\hat A_i$ of the random vectors $A_i$ are found  minimizing
with respect to $a_1,\dots,a_N$ the quadratic form
\beq
\sum_{i=1}^N(Y_i-\mu_i(a_i))^T\;\Sigma_i^{-1}\;(Y_i-\mu_i(a_i)).
\eeq
\par Suppose that the function $g(t,a)$ is differentiable with respect to $a$ and set
 $m=\sum_{i=1}^{N}j_i$.
The estimator $\hat \sigma^2$ of the parameter $\sigma^2$ is found  maximizing
with respect to $\sigma^2$ the conditional likelihood
function
\beq
L(\sigma^2\mid \hat A_1,\dots,\hat A_N)=
$$
$$
\frac1{(2\pi)^{m/2}\sigma^m}\prodl_{i=1}^N\mid\Sigma_i\mid^{-1/2}
\exp\left\{-\frac1{2\sigma^2}\sum_{i=1}^N(Y_i-\mu_i(\hat A_i))^T\;\Sigma_i^{-1}\;(Y_i-\mu_i(\hat A_i))\right\}
.
\eeq
The minimization of the quadratic form (42) and the
maximization of the conditional likelihood function gives the following equations
for $\hat \sigma^2$ and $\hat A_i$ computing:
\beq
\suml_{k=1}^{j_i}\frac{\partial}{\partial \hat A_{i}}\ln g(t_{ik},\hat A_i)
\suml_{l=1}^{j_i}\{Y_{il}-\ln g(t_{il},\hat A_i)\}b_{ikl}=0;
\eeq
\beq
\hat \sigma^2=\frac{\hat c}
{m}
\eeq
where
\beq
\hat c=\sum_{i=1}^N(Y_i-\mu_i(\hat A_i))^T\;\Sigma_i^{-1}\;(Y_i-\mu_i(\hat A_i))
.
\eeq
Let us consider the following model for the paths $g(t,A_i)$:
\beq
g(t, A_i)=e^{A_{i1}}(1+t)^{A_{i2}}.
\eeq
Set ${\bf 1}=(1,\dots,1)^T_{j_i}$, $C_i=(c_{i1},\dots,c_{i,j_i})^T$. In such a  case  we have
\beq
\hat A_{i1}=\frac{c_id_i-e_if_i}
{c_i^2-b_ie_i },
\quad
\hat A_{i2}=\frac{c_if_i-b_id_i }
{c_i^2-b_ie_i },
\eeq
and
\beq
 \hat \sigma^2=\frac{\hat c}m,
\eeq
where
$$
\hat c=\suml_{i=1}^N(g_i+b_i\hat A_{1i}^2+e_i\hat A_{2i}^2+2c_i\hat A_{1i}\hat A_{2i}-2f_i\hat A_{1i}
-2d_i\hat A_{2i})
$$
$$
b_i={\bf 1}^T\Sigma^{-1}_i{\bf 1},
\quad
c_i=C_i^T\Sigma^{-1}_i{\bf 1},
\quad
d_i=Y_i^T\Sigma^{-1}_iC_i,\quad
$$
\beq
e_i=C_i^T\Sigma^{-1}_iC_i
,\quad
 f_i=Y_i^T\Sigma^{-1}_i{\bf 1},
\quad
g_i=Y_i^T\Sigma^{-1}_iY_i.
\eeq
The conditional mean  of the estimator $\hat \sigma^2$ given $N$ and $j_i$ is:
$$
E(\hat \sigma^2\mid j_i, i=1,\dots, N)=\frac {m+2}m\sigma^2,
$$
so $\hat \sigma^2$ is a consistent estimator of $\sigma^2$.

Set
$$
V_{ij}=e_{ij}\quad \mbox{(first model)},\quad V_{ij}=\sigma W_i(c_{ij})
\quad \mbox{(second model)},
$$
$$
V_{i}=(V_{i1},\dots,V_{i,j_i})^T.
$$
Replacing $Y_{ij}$  by their expressions
$Y_{ij}=A_{i1}+A_{i2}c_{ij}+V_{ij}$ in the formulas (48), we
obtain that
$$
\hat A_i=A_{i}+\varepsilon_i,
$$
where
$$
\varepsilon_i=(\varepsilon_{i1},\varepsilon_{i1}),\quad
\varepsilon_{i1}=\frac{c_iv_i-e_iu_i}{c_i^2-b_ie_i },\quad
\varepsilon_{i2}=\frac{c_iu_i-b_iv_i}{c_i^2-b_ie_i },
$$
$$
u_i=V_i^T\Sigma^{-1}_i{\bf 1},
\quad
v_i=V_i^T\Sigma^{-1}_iC_i.$$
Given $j_i$ the vector $V_{i}\sim N(0,\sigma^2\Sigma_i)$,
so the bi-dimensional vector $\varepsilon_i$ is normally
distributed with zero mean and the density
$$\varphi_i(x;\sigma^2)=\frac1{2\pi \sigma_{i1}\sigma_{i2}\sqrt{1-\rho_i^2}}\exp\left\{-\frac1{2(1-\rho_i^2)}\left[
\frac{x_1^2}{\sigma_{i1}^2}-2\rho_i \frac{x_1x_2}{\sigma_{i1}\sigma_{i2}}+
\frac{x_2^2}{\sigma_{i2}^2}\right]\right\},$$
which depends only on $\sigma^2$, because
$$
\sigma^2_{i1}=Var(\varepsilon_{i1})=\frac{e_i}{b_ie_i-c_i^2}\sigma^2,
\quad
\sigma^2_{i2}=Var(\varepsilon_{i2})=\frac{b_i}{b_ie_i-c_i^2}\sigma^2,
$$
$$
cov(\varepsilon_{i1},\varepsilon_{i2})=-\frac{c_i}{b_ie_i-c_i^2}\sigma^2,
$$
and the correlation coefficient
$$
\rho_i=-\frac{c_i}{\sqrt{b_ie_i}}.
$$
Note that the means of the random variables $A_i$ and $\hat A_i$ are equal:
$$E(\hat A_i)=E(A_i),
$$
and
$$
Eg(t,\hat A_i)=m_{r0}(t)\exp\{\frac12 [\sigma_{i1}^2+2\sigma_{i1}\sigma_{i2}\rho_i\ln(1+t)+\sigma_{i2}^2\ln^2 (1+t)]\}.
$$
So a consistent estimator of the mean real degradation $m_{r0}(t)$  is
$$
\hat m_{r0}(t)=
\frac { \suml_{i=1}^N g(t,\hat A_i)}{\suml_{i=1}^N
 \exp\{\frac12 [\hat \sigma_{i1}^2+2\hat\sigma_{i1}\hat\sigma_{i2}\hat\rho_i\ln(1+t)+\hat\sigma_{i2}^2\ln^2 (1+t)]\}},
$$
and a consistent estimator of the mean observed degradation $m_{o0}$  (for the model 2) is
$$
\hat m_{o0}(t)=e^{\frac12 \hat \sigma^2 c^2(t)}\hat m_{r0}(t),
$$
where
$$
\hat\sigma^2_{i1}=\frac{e_i}{b_ie_i-c_i^2}\hat\sigma^2,
\quad
\hat\sigma^2_{i2}=\frac{b_i}{b_ie_i-c_i^2}\sigma^2,
\quad
 \hat \rho_i=-\frac{c_i}{\sqrt{b_ie_i}}.
$$
{\bf Lemma 1}. {\it If  $t_{ij}=q_i(j/m_i)$, $j=1,\dots,m_i$, where $q_i$ is a continuous increasing on $[0,1]$ function,
$q_i(0)=0, q_i(1)\leq c<\infty$.
 Under the first model almost surely}
$$
\lim_{m_i\to \infty}\sigma^2_{i1}=\lim_{m_i\to \infty}\sigma^2_{i2}= 0.
$$
Proof. It is sufficient to prove that almost surely
$$
b_i-\frac{c_i^2}{e_i}=\suml_{j=1}^{j_i}c_{ij}^2-\frac1n(\suml_{j=1}^{j_i}c_{ij})^2\stackrel{a.s.}\rightarrow\infty,\quad
e_i-\frac{c_i^2}{b_i}=j_i-\frac{(\suml_{j=1}^{j_i}c_{ij})^2}{\suml_{j=1}^{j_i}c_{ij}^2}\stackrel{a.s.}\rightarrow \infty.
$$
The function $r_i(x)=c(q_i(x))$ is continuous and increasing on $(0,1)$, $r_i(0)=0$. We have
$$
\frac1{j_i} (b_i-\frac{c_i^2}{e_i})=\frac1{j_i}
\suml_{j=1}^{j_i}r^2_i(j/{j_i})-(\frac1{j_i^2} \suml_{j=1}^{j_i}r_i(j/j_i))^2\stackrel{a.s.}\rightarrow
$$
$$
\intl_0^{a_i}r^2_i(x)dx-(\intl_0^{a_i}r_i(x)dx)^2>0,
$$
where
$$
a_i=q_i^{-1}(E(\tau_i\wedge q_i(1)).
$$
So  $b_i-{c_i^2}/{e_i}\stackrel{a.s.}\rightarrow \infty$.

Analogously
$$
\frac1{j_i} (e_i-\frac{c_i^2}{b_i})\stackrel{a.s.}\rightarrow 1-\frac{(\intl_0^{a_i}r_i(x)dx)^2}{\intl_0^{a_i}r^2_i(x)dx}\in (0,1),
$$
so $e_i-{c_i^2}/{b_i}\stackrel{a.s.}\rightarrow \infty$.

The proof is complete.

\vskip 0.2cm
 \noindent{\bf Remark 1.} {\it The lemma is true in the particular case $q_i(x)=x$, i.e. when $t_{ij}=j/m_i$ are equidistant}.
\vskip 0.2cm
Set
\beq
  \tilde F_A(a)
  =\frac{1}{n}\sum_{i=1}^n\Un_{\{A_i\le a\}}
  =\frac{1}{n}\sum_{i=1}^n
    \Un_{\{A_{i1}\le a_1,\dots,A_{ir}\le a_r\}}.
\eeq
and let us consider the statistic
\beq
 \hat F_A (a)=\frac 1n \suml_{i=1}^n \Un_{\{\hat
A_{i}\leq a\}}
\eeq
as the estimator of $F_A$. If the conditions of the Lemma 1 are verified then
 under the model 1  the difference $\hat F_A-\tilde F_A$ converges in probability to
 zero uniformly on ${\cal A}$ as $\min m_i\to \infty$.

Simulation results show that if the noise is not very large (i.e. the variance $\sigma^2$ is  not large)
 then the estimators $\hat F_A$ and $\tilde F_A$ are close and for the model 2.

The estimator $\tilde F_A$
 is the best estimator of the c.d.f. $F_A$. If $m_i$ are not very small (1 model) or $\sigma^2$ is not large (models 1 and 2),
then $\hat F_A$ is a good estimator of $F_A$ and the c.d.f.  of the real degradation at the moment $t$ may be estimated by
$$
\hat F_{r0}(z\mid t)=
\frac 1n \suml_{i=1}^n\Un_{\{g(t,\hat A_i)\leq z\}},
$$
Let us consider estimation of $\Lambda$. Suppose at first that $\sigma=\alpha=0$.

Denote by $\tau_i=T_i\wedge T_i^0$ the failure time of the $i$th individual, $T_i$, $T_i^0$
being  the traumatic and non-traumatic failure moments. Set
 $$
X_i=\min (\tau_i,t_{i,m_i}),\quad  \delta_{i}=\left\{ \begin{array}{c}
{1},\quad \mbox{if $X_i=T_i$},\\
{0},\quad  \mbox{otherwise}.
\end{array}\right.
$$
If $\sigma=0$ and $j_i\geq r$ then
 the values of $A_i$ can be found and   the degradation value $g(T_i,A_i)$ just prior to the
 failure also can be found.  Set
$$
 Z_{i}=
g(X_i,A_i),
$$
and on $[z_*,z_0]$ define the following counting processes:
 $$
N_i(z)=\Un_{\{Z_i\leq z, \delta_i=1\}}, \quad N(z)=\suml_{i=1}^n N_i(z).
$$
$N(z)$ is the number of units having a failure before or at the moment when the degradation attains the
level $z$.

Let ${\cal F}_z$ be the $\sigma$-algebra generated by the random
variables
$$
A_1, \dots,A_n \quad  \mbox{and}  \quad N_1(y),\dots ,N_n(y) \quad \mbox{with} \quad y\leq
z.
$$
 Then (see Bagdonavi\v{c}ius,V.,  Bikelis, A., Kazakevi\v{c}ius, A. and  Nikulin, M.  (2002, 2003))) the process $N_i(z)$ can be written as the sum
\beq
  N_i(z)=\int_{z_*}^z\lambda(y)Y_i(y)dy+M_i(z),
\eeq
where
\beq
  Y_i(z)=\Un_{\{Z_i\geq z\}}h'(z,A_i),
\eeq
and $M_i$ is a martingale with respect to the filtration
$({\cal F}_z\mid z\geq 0)$.

This implies that the optimal estimator of the cumulative
intensity $\Lambda(z)$ is of Nelson-Aalen type (see Andersen, Borgan, Gill and Keiding
(1993), pp.177-178):
\beq
  \tilde\Lambda(z)=\int_{z_*}^zY^{-1}(y)\,dN(y)
  =\sum_{Z_i\leq z,\delta_i=1}
    Y^{-1}(Z_i)=\sum_{Z_i\leq z,\delta_i=1}\left(\sum_{Z_j\geq Z_i}h'(z,A_i)\right)^{-1},
\eeq
where $Y(z)=\suml_{j=1}^n Y_j(z)$.

The estimator is correctly defined if $Z_i\leq z$ and $\delta_i=1$ for some $i$. If
such $i$ do not exist then the estimator is defined as $0$.


To get asymptotic properties of $\hat\Lambda$ assumptions about the choice of $t_{i,m_i}$
 are needed. We assume that $t_{i,m_i}$ are the realizations of  (censoring) random
 variables $C_i$ which are mutually independent, identically distributed and conditionally
 (given $A_i=a$) independent of $(\tau_i,\delta_i)$. Let $S_c(t\mid a)$
 denote the conditional survival function of $C_i$:
 $$
S_c(t\mid a)=P(C_i>t\mid A_i=a).
$$
Two
additional assumptions are needed. Set
$$
  c_0(a)=\sup_{z\le z_0}h'(z,a),\quad
  c_1(a)=\sup_{z_1<z_2\le z_0}
    \frac{\mid h'(z_2,a)-h'(z_1,a)\mid}{z_2-z_1}
$$
and
$$
  b(z)=E[h'(z,A_i)\Un_{\{Z_i\ge z\}}]=\int h'(z,a)S_c(h(z,a)\mid a)\,S_T(h(z,a)\mid a)\,dF_A(a).
$$
\newtheorem{monass}{Assumption}
\begin{monass}
$E [c_0(A)+c_1(A)]<\infty$.
\end{monass}


\begin{monass}
$\inf_{z\le z_0}b(z)>0$.
\end{monass}

Similarly as in Bagdonavicius et al (2002,2003) we obtain:

\newtheorem{montheo}{Theoreme}
\begin{montheo}
Under Assumptions 1-2, the estimator $\hat\Lambda$ is uniformly
consistent, i.e.
$$
  \sup_{z\le z_0}\mid\hat\Lambda(z)-\Lambda(z)\mid \stackrel{P}\rightarrow 0,
$$
as $n\to\infty$.
\end{montheo}

\begin{montheo}
 If
Assumptions 1-2 hold, then  the random vector function $\sqrt
n(\hat\Lambda-\Lambda)$ tends in distribution in the space
$D[z_*;z_0]$ to a
zero mean Gaussian process $W$ with the covariance function
$$
  E[W(z)W(z')]=\sigma^2(z\wedge z'),
$$
where
$$
  \sigma^2(z)=\int_{z_*}^z\frac{\lambda(y)}{b(y)}dy.
$$
\end{montheo}

The c.d.f $F_A$ is estimated by $\tilde F_A$ given by (51). It is well-known that
almost surely
$$
  \sup_a \mid \tilde F_A(a)- F_A(a)\mid \to 0,
$$
as $n\to \infty$. Moreover, if the function $F_A$ is absolutely
continuous then the random function $\sqrt{n}(\hat F_A-
F_A)$ tends in distribution in the Skorokhod space $D^r[0;\infty]$
to a zero mean Gaussian field $W_0$ with the covariance function
$$
  E[W_0(a)W_0(a')]=F_A(a\wedge a')-F_A(a)\,F_A(a');
$$
where $a\wedge a'=(a_1\wedge a_1',\dots,a_r\wedge a_r')$.
\vskip 0.3cm
\begin{montheo}
If Assumptions 1-2 hold and the function $F_A$ is absolutely
continuous then the random vector function
$\sqrt{n}(\hat F_A-F_A,\,\hat\Lambda-\Lambda)$ tends in
distribution in the space $D^r[0;\infty]\times D[z_*;z_0]$ to a
random vector function with independent components.
\end{montheo}

Estimating the survival and degradation characteristics  we replace the functions $F_A$ and $\Lambda$ in
the expressions of these characteristics by $\tilde F_A$
and $\hat\Lambda$. Asymptotic properties of such estimators can be easily found because
they are smooth functionals of the pair $(\tilde F_A,\hat\Lambda)$.


If $\alpha\neq 0$ then two step estimation of $\Lambda$ is possible. If $\alpha$ would be known then
similarly as in the case $\alpha\neq 0$ the following estimator of $\Lambda$  could be
used:
$$
  \Lambda^*(z,\alpha)=\int_{z_*}^zY^{-1}(y,\alpha)\,dN(y)
  =\sum_{Z_i\leq z,\delta_i=1}
    Y^{-1}(Z_i,\alpha),
$$
where $Y(z)=\suml_{j=1}^n Y_j(z)$, and
$$
Y_j(z)=\Un_{\{Z_j\geq z\}}\lambda_0(h(z,A_j),\alpha) h'(z,A_j).
$$

If $\alpha$ is unknown, the estimator $\hat\alpha$ is obtained by maximising the loglikelihood
function
$$
\ln L(\alpha)\asymp \suml_{i=1}^n \delta_i\ln \lambda_0(X_i,\alpha)+\suml_{i=1}^n\delta_i\ln \lambda^* (Z_i,\alpha)
-\hskip 5cm
$$
$$
\hskip 5cm \suml_{i=1}^n \intl_{g(0,A_i)}^{g(X_i,A_i)}
\lambda_0(h(z,A_i)) h'(z,A_i)\frac{d N(z)}{Y(z,\alpha)},
$$
where
$$
\lambda^* (z,\alpha)=\suml_{j=1}^n \frac{\delta_j\Un_{\{z=Z_j\}}}{Y(z,\alpha)}.
$$
The final estimator of $\Lambda$ is
$$
\tilde \Lambda(z)= \Lambda^*(z,\hat\alpha).
$$
\noindent\par If $\sigma\neq 0$ then the values of
stochastic processes $Z_i(t)$ instead of $g(t,A_i)$ are observed.

 Using the values of
 $Z_i(t)$  the predictors $\hat A_i$ of
the random variables $A_i$ are found using the equations (48). Replacing $A_i$
by their predictors $\hat A_i$  we obtain the following
estimator of the cumulative intensity $\Lambda$:
\beq
  \hat\Lambda(z)=\int_{z_*}^z\frac{d\hat N(y)}{\hat {Y}(y)}
  =\sum_{\hat Z_i\leq z,\delta_i=1}
    \frac{1}{\hat {Y}(\hat Z_i)},
\eeq
    where
     $$
\hat Z_i=g(X_i,\hat A_i),\quad \hat N(z)=\suml_{i=1}^n
 \hat N_i(z),\quad \hat N_i(z)=\Un_{\{\hat Z_i\leq z, \delta_i=1\}},
$$
$$
\hat Y(z)=\tilde Y(z,\hat\alpha), \quad \tilde Y(z,\alpha)=\suml_{i=1}^n \tilde Y_i(z,\alpha),
$$
$$
 \tilde Y_i(z,\alpha)
=\Un_{\{\hat Z_i\geq z\}}
\lambda_0(h(z,\hat A_i),\alpha){h'(z,\hat A_i)},
$$
and $\hat\alpha$ is obtained by maximizing the loglikelihood
function
$$
\ln L(\alpha)\asymp \suml_{i=1}^n \delta_i\ln \lambda_0(X_i,\alpha)+\suml_{i=1}^n\delta_i\ln \tilde\lambda (\hat Z_i,\alpha)
-
$$
$$
\suml_{i=1}^n \intl_{g(0,\hat A_i)}^{g(X_i,\hat A_i)}
\lambda_0(h(z,\hat A_i),\alpha) h'(z,\hat A_i)\frac{d \hat N(z)}{\tilde Y(z,\alpha)}=
$$
\beq
\suml_{i=1}^n \delta_i\ln \lambda_0(X_i,\alpha)+\suml_{i=1}^n\delta_i\ln \tilde\lambda (\hat Z_i,\alpha)
-\suml_{i=1}^n \suml_{\hat Z_j\leq g(X_i,\hat A_i), \delta_j=1}
\frac{\lambda_0(h(\hat Z_j,\hat A_i),\alpha) h'(\hat Z_j,\hat A_i)}{\tilde Y(\hat Z_j,\alpha)},
\eeq
where
$$
\tilde\lambda (z,\alpha)=\suml_{j=1}^n \frac{\delta_j\Un_{\{z=\hat Z_j\}}}{\tilde Y(z,\alpha)}.
$$
Note that
\beq
h(z,A_j)=e^{-A_{1j}/A_{2j}}z^{1/A_{2j}}-1,\quad
h'(z,A_j)=\frac1{A_{2j}}e^{-A_{1j}/A_{2j}}z^{1/A_{2j}-1}.
\eeq

The formula (13) implies  the following estimator of the  probability
 $
S_\tau(t)$ to survive time $t$:
$$
\hat S_\tau(t)=\intl_{g(t,a)\leq
z_0}\hat S_T(t\mid a)d\hat F_A(a)=\frac1n \suml_{i:g(t,\hat A_i)\leq z_0}\hat S_T(t\mid \hat A_i),
$$
where
$$
\hat S_T(t\mid \hat A_i)=\exp\Bigl\{-\int_{g(0,\hat A_i)}^{g(t,\hat A_i)}h'(z,\hat A_i)\,d\hat\Lambda(z)\Bigr\}=
$$
$$
\exp\{-\suml_{j=1}^n\Un_{\{\delta_j=1,g(0,\hat A_i)\leq\hat Z_j<g(t,\hat A_i)\}}
\frac{h'(\hat Z_j,\hat A_i)}{\hat Y(\hat Z_j)}\}.
$$
The probabilities $P^{(tr)}(t)$ and $P^{(0)}(t)$ are estimated by
\beq
\hat P^{(tr)}(t)=
1-\frac 1n \suml_{i=1}^n\hat S_T(t\wedge h(z_0,\hat A_i)\mid \hat A_i))
\eeq
and
\beq
\hat P^{(0)}(t)=\frac 1n \suml_{g(t,\hat A_i)\geq z_0}
\hat S_T(h(z_0,\hat A_i)\mid \hat A_i),
\eeq
The c.d.f. of the real degradation at the moment $t$ given the survival until this moment is estimated by
$$
\hat F_{r}(z\mid t)=\frac1{\hat S_\tau(t)}\intl_{g(t,a)\leq z}\hat S_T(t\mid a)d\hat F_A (a)
=\frac1n \suml_{i:g(t,\hat A_i)\leq z}\hat S_T(t\mid \hat A_i)/\hat S_\tau(t),
$$
and the mean real degradation attained at the moment $t$ given the survival until this moment is estimated by
$$
\hat m_{r}(t)=\frac1{\hat S_\tau(t)}\intl_{\cal A} (g(t,a)\wedge z_0-z_*) \hat S_T(t\mid a)d\hat F_A (a)
=\frac1n \suml_{i=1}^n(g(t,\hat A_i)\wedge z_0-z_*)\hat S_T(t\mid \hat A_i)/\hat S_T(t).
$$
The probability that at the moment
$t$ the value of the real degradation will be in the set $B$ given that at the moment $s$ it is
in the set $B_0$, is estimated by
$$
\hat P(g(t,A)\in B\mid g(s,A)\in B_0, \tau>t)
$$
$$
=
{\intl_{g(t,a)\in B,g(s,a)\in B_0}\hat S_T(t\mid a)d
\hat F_A (a)}/{\intl_{g(s,a)\in B_0}\hat S_T(t\mid a)d\hat F_A (a)}=
$$
$$
\frac{\suml_{i:g(t,\hat A_i)\in B,g(s,\hat A_i)\in B_0}
\hat S_T(t\mid \hat A_i)
 }{\suml_{i:g(s,\hat A_i)\in B_0} \hat S_T(t\mid \hat A_i)}.
$$
The conditional survival function of
 $T(z_0)$ and the conditional mean given the survival until  the degradation attains the level $z_0$ are estimated by
are estimated by
$$
\hat P(T^0(z)>t\mid \tau>T^0(z))=
\frac{\intl_{g(t,a)<z}\hat S_T(h(z,a)\mid a)
d\hat F_A (a)}{\intl_{{\cal A}}\hat S_T(h(z,a)\mid a)
d\hat F_A (a)}=
\frac{\suml_{i:g(t,\hat A_i)<z}
\hat S_T(h(z,\hat A_i)\mid \hat A_i)
 }{\suml_{i=1}^n\hat S_T(h(z,\hat A_i)\mid \hat A_i)},
$$
and
$$
\hat E(T^0(z)\mid \tau>T^0(z))=
\frac{\intl_{{\cal A}}h(z,a)\hat S_T(h(z,a)\mid a)
d\hat F_A (a)}{\intl_{{\cal A}}\hat S_T(h(z,a)\mid a)
d\hat F_A (a)}=
\frac{\suml_{i=1}^nh(z,\hat A_i)
\hat S_T(h(z,\hat A_i)\mid \hat A_i)
 }{\suml_{i=1}^n\hat S_T(h(z,\hat A_i)\mid \hat A_i)},
$$
respectively.


\vskip 0.5cm

\centerline{\bf 6. Estimation of degradation and survival characteristics:}
\centerline{\bf models with covariates}
\vskip 0.5cm
Assume that the c.d.f. $F_A$ and the intensities $\lambda$ are completely unknown.

Suppose that $n$ units are tested , and the $i$th of them is
observed under the covariate $x^{(i)}$. Fix the moments of degradation measurements
$t_{i1},\dots,t_{i,m_i}$ of this unit. If the failure time
 $\tau_i$ of the $i$th unit occurs in the interval $[t_{i,j_i},t_{i,j_i+1})$ $(j_i=1,\dots,m_i;t_{m_{i}+1}=\infty)$
 then
 the values $Z_{i1},\dots,Z_{i,j_i}$
of the degradation process $Z_i$ of the $i$th unit are observed at the time moments
$t_{i1},\dots,t_{i,j_i}$.

 Set
$$
Y_{ij}=\ln Z_{ij},
\quad
Y_i=(Y_{i1},\dots,Y_{i,j_i})^T.
$$
Then given $A_i=a_i$
$$
Y_i\sim N(\mu_i,\sigma^2\Sigma_i),
$$
where
$$
\mu_i=(\mu_{i1},\dots,\mu_{i,j_i})^T,\quad
\mu_{ij}=\mu_{ij}(a_i,\beta)=\ln g(\varphi(t_{ij},\beta,x^{(i)}),a_i),
$$
$
 \Sigma_i$
 is given in (41). Denote by $N$ the number of units such that $j_i\geq r$.

The estimators $\hat \sigma^2$, $\hat\beta$ of the parameters $\sigma^2$, $\beta$ and the
predictors $\hat A_i$ of the random vectors $A_i$ are found by maximizing
with respect to $\sigma^2, \beta,a_1,\dots,a_n$ the conditional likelihood
function
\beq
L(\sigma^2,\beta, a_1,\dots,a_n)=
$$
$$
\frac1{(2\pi)^{m/2}\sigma^m}\prodl_{i=1}^N\mid\Sigma_i\mid^{-1/2}
\exp\left\{-\frac1{2\sigma^2}\sum_{i=1}^N(Y_i-\mu_i(a_i,\beta))^T\;\Sigma_i^{-1}\;(Y_i-\mu_i(a_i,\beta))\right\}.
\eeq
Suppose that the function $g(\varphi(t,\beta,x),a)$ is differentiable with respect to $\beta$ and $a$.
The maximization of the likelihood function gives the following equations for $\hat \sigma^2$, $\hat\beta$, and $\hat A_i$ computing:
$$
\suml_{k=1}^{j_i}\frac{\partial}{\partial A_{i}}\ln g(\varphi(t_{ik},\hat\beta,x^{(i)}),\hat A_i)
\suml_{l=1}^{j_i}\{Y_{il}-\ln g(\varphi(t_{ij},\hat\beta,x^{(i)}),\hat A_i)\}b_{ikl}=0,
$$
$$
\suml_{i=1}^{N}\suml_{k=1}^{j_i}\frac{\partial}{\partial \beta}\ln g(\varphi(t_{ik},\hat\beta,x^{(i)}),\hat A_i)
\suml_{l=1}^{j_i}\{Y_{il}-\ln g(\varphi(t_{ij},\hat\beta,x^{(i)}),\hat A_i)\}b_{ikl},
$$
\beq
\hat \sigma^2=\frac{c(\hat\beta)}
{m}
\eeq
where
$$
c(\hat\beta)=\sum_{i=1}^N(Y_i-\mu_i(\hat A_i,\hat\beta))\;\Sigma_i^{-1}\;(Y_i-\mu_i(\hat A_i,\hat\beta))^{T}
.
$$
Set
$$c_{ij}(\beta)=c(\varphi(t_{ij},\beta,x^{(i)})).$$
For the  model (47) we have
\beq
\hat A_{i1}=\frac{c_i(\hat\beta)d_i(\hat\beta)-e_i(\hat\beta)f_i}
{c_i^2(\hat\beta)-b_ie_i(\hat\beta) },
\quad
\hat A_{i2}=\frac{c_i(\hat\beta)f_i-b_id_i(\hat\beta) }
{c_i^2(\hat\beta)-b_ie_i(\hat\beta) },
\eeq
\beq
 \hat \sigma^2=\frac{c(\hat\beta)}m,
\eeq
and $\hat \beta$ verifies the system of equations
$$
\suml_{i=1}^{N}\hat A_{2i}\suml_{k=1}^{j_i}\psi_{iks}(\hat\beta)
\suml_{l=1}^{j_i}\{Y_{il}-\hat A_{1i}-\hat A_{2i}c_{il}(\hat\beta)\}b_{ikl}=0,
$$
where
$$
c(\hat\beta)=\suml_{i=1}^n(g_i+b_i\hat A_{1i}^2+e_i(\hat\beta)\hat A_{2i}^2+2c_i(\hat\beta)\hat A_{1i}\hat A_{2i}
-2f_i\hat A_{1i}
-2d_i(\hat\beta)\hat A_{2i})
$$
$$
c_i(\hat\beta)=C_i^T(\hat\beta)\Sigma^{-1}_i{\bf 1},
\quad
d_i(\hat\beta)=Y_i^T\Sigma^{-1}_iC_i(\hat\beta),\quad
$$
\beq
e_i(\hat\beta)=C_i^T(\hat\beta)\Sigma^{-1}_iC_i(\hat\beta),\quad
\psi_{iks}(\hat\beta)=
\frac{\int_0^{t_{ij}}x^{(i)}_s(u)e^{ x^{(i)}(u)\hat\beta^T}du}{1+\int_0^{t_{ij}}e^{x^{(i)}(u)\hat\beta^T}du};
$$
$$
{\bf 1}=(1,\dots,1)^T_{j_i},\quad  C_i(\hat\beta)=(c_{i1}(\hat\beta),\dots,c_{i,j_i}(\hat\beta))^T,
\eeq
and $b_i,f_i,g_i$ are given by (50).

If the vector of covariates $x^{(i)}$ is a step-function , and has the form
  $$
x^{(i)}(t)=x^{(ik)},\quad \mbox{if}\quad t\in [t^*_{i,k-1},t^*_{ik})\quad (k=1,\dots l_i,\; t^*_{i0}=0),
$$
then
$$
\psi_{iks}(\hat\beta)=
\frac{\suml_{l=1}^{\nu_{ik}-1}x^{(il)}_se^{ x^{(ik)}\hat\beta^T}(t^*_{il}-t^*_{i,l-1})
+x^{(i,\nu_{ik})}_se^{ x^{(i\nu_{ik})}\hat\beta^T}(t^*_{i,\nu_{ik}}-t^*_{i,\nu_{ik}-1})}{1+
\suml_{k=1}^{\nu_{ij}-1}e^{ x^{(ik)}\hat\beta^T}(t^*_{ik}-t^*_{i,k-1})+
e^{ x^{(i\nu_{ik})}\hat\beta^T}(t^*_{i,\nu_{ik}}-t^*_{i,\nu_{ik}-1})},
$$
where $\nu_{ik}=\{l:t_{ik}\in [t^*_{i,l-1},t^*_{il})\}$.

In particular, if $x^{(i)}$ has only one jump ($l_i=2$) then
$$
\psi_{iks}(\hat\beta)
=\frac{x^{(i1)}_se^{x^{(i1)}\beta^T}\min(\tau_i,t^*_{ik})+x^{(i2)}_s
e^{x^{(i2)}\beta^T}\max(t^*_{ik}-\tau_i,0)}{1+e^{x^{(i1)}\beta^T}
\min(\tau_i,t^*_{ik})+e^{x^{(i2)}\beta^T}\max(t^*_{ik}-\tau_i,0)}.
$$
The estimator of the c.d.f. $F_A(a)$ has the form  (52).

The estimators of the
c.d.f. and the mean of the real degradation at the moment $t$  given the covarate $x=x(s),0\leq s \leq t$ is
$$
\hat F_{r0}(z\mid t,x)=
\frac 1n \suml_{i=1}^n\Un_{\{g(\varphi(t,\hat\beta,x),\hat A_i)\leq z\}}
$$
the mean real degradation attained at the moment $t$ is estimated by
$$
\hat m_{r0}(t,x)=\frac 1n \suml_{i=1}^n g(\varphi(t,\hat\beta,x),\hat A_i).
$$
Similarly as in the case without covariates we define the estimators
$$
\hat\Lambda(z)=\tilde\Lambda(z,\hat\alpha),
$$
where
$$
  \tilde\Lambda(z,\alpha)=\int_0^z\hat {Y}^{-1}(y,\alpha)\,d\hat N(y)
  =\sum_{\hat Z_i\leq z,\delta_i=1}
    \tilde {Y}^{-1}(\hat Z_i,\alpha),
    $$
        $$
\hat Z_i=g(\varphi(X_{i},\hat\beta,x^{(i)}),\hat A_i),\quad \hat N(z)=\suml_{i=1}^N \hat N_i(z),
\quad \hat N_i(z)=\Un_{\{\hat Z_i\leq z, \delta_i=1\}},
$$
$$
\tilde Y(z,\alpha)=\suml_{i=1}^N \tilde Y_i(z,\alpha),\quad \tilde Y_i(z,\alpha)
=\Un_{\{\hat Z_i\geq z\}}\lambda_0(h(z,\hat A_i),\alpha)\psi'(h(z,\hat A_i),\hat\beta,x^{(i)})h'(z,\hat A_i),
$$
and $\hat\alpha$ is obtained by maximizing the loglikelihood
function
$$
\ln L(\alpha)\asymp \suml_{i=1}^N \delta_i\ln \lambda_0(\varphi(X_i,\hat\beta,x^{(i)}),\alpha)
+\suml_{i=1}^n\delta_i\ln \tilde\lambda (\hat Z_i,\alpha)
-
$$
$$
\suml_{i=1}^N \intl_{g(\varphi(0,\hat\beta,x^{(i)},\hat A_i)}^{g(\varphi(X_i,\hat\beta,x^{(i)},\hat A_i)}
\lambda_0(h(z,\hat A_i),\alpha) \psi'(h(z,\hat A_i),\hat\beta,x^{(i)})h'(z,\hat A_i)\frac{d \hat N(z)}{\tilde Y(z,\alpha)}=
$$
\beq
\suml_{i=1}^N \delta_i\ln \lambda_0(\varphi(X_i,\hat\beta,x^{(i)}),\alpha)
+\suml_{i=1}^n\delta_i\ln \tilde\lambda (\hat Z_i,\alpha)
$$
$$
-\suml_{i=1}^N \suml_{g(\varphi(0,\hat\beta,x^{(i)},\hat A_i)\geq\hat Z_j\leq g(\varphi(X_i,\hat\beta,x^{(i)},\hat A_i), \delta_j=1}
\frac{\lambda_0(h(\hat Z_j,\hat A_i),\alpha)
\psi'(h(\hat Z_j,\hat A_i),\hat\beta,x^{(i)})
h'(\hat Z_j,\hat A_i)}{\tilde Y(\hat Z_j,\alpha)},
\eeq
where
$$
\tilde\lambda (z,\alpha)=\suml_{j=1}^N \frac{\delta_j\,\Un_{\{z=\hat Z_j\}}}{\tilde Y(z,\alpha)}.
$$
The estimators of the survival function
\beq
\hat S_\tau(t\mid x)=\frac1n \suml_{ig(\varphi(t,\hat\beta,x)\hat A_i)<z_0}\hat S_T(t\mid \hat A_i,x),
\eeq
where
$$
\hat S_T(t\mid \hat A_i,x)=
$$
$$
\exp\{-\suml_{\delta_j=1,g(\varphi(0,\hat\beta,x),\hat A_i)\leq\hat Z_j<g(\varphi(t,\hat\beta,x),\hat A_i)}
\lambda_0(h(\hat Z_j,\hat A_i),\hat\alpha)\,\psi'(h(\hat Z_j,\hat A_i),\hat\beta,x)
\,\frac{h'(\hat Z_j,\hat A_i)}{\hat Y(\hat Z_j)}\}.
$$
The c.d.f. and the mean of the real degradation at the moment $t$ given the survival until this moment are estimated by
\beq
\hat F_{r}(z\mid t,x)
=\frac1N  \suml_{i:g(\varphi(t,\hat\beta,x),\hat A_i)\leq z}\hat S_T(t\mid \hat A_i,x)/\hat S_T(t\mid x),
\eeq
\beq
\hat m_{r}(t,x)=\frac1N \suml_{i=1}^ng(\varphi(t,\hat\beta,x),\hat A_i)\hat S_T(t\mid \hat A_i,x)/\hat S_T(t\mid x).
\eeq
 The
probabilities $P^{(tr)}(t\mid x)$ and $P^{(0)}(t\mid x)$ are estimated by
\beq
\hat P^{(tr)}(t\mid x)=
1-\frac 1N\suml_{i=1}^N\hat S_T(t\wedge \psi (h(z_0,\hat A_i),\hat\beta,x))\mid \hat A_i,x),
\eeq
\beq
\hat P^{(0)}(t\mid x)=\frac 1N\suml_{g(\varphi (t,\hat\beta,x)),\hat A_i)\geq z_0}
S_T(\psi (h(z_0,\hat A_i),\hat\beta,x))\mid \hat A_i,x).
\eeq
The probability that at the moment
$t$ the value of the real degradation will be in the set $B$ given that at the moment $s$ it is
in the set $B_0$ and given $x(u),0\leq u\leq t$, is estimated by
$$
\hat P(g(\varphi(t,\beta,x),a)\in B\mid g(\varphi(s,\beta,x),a)\in B_0, \tau>t, x(u),0\leq u\leq t)
$$
\beq
=
\frac{\suml_{i:g(\varphi(t,\hat\beta,x),\hat A_i)\in B,g(\varphi(s,\hat\beta,x),\hat A_i)\in B_0}
\hat S_T(t\mid \hat A_i,x) }{\suml_{i:g(\varphi(s,\hat\beta,x),\hat A_i)\in B_0}\hat S_T(t\mid \hat A_i,x)}.
\eeq
The conditional survival
function of
 $T^0(z,x)$ given the survival until  the degradation attains the level $z$ and the surviving until the moment $t_1$,
 and given $x(u),0\leq u\leq t$, is
 estimated by
\beq
\hat P(T^0(z)>t\mid \tau>T^0(z),x(u),0\leq u\leq t)=
$$
$$
\frac{\suml_{i:g(\varphi(t,\hat\beta,x),\hat A_i)<z}
\hat S_T(\psi(h(z,\hat A_i),\hat \beta,x)\mid \hat A_i,x)
 }{\suml_{i=1}^n\hat S_T(\psi(h(z,\hat A_i),\hat \beta,x)\mid \hat A_i,x)},
\eeq
and
$$
\hat E(T^0(z)\mid \tau>T(z_0),x(u),0\leq u\leq t)=
$$
$$
\frac{\suml_{i=1}^n\psi(h(z,\hat A_i),\hat\beta,x)
\hat S_T(\psi(h(z,\hat A_i),\hat \beta,x)\mid \hat A_i,x)
 }{\suml_{i=1}^n\hat S_T(\psi(h(z,\hat A_i),\hat \beta,x)\mid \hat A_i,x)},
$$
respectively.

\vskip 1cm
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\end{document}
$$
P(C\mid B)=\int P(C\mid B,X=x)dF_X(x\mid B).
$$
