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\begin{document}


\centerline{\large \bf Statistical Analysis of Semiparametric Models}
\centerline{\large \bf  in Accelerated Life Testing}
\vspace{1cm}

\leftline{\scriptsize VILIJANDAS BAGDONAVI\v{C}IUS  }

\leftline{\scriptsize \it Department of Statistics, Vilnius University, Lithuania}
\bigskip

\leftline{\scriptsize MIKHAIL S. NIKULIN }


\leftline{\scriptsize \it  Steklov Mathematical Institute, 27 Fontanka, 191011,
Saint Petersburg, Russia  }
\bigskip



\noindent
{\bf Abstract:}
Several accelerated life models are introduced. Their relationships are investigated. Semiparametric estimation procedures are proposed. Properties of estimators are discussed

\bigskip

\noindent
{\bf Keywords and phrases:}
Accelerated life models, additive accumulation of damages, generalized proportional hazards, semiparametric estimation
\vspace*{30pt}





\leftline{\bf 1. Models}
\vspace{0.5cm}

\par Suppose  that stresses are deterministic time functions:
$$
x(\cdot)=(x_1(\cdot),...,x_m(\cdot))^T\,:[0,\infty)\to B\in {\bf R^m}.
$$
Let $\Omega$ be a population of items and suppose that the time-to-failure of items under
the stress $x(\cdot)$ is defined by
a non-negative absolutely continuous random
variable $T_{x(\cdot)}=T_{x(\cdot)}(\omega),\omega \in \Omega$, with the survival function $S_{x(\cdot)}(t) = {\bf P}\{T_{x(\cdot)}>t\}$ and the cummulative distribution function $F_{x(\cdot)}(t)=1-S_{x(\cdot)}(t)$.
If $x(\cdot)$ is constant in time, we'll write $x$ instead of $x(\cdot)$
in all formulae. We shall consider here only the deterministic
stresses. The case when the stress is a stochastic process
X(t),can be done by the same way as in Bagdonavi\v{c}ius and  Nikulin (1996, 1997c, 1997d)
for the generalized additive and additive-multiplicative
semiparametric models with random covariates.



\par The proportion $F_{x(\cdot)}(t)$ of items from $\Omega$ which fail until the moment
$t$ under the stress $x(\cdot)$ is called the {\it resource used until the
moment $t$}.
\par Let
$$
\alpha_{x(\cdot)}(t)
=\lim_{h\downarrow 0}\, \frac{1}{h}\, \PP \{T_{x(\cdot)}\in (t,t+h] \mid
T_{x(\cdot)}>t\}=-\frac{S^{\prime}_{x(\cdot)}(t)}{S_{x(\cdot)}(t)},
\, t \geq 0,
$$
be the hazard rate function under the stress $x(\cdot)$.
Denote by
$$
A_{x(\cdot)}(t)=\int^t_0
\alpha_{x(\cdot)}(u)du=-ln\{S_{x(\cdot)}(t)\},
\, t \geq 0,
$$
the accumulated hazard rate under $x(\cdot)$.

The first idea which comes by modelling the influence of a stress on lifetime distribution is to suppose that
the hazard rate at any moment $t$ depends on the value of the stress
at this moment and the resource used until $t$:
\par {\it The generalized Sedyakin's }(GS) model (Bagdonavi\v{c}ius  (1978),Bagdonavi\v{c}ius and Nikulin (1998) )
holds on the set of stresses $E $, if for all $x(\cdot)\in E$
$$
\alpha_{x(\cdot)}(t) = g \left(x(t),A_{x(\cdot)}(t) \right).
$$
\par Consider the following important particular case of the GS model. Suppose that
the hazard rate at any moment $t$ is proportional to a function of the stress used until
this moment and to a function of the resource used until $t$:
\par {\it The additive accumulation of damages }(AAD) model
(Bagdonavi\v{c}ius (1978))
 holds on the set of stresses $E$, if for all $x(\cdot)\in E$
$$
\alpha_{x(\cdot)}(t) = r \{x(t)\}\,q\{A_{x(\cdot)}(t) \}.
$$
\par {\bf Proposition 1}. {\it The AAD model holds on $E$ iff there exists a survival function $G$ such that}
$$
S_{x(\cdot)}(t) = G\left(\int^{t}_{0} r\{x(\tau)\}\;d\tau\right). \eqno (1)
$$
\par The AAD model written in the form (1) is also called
the {\it accelerated failure time} (AFT)
model (Cox and Oakes (1984)).
\par Cox (1972) considered the model which means that the hazard rate at any
moment $t$ is proportional to some function of a
stress applied at this moment and to a baseline rate which doesn't depend on the stress:
\par The {\it proportional hazards} (PH) model holds on $E$ if for all $x(\cdot) \in {E}$
$$
\alpha_{x(\cdot)}(t)=r\{x(t)\}\,\,\alpha_0(t). \eqno(2)
$$
\par When the PH model is also the AAD model? The answer is
given in the following two propositions.
\par {\bf Proposition 2} (Cox and Oakes (1984)). {\it Suppose that the PH model holds on a set $E_0$ of constant stresses
such that the set $r(E_0)$ has an interior point.
\par The AAD model also holds on $E_0$ iff the time-to-failure
distribution is Weibull for all}
$x \in E_0$.
\par Suppose that $E_0$ is the set of constant stresses defined in Proposition 2,
$x_1,x_2\in E_0$ are two fixed constant stresses and a step-stress
$x_s(\cdot)$ has the form
$$
x_s(\tau)=\left\{\begin{array}{cc}
x_1,& 0 \leq \tau < s,\\
x_2,& \tau \geq s,
\end{array} \right. \eqno (3)
$$
where $s$ is a fixed positive number.

\par {\bf Proposition 3}. {\it Suppose that  the PH model holds on the
set $E$ including $ E_0$ and ${x_s(\cdot)}$. The AAD model also holds on
${E}$ iff
 the time to-failure is exponential for all $x\in {E}_0$.}
\par Proof. The Proposition 2 implies that for all $x\in E_0$
$$
S_{x}(t)=e^{-(\frac{t}{\theta (x)})^{\alpha}}. \eqno (4)
$$
Put $\theta_i=\theta(x_i),\,i=1,2$. Then
$$
S_{x_i}(t)=e^{-(\frac{t}{\theta_i})^{\alpha}},\quad
\alpha_{x_i}(t)=\frac{\alpha}{\theta_i^{\alpha}}t^{\alpha-1}.
$$
The PH model implies that
$$
\alpha_{x_s(\cdot)}(t)=\left\{\begin{array}{cc}
\alpha_{x_1}(t),& 0 < \tau < s,\\
\alpha_{x_2}(t),& \tau \geq s,
\end{array} \right.
$$
and for all $t>s$
$$
S_{x_s(\cdot)}(t)=\exp\{-\int_0^t \alpha_{x_s(\cdot)}(u)du\}
=\exp\{ -\int_0^s \alpha_{x_1}(u)du-\int_s^t \alpha_{x_2}(u)du\} =
$$
$$
\exp \left\{ -\left(\frac{s}{\theta_1}\right)^{\alpha}
-\left(\frac{t}{\theta_2}\right)^{\alpha}+
\left(\frac{s}{\theta_2}\right)^{\alpha} \right\}. \eqno (5)
$$
\par 1) Suppose that both the PH and the AAD models hold on $E$. Then (1) and (4)
imply that for all $t>s$
$$
S_{x_s(\cdot)}(t)=\exp \left\{ -\left(\frac{s}{\theta_1}+
\frac{t-s}{\theta_2}\right)^{\alpha} \right\} \eqno (6)
$$
The equalities (5) and (6) imply that for all $t>s$
$$
\exp \left\{ -\left(\frac{s}{\theta_1}\right)^{\alpha}-\left(\frac{t}{\theta_2}\right)^{\alpha}+
\left(\frac{s}{\theta_2}\right)^{\alpha} \right\} =
\exp \left\{ -\left(\frac{s}{\theta_1}+
\frac{t-s}{\theta_2}\right)^{\alpha} \right\} . \eqno (7)
$$
If $\alpha=1$, this equality is verified. Suppose that $\alpha \neq 1$. For all $t>s$ put
$$
g(t)= \exp\left\{-\left(\frac{s}{\theta_1}\right)^\alpha-\left(\frac{t}{\theta_2}\right)^\alpha
+\left(\frac{s}{\theta_2}\right)^\alpha\right\}-
\exp\left\{-\left(\frac{s}{\theta_1}+\frac{t-s}{\theta_2}\right)^\alpha\right\}.
$$
This function is strictly monotone
for all $t>s$ with fixed $\theta_1\neq \theta_2$ and $\alpha\neq 1$.
So the function $g$ is  not constant in $t$ which
contradicts to the equality (7). The assumption $\alpha \neq 1$ was false. Thus
$\alpha =1$, and the
lifetime distribution under any $x \in E_0$ is exponential.
\par 2) Suppose that the PH model holds on $E$ and the time-to-failure is exponential for all $x \in E_0$. The formula (2)
implies that for all $x \in E_0$
$$
S_x(t) = \exp\{-r(x)\,A_0(t)\}, \quad A_0(t)=\int_0^t \alpha_0(v)dv.
$$
Exponentiality of the times-to-failure under $x\in E_0$ and the last formula imply that $A_0(t)=c\,t$.
The constant $c$ can be included in $r(x)$, so we have $A_0(t)=t$. The formula (2) implies that
$$
S_{x(\cdot)}(t)=\exp{\left\{-\int^t_0r\{x(u)\}du\right\}},
$$
i.e. the AAD model holds on $E$.
\par The proof is complete.

\par The GS model is more general then the AAD model. When the
PH model is also the GS model? It is given in the following proposition.
\par {\bf Proposition 4}. {\it Suppose that  the PH model holds on the set
${E}$ including $E_0$ and all the stresses of the form (3) with $s<\delta$,
where $\delta$ is any positive number. The GS model also holds on
${E}$ iff
 the time to-failure is exponential for all $x\in {E}_0$.}
\par Proof. The PH model implies that for all $s<\delta $
$$
\alpha_{x_s(\cdot)}(t)=\alpha_{x_2}(t),\; t > s.
$$
\par 1) If the GS model also holds on ${E}$, then for all $s<\delta$
$$
\alpha_{x_s(\cdot)}(t)=\alpha_{x_2}(t-s+\varphi (s)),\; t > s,
$$
where
$$
\varphi (s)=A^{-1}_{x_2}(A_{x_1}(s))
$$
is an increasing function.
It implies that if both the GS and PH models hold on $E$ then
for all $s_1<\delta$ and $s_2<\delta$
$$
\alpha_{x_2}(t-s_1+\varphi (s_1))=\alpha_{x_2}(t-s_2+\varphi (s_2)),\; t>max(s_1,s_2).
$$
Any function $\alpha_{x_2}(t)\equiv $const verifies this.
Assume that the function $\alpha_{x_2}(t)$ is not constant.
Then  $\varphi (s)-s=c=$const for all $s>0$, because the function $\alpha_{x_2}(t)$
cannot be two or more-periodic. Note that $c\neq 0$, because
$$
A_{x_2}(\varphi(s))=A_{x_1}(s)\neq A_{x_2}(s).
$$
The equalities
$$
\lim_{s\to 0} A_{x_2}(\varphi (s))=\lim_{s\to 0} A_{x_1}(s)=0
$$
and the monotonicity of $\varphi (s)$ imply that $\lim_{s\to 0} \varphi (s)=0.$ So exists $\delta_0 \in (0,\delta)$ such that
$\mid \varphi (s)-s \mid<\mid c \mid $ when $0<s<\delta_0$. It contradicts
the implication that  $\varphi (s)-s=c$ for any $s>0$. It means that the
assumption, that $\alpha_{x_2}(t)$ is not constant, was false.
So $\alpha_{x_2}(t)=\alpha=$const which implies that $S_{x_2}(t)=e^{-\alpha_2 t}$.
The PH model implies that for all $x\in E_0$
$$
S_{x}(t)=S_0(t)^{r(x)}=e^{-r(x)t},
$$
i.e. the time-to-failure distribution is exponential for all $x\in E_0$.
\par 2) Suppose that for all $x\in E_0$ the time-to-failure distribution is exponential and the PH model holds.
The proof of the Proposition 2.8 implies that the AAD model and consequently
the GS model also holds on $E$.
\par The proof is complete.

\par The AAD and the PH models are rather restrictive. In the case of the AAD model  the stress changes locally
only the  scale.
\par Under the PH model the hazard rate under the stress $x(\cdot)$ at the moment $t$
{\it doesn't depend on the
resource used until $t$}. It is not very natural if items are aging. Indeed,
let ${E}_0 \subset {E}$ be a set of constant in time stresses, $x_0$ be an
usual stress, $x_1$ be an accelerated with
respect to $x_0$ stress, $x_0,x_1 \in {E}_0$, i.e. $S_{x_0}(t) > S_{x_1}(t)$ for
all $t \geq 0$, and $E_1$ be a set of simple step-stresses of the form
$$
x(t)=\left\{
\begin{array}{ll}
x_1,& 0 \leq t < t_1,\\
x_0,& t \geq t_1.
\end{array} \right.
$$
If the PH model holds on $E_0\cup E_1$ then for all $ t_1 > 0$, $t > t_1$ we have $\alpha_{x(\cdot)}(t)=\alpha_{x_0}(t).$
\par If two
groups of items are tested under the usual stress $x_0$ and the accelerated stress $x_1$, respectively, until a moment $t_1$ and after this moment both groups are observed under the same
usual stress $x_0$, the failure rate after the moment $t_1$ is the same for both groups under the PH model.
It is natural only for aging items.

\par A  generalization of AAD and PH models is obtained by supposing that
the hazard rate  at any moment $t$ is proportional to some function of the
stress applied at
this moment, to some function of the resource, used until $t$, and to some baseline rate.
 This is formalized by the following definition:

\par {\it The first generalized proportional hazards} (GPH1)
model ( Bagdonavi\v{c}ius  and Nikulin (1999c)) holds on $E$ if for
all $x(\cdot) \in {E}$
$$
\alpha_{x(\cdot)}(t)=r\{x(t)\}\;q\{A_{x(\cdot)}(t)\}\;\alpha_{0}(t).
\eqno(8)
$$
\par The particular cases of the GPH1 model are the PH model ($q(u)\equiv 1$) and the
AAD model ($\alpha_0(t) \equiv \alpha_0=const$).
\par A generalization of the GS and PH models is the following model:

\par {\it The second generalized proportional hazards} (GPH2)
model  holds on $E$ if for
all $x(\cdot) \in {E}$
$$
\alpha_{x(\cdot)}(t)=u\{x(t),A_{x(\cdot)}(t)\}\,\alpha_{0}(t). \eqno(9)
$$
We have inclusions
$$
AAD\subset GS \subset GPH1 \subset GPH2, \quad PH\subset GPH1 \subset GPH2.
$$
 If $AAD=PH$ on $E_0$ then the lifetime distribution is Weibull under any stress $x \in E_0$. If $AAD=PH$ on $E_1$ then
 the lifetime distribution is exponential under any stress $x \in E_0$


\par Particular cases of the GPH models give important accelerated
life models. Numerous examples of real data show that taking two constant in time
stresses, say $x_1$
and $x_2$, the ratio $\alpha_{x_2}(t)/\alpha_{x_1}(t)$ (which is constant under the
PH model) can be increasing or decreasing in time
and even a  cross-effect of hazard rates can be observed, see Bagdonavi\v{c}ius and  Nikulin (1999e).

Such data can be modelled by
submodels of the GPH1 or of more general the GPH2 model.
\par {\bf Example 1.} Consider the following parametrization of $q$ in the GPH1 model: $q(u)=(1+u)^{\gamma+1}, $
where $\gamma \in I\!\!R$ is an unknown scalar parameter. We have
the model
$$
\alpha_{x(\cdot)}(t)=r\{x(t)\}(1+A_{x(\cdot)}(t))^{\gamma+1}\alpha_{0}(t).
\eqno(10)
$$
If $\gamma=-1$, it is the PH model.
\par Suppose that $\gamma<0$ and $c_0=r(x_2)/r(x_1)$. Then
$$
\alpha_{x_2}(t)/\alpha_{x_1}(t)
=c_0
\left\{ \frac{1-\gamma\, r(x_2)\, A_0(t)}{1-\gamma\, r(x_1)\, A_0(t)}\right\}^{-1-\frac{1}{\gamma}}.
$$
The ratio $\alpha_{x_2}(t)/\alpha_{x_1}(t)$ has the following
properties:
\par a) if $-1<\gamma<0$, then the ratio  $\alpha_{x_2}(t)/\alpha_{x_1}(t)$  increases from the value $c_0$ until
the value $c_{\infty}=\lim_{t\to \infty}\alpha_{x_2}(t)/\alpha_{x_1}(t)$, where the constant $c_{\infty}$ can take any value in the interval $(c_0, \infty)$;
\par b) if $\gamma=-1$ (PH model), the ratio $\alpha_{x_2}(t)/\alpha_{x_1}(t)$ is constant in time.
\par c) if $\gamma<-1$, then the ratio $\alpha_{x_2}(t)/\alpha_{x_1}(t)$  decreases from the
value $c_0$ until the value $c_{\infty}\in (1, c_0)$.
\par {\bf Example 2.} To obtain a cross-effect of hazard rates consider the following submodel of GPH2:
$$
\alpha_{x(\cdot)}(t)=r( x(t))(1+A_{x(\cdot)}(t))^{\gamma^T x(t)+1}\,
\alpha_{0}(t).\eqno(11)
$$
Suppose that $c_0=r(x_2)/r(x_1)>1$ and $\gamma^T x_2<\gamma^T x_1<0$. Then
$$
\alpha_{x_2}(t)/ \alpha_{x_1}(t)=c_0
\frac{(1-\gamma^T x_2\, r( x_2) A_0(t))^{-1-\frac{1}{\gamma^T x_2}}}
{(1-\gamma^T x_1\, r(x_1)\, A_0(t))^{-1-\frac{1}{\gamma^T x_1}}}
$$
and
$$
\alpha_{x_2}(0)/ \alpha_{x_1}(0)=c_0>1,\;\;
 \lim_{t\to \infty}\alpha_{x_2}(t)/\alpha_{x_1}(t)=0.
$$
So we have the cross-effect of the hazard rates.\\
\par {\bf Example 3.} Taking parametrization $q(u)=e^{\gamma u}, \; \gamma \in {\bf R},$
we obtain the model
$$
\alpha_{x(\cdot)}(t)=r({x}(t))\,e^{\gamma A_{{x}(\cdot)}(t)}\,\alpha_0(t).
\eqno(12)
$$
 This model is a generalization of the gamma frailty model (Vaupel et al. (1979)). If $\gamma =0$, it becomes the usual PH model.

\par {\bf Example 4}. Taking $q(u)=1/(1+\gamma u), \quad \gamma > 0$,
we obtain the model
$$
\alpha_{x(\cdot)}(t)=r({x}(t))\,\frac{1}{1+\gamma A_{{x}(\cdot)}(t)}\,\alpha_0(t).
$$

\par Consider now an important model which doesn't lie in the class of the GPH models but includes the AAD model as the particular case:
\par {\it Locally changing shape and scale} (LCSS) model (\, Bagdonavi\v{c}ius and Nikulin (1998, 1999a)) holds  on $E$ if
$$
S_{x(\cdot)}(t)=G\left(\int^t_0r\{x(\tau)\}\tau^{\nu(x(\tau))-1}d\tau\right). \eqno(13)
$$
Schabe and Viertl (1995) considered an axiomatic approach to model building.
\par {\bf Proposition 4} (Schabe and Viertl (1995)). {\it Suppose
that exists a functional $a:E\times E\times [0, \infty )\rightarrow [0, \infty )$ such that for any $x_1(\cdot),x_2(\cdot) \in E$ it is differentiable and increasing in $t$, $a(x_1(\cdot),x_2(\cdot),0)=0$ and
 $T_{x_2(\cdot)} \sim a(x_1(\cdot),x_2(\cdot),T_{x_1(\cdot)})$,
 where $\sim $ denotes equality in distribution.
\par For any differentiable on $[0, \infty )$ survival function $S$ exists a functional}
$b:E\times [0, \infty ) \rightarrow [0, \infty )$ {\it such
that for all} $x(\cdot)\in E$
$$
S_{x(\cdot)}(t)=S\left(\int^t_0b\{x(\cdot),u\}\,du\right).
$$
All above considered models are particular cases of this general Schabe-Viertl
model. See also Bagdonavi\v{c}ius and  Nikulin (1999d).
\par The GS and AAD models
are not appropriate when stresses are periodic with quick change of
their values. Greater is the number of stress cycles, shorter is the life of
items. So the effect of cycling must be included in the model.
\par Suppose that a periodic stress is differentiable. Then the
number  of cycles in the interval $[0,t]$ is
$$
n(t)=\int^t_0 \mid d \Un \{x'(u)> 0\}\mid .
$$
Generalization of the GPH1  model  has the form
$$
S_{x(\cdot)}(t)=G\left\{\int_0^t r_1\{x(u)\}\,dH(S_0(u))+\int_0^t  r_2\{x(u)\}\mid d
\Un \{x'(u)> 0\}\right\}.
$$
The second integral includes the effect of cycling on resource usage.
\par  Taking $H=S_0^{-1}$, we obtain a generalization of the AAD model which includes the effect of cycling.
The GS and AAD models are also not appropriate if $x(\cdot)$ is
a step stress with many switch on's and switch off's which shorten the life of items.
In this case the following model can be considered:
$$
S_{x(\cdot)}(t)=\int_0^t r_1\{x(u)\}\,dH(S_0(u))+\int_0^t  r_2\{x(u)\}
\Un (\Delta x(u)>0)\frac{\mid d x(u)\mid}{\mid \Delta x(u)\mid }
$$
$$
+\int_0^t  r_3\{x(u)\}
\Un (\Delta x(u)<0)\frac{\mid d x(u)\mid}{\mid \Delta x(u)\mid } .
$$
The second and the third terms include the effect of switch-on's and switch-off's on resource usage. Taking
$H=S_0^{-1}$, we obtain a generalization of the AAD model which includes this effect.






\vspace*{30pt}

\leftline{\bf 2. Semiparametric estimation}
\vspace{0.5cm}

Semiparametric estimation for the \, AAD \, model was considered by Basu and Ebrahimi (1982), Sethuraman and Singpurwalla (1982),
 Shaked and Singpurwalla (1983), Schmoyer (1986, 1991), Robin and Tsiatis (1992), Lin and Ying (1995),
 Bagdonavi\v{c}ius and Nikulin (1999a,b).
\par Estimation for the PH model was developed by Cox (1972), Tsiatis (1981),  Andersen and Gill (1982).
\par Particular cases of the GPH1 model with parametrization of $q$ was considered by Andersen, Borgan, Gill and
Keiding (1993), with completely known $q$ by Dabrowska and Doksum (1988a,b), Cheng
and others (1995), Murphy and others (1997), Bagdonavi\v{c}ius and Nikulin (1997a,b).
\par Estimation in LCSS model was discussed by Bagdonavi\v{c}ius and Nikulin (1998, 1999a,b,c).
\par We consider the most general GPH2 model
$$
\alpha_{x(\cdot)}(t)=u\{x(t),A_{x(\cdot)}(t),\theta\}
\,\alpha_{0}(t). \eqno(14)
$$
with any specified parametrization $u=u(x,s,\theta)$ via the parameters $\theta$ and unknown baseline function $\alpha_{0}(t)$.
\par Suppose that $m$ groups of items are tested. The ith group of $k_i$ items is tested under the accelerated stress $x_i(\cdot)$. Denote by $T_{ij}$ and $C_{ij}$ the failure and
censoring times for the jth item of the ith group and let
$$
X_{ij}=T_{ij}\wedge C_{ij}, \, \delta_{ij}=I_{\{T_{ij} \leq C_{ij}\}}, \,
N_{ij}(t)=I_{\{T_{ij}\leq t, \delta_{ij}=1\}}, \, Y_{ij}(t)=I_{\{X_{ij} \geq t\}},
$$
where $I_A$ denotes the indicator of the event $A$. Then
$$
N_i(t)=\sum^{k_i}_{j=1}N_{ij}(t)\quad \mbox{ and}\quad Y_i(t)=\sum^{k_i}_{i=1}Y_{ij}(t)
$$
are the numbers of observed
failures in the interval $[0,t]$ and items "at risk" just prior to the moment $t$, respectively, for the ith group of items.
\par We suppose that times-to-failures $T_{ij}$ are absolutely continuous random variables and censoring is independent right censoring.
\par  Suppose that maximal time given for experiment is $\tau \in (0,\infty)$ and all items which  did
not fail and were not censored before $\tau$, are censored at this moment.

\par The partial likelihood function (see Andersen and others (1993))
$$
L(\theta)=\prod_{i=1}^m \prod _{j=1}^{k_i}\left(\int_0^{\tau}\frac{u\{x_i(v),A_i(v),\theta\}}{\sum_{l=1}^m Y_l(v)\,u\{x_l(v),A_l(v),\theta\}}dN_{ij}(v)\right)^{\delta_{ij}} \eqno(15)
$$
depend on unknown accumulated hazard rates $A_i(t)=\int_0^t \alpha_{x_i}(u)du.$
\par Modify the partial likelihood function in the following manner: replace the functions $A_i(t)$ in (15) by the Nelson-Aalen estimators:
$$
\hat A_i(t)=\int_0^t \frac{dN_i(s)}{Y_i(s)}.
$$
The modified partial likelihood function
$$
\tilde L(\theta)=\prod_{i=1}^m \prod _{j=1}^{k_i}\left(\int_0^{\tau}\frac{u\{x_i(v),\hat A_i(v),\theta\}}{\sum_{l=1}^m Y_l(v)\,u\{x_l(v),\hat A_l(v),\theta\}}dN_{ij}(v)\right)^{\delta_{ij}} \eqno(16)
$$
Put $g(x,t,\theta)=\frac{\partial}{\partial \theta} u(x,t,\theta)$.
The score function has the form
$$
U(\theta)=\sum_{i=1}^m\int_0^{\tau}\left(\frac{g\{x_i(v),\hat A_i(v),\theta\}}{u\{x_i(v),\hat A_i(v),\theta\}}-\frac {\sum_{l=1}^m Y_l(v)\,g\{x_l(v),\hat A_l(v),\theta\}}{\sum_{l=1}^m Y_l(v)\,u\{x_l(v),\hat A_l(v),\theta\}}\right)dN_i(v).\eqno(17)
$$
If $k_i$ are small, the Nelson-Aalen estimators can be not very efficient and the functions $A_i$ should be replaced by some other estimator which is a function of all data. This can be done in the following manner.
\par The model (14) implies that
$$
dA_i(t)=u\{x_{i}(t),A_i(t),\theta\}\,dA_{0}(t).
$$
Using considerations similar to considerations used to obtain the Nelson-Aalen estimator for the PH model we obtain a pseudoestimator $\tilde A_0(t,\theta)$ (depending on $\theta$ ) for $A_0(t)$:
$$
\tilde A_0(t,\theta)=
\int^t_0 \frac{dN(u)}{\sum^m_{i=1}
u\{x_i(u),\tilde {A}_i(u-;\theta),\theta\}\,Y_i(u)}.
$$
The last two equalities give the recurrent equations for a pseudoestimator $\tilde {A}_i(t;\theta)$ of  ${A}_i(t)$:
$$
\tilde {A}_i(t;\theta)=
\int^t_0 \frac{u\{x_i(u),\tilde{A}_i(u-,\theta),\theta\}\,dN(u)}{\sum^m_{l=1}
u\{x_l(u),\tilde {A}_l(u-;\theta),\theta\}\,Y_l(u)},\,\,\tilde {A}_i(0;\theta)=0.  \eqno(18)
$$
\par Modify the score function (17) by considering  the score function
$$
\tilde U(\theta)=
\sum_{i=1}^m\int_0^{\tau}\left(\frac{g\{x_i(v),\tilde A_i(v,\theta),\theta\}}{u\{x_i(v),\tilde A_i(v,\theta),\theta\}}-\frac {\sum_{l=1}^m Y_l(v)\,g\{x_l(v),\tilde A_l(v,\theta),\theta\}}{\sum_{l=1}^m Y_l(v)\,u\{x_l(v),\tilde A_l(v,\theta),\theta\}}\right).\eqno(19)
$$
Note that to find an initial estimator $\hat \theta^{(0)}$ verifying  the equations (19) you need  initial estimators $\tilde{A}^{(0)}_i(t,\theta)$ and {\it vice versa}: to find initial estimators $\tilde{A}^{(0)}_i(t,\theta)$ you need an initial estimator $\hat \theta^{(0)}$. This problem can be solved by taking the initial estimator $\hat \theta^{(0)}$ as the solution of the equations $U(\theta)=0$, where the function $U(\theta)$ is defined by the formula (17). Then the initial estimators $\tilde{A}^{(0)}_i(t,\theta^{(0)})$ are obtained recurrently from the equation (18). The estimator  $\hat \theta^{(1)}$ is obtained by solving the equations $\tilde U(\theta)=0$, where $\tilde U(\theta)$ is given  by (19) with $\tilde{A}^{(0)}_i(t,\theta^{(0)})$ taking place of $\tilde{A}_i(t,\theta)$. And so on. Given the estimator $\hat \theta^{(l)}$, the estimator  $\tilde{A}^{(l)}_i(t,\theta^{(l)})$ is right-continuous step-function with jumps at the observed failure moments  $T_{(j)}$ !
!
(whi
ch are the jump points of $N(t)$).
\par The estimator of the survival function $S_{x_0}(t),\,t\leq \tau$ under the usual stress is $\hat S_{x_0}(t)=\exp \{-\hat A_{x_0}(t)\}$, where $\hat A_{x_0}(t)$ is found recurrently from the equation
$$
\hat A_{x_0}(t)=
\int^t_0 \frac{u\{x_0,\hat{A}_{x_0}(u-),\hat \theta\}\,dN(u)}{\sum^m_{l=1}
u\{x_0,\tilde {A}_l(u-;\hat \theta),\hat \theta\}\,Y_l(u)},\,\,\hat A_{x_0}(0)=0.
$$

\vspace*{30pt}



\leftline{\bf 3. Properties of the estimators}
\vspace{0.5cm}

 Suppose that the  covariates $x_i(\tau)\equiv x_i$ are constant in time. Denote by $\theta_0$ the true value of $\theta$ under the GPH2 model, $\|A\|=\sup_{i,j}|a_{ij}|$ - the norm
of the matrix $A=(a_{ij})$, $A^{\otimes 2}$ the product $A{A}^T$,
$$
w_j(v;\theta)=\frac{g\{x_j,A_j(v),\theta\}}{u\{x_j,A_j(v),\theta\}} \quad
S^{(0)}(v;\theta)=\sum^m_{j=1}Y_j(v)\,u\{x_j,A_j(v),\theta\},
$$
$$
S^{(1)}(v;\theta)=\sum^m_{j=1}Y_j(v)\,g\{x_j,A_j(v),\theta\}, \quad
 E(v;\theta)=\frac{S^{(1)}(v;\theta)}{S^{(0)}(v;\theta)},
$$
$$
S^{(0)}_*(v;\theta)=\sum^n_{i=1}Y_i(v)\,u\{x_j,A_j(v),\theta\}u_2\{x_j,A_j(v),\theta\},
$$
$$
S^{(1)}_*(v;\theta)=\sum^n_{j=1}g\{x_j,A_j(v),\theta\}\,u_2\{x_j,A_j(v),\theta\},
$$
$$
S^{(2)}(v;\theta)=\sum^n_{j=1}\frac{\partial w_j(v;\theta)}{\partial \theta}Y_j(v)\,u\{x_j,A_j(v),\theta\},
$$
$$
S^{(2)}_*(v;\theta)=\sum^n_{j=1}u\{x_j,A_j(v),\theta\}\,g_2\{x_j,A_j(v),\theta\},
$$
where $u_2$ denotes the partial derivative of $u$ with respect to the second argument.
\par {\bf Assumptions A:}\\
\noindent a).  Suppose that exist  a neighbourhood $\theta$ of  $\theta_0$ and  continuous on $\theta$
uniformly in $t \in [0,\tau]$ and bounded on $\theta \times [0,\tau]$   functions $\, s^{(i)}(v;
\theta)$,  $s^{(i)}_*(v;\theta)$, such that $ s^{(0)}(v;\theta_0)>0$ on $[0,\tau]$ and
$$
\sup_{\theta \in \theta,\, v\in [0,\tau]}\|\frac{1}{n}S^{(i)}(v;\theta)-s^{(i)}(v;\theta)\| \rightarrow 0 \quad \mbox{as} \quad
n \rightarrow \infty,,
$$
$$
\sup_{\theta \in \theta,\, v\in [0,\tau]}\|\frac{1}{n}S^{(i)}_*(v;\theta)-s^{(i)}_*(v;\theta)\| \rightarrow 0 \quad
\mbox{as} \quad n \rightarrow \infty, \,\, (i=0,1,2),
$$
$$
 \sup_{\theta \in \theta, \, v \in [0,\tau]} \left\|\frac{\partial E(v;\theta)}{\partial \theta}-\frac{\partial e(v;\theta)}{\partial \theta}\right\|
\stackrel{{\bf P}}{\rightarrow}0,\quad e(v;\theta)=\frac{s^{(1)}(v;\theta)}{s^{(0)}(v;\theta)}.
$$
\noindent b).   $A_0(\tau) < \infty$.
\par Put  $e_*(v;\theta)=s_*^{(0)}(v;\theta)/s^{(0)}(v;\theta)$, $h(t;\theta)=exp\{ \int^t_0e_*(v;\theta)dA_0(v)\}$ ,
$$
h_1(v;\theta)=\frac{s^{(1)}(v;\theta) s^{(0)}_*(v;\theta)-s^{(0)}(v;\theta) s^{(1)}_*(v;\theta)}{s^{(0)}(v;\theta)}-s^{(2)}_*(v;\theta),
$$
$$
w(v;\theta)=e(v;\theta)-\frac{1}{h(v;\theta)s^{(0)}(v;\theta)}\int^{\tau}_vh_1(s;\theta)h(s;\theta)dA_0(s).
$$
\noindent c). $\frac{1}{n}\sum^n_{i=1}\int^{\tau}_0J(v) (w_i(v;\theta)-w(v;\theta)^{\otimes 2}u\{x_j,A_j(v),\theta\}
Y_i(v)dA_0(u)\stackrel{{\bf P}}{\rightarrow} \SIGMA(\theta). $\\
\noindent d).The matrix
$$
\SIGMA_1(\theta_0)=-\int^t_0\left(s^{(2)}(u;\theta_0)-
\frac{\partial e(u;\theta_0)}{\partial \theta}s^{(0)}(u;\theta_0)\right)dA_0(u)
$$
is positive definite.
\par {\bf Theorem 1}. {\it Under Assumptions A}
$$
n^{1/2}(\hat{\theta}-\theta_0)\stackrel{{\cal D}}{\rightarrow} N(0,\SIGMA_1^{-1}(\theta_0)
\SIGMA(\theta_0)\left(\SIGMA_1^{-1}(\theta_0)\right)^{T}).
$$
\par Consider the asymptotic distribution of the survival function $S_{x_0}(t)$ under the usual stress $x_0$. Put
$$
C(\theta_0)=-\int^t_0\frac{\partial}{\partial \theta}\ln{S^{(0)}(v;\theta_0)}dA_0(v),
$$
$$
H_i(v;\theta_0)=J(v)\{C(\theta_0)\SIGMA_1^{-1}(\theta_0)(w_i(v;\theta_0)-w(v;\theta_0))+\frac{1}{s^{(0)}
(v;\theta_0)}\},
$$
\par {\bf Assumptions B}.
$$
\frac{1}{n}\sum^m_{i=1}\int^t_0H_{i}^2(v;\theta_0)\,Y_i(v)\,u\{x_i,A_i(v),\theta\}\,dA_0(v)\stackrel{{\PP}}{\rightarrow} \sigma^2(t).
$$
Put
$$
\sigma_{x_0}^2(t) = S^2_{x_0}(t)\,u^2\{x_0,A_{x_0}(t),\theta\}\,\sigma^2(t).
$$
\par {\bf Theorem 2}. {\it Under Assumptions A and B}
$$
\sqrt{n}(\hat{S}_{x_0}(t)-S_{x_0}(t))\stackrel{\cal D}{\rightarrow}N(0,\sigma_{x_0}^2(t)).
$$



\vspace{0.5cm}



\leftline{\bf References}
\vspace{0.5cm}

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\end{enumerate}



\end{document}
