\documentclass[12pt,A4]{article} \usepackage[dvips]{graphics} \usepackage{amssymb} %\renewcommand{\baselinestretch}{1.5} %\usepackage{latexsym} %\let\square=\Box %\documentclass{royal1} \def \R{I\!\!R} %space \def \N{I\!\!N} %the set of natural numbers \def \X{X\!\!\!\!\!\!X} %sample \def \Z{Z\!\!\!Z} %the set of whole numbers \def \F{I\!\!F} \def \iint{\mathop{\int\!\!\!\int}\limits} %double integral \def \idotsint{\mathop{\int\!...\!\int}\limits} %multiply integral %______________for_UNIX_______________________________ %\documentstyle[12pt,amssymbols]{article} %\input amssym.def %\input amssym.tex %\def \R{{\displaystyle\Bbb R}} %\def \N{{\displaystyle\Bbb N}} %\def \X{{\displaystyle\Bbb X}} %\def \Y{{\displaystyle\Bbb Y}} %\def \Z{{\displaystyle\Bbb Z}} %\def \F{{\displaystyle\Bbb F}} %\def\blacksquare{\hbox{\vrule width 4pt height 4pt depth 0pt}} %from "begla" %\def\eqnum#1{\eqno(#1)} %_____________ %_____LAYOUT____________________________ \oddsidemargin 0cm \textwidth 15.5575cm \textheight 23.495cm \topmargin -1.5cm %___roman____________________________________________ \def \cov{{\bf Cov\,}} %Covariance \def \var{{\bf Var\,}} %Variance %___roman + bold_____________________________________ \def \PP{{\bf P}} %Probability \def \EE{{\bf E}} %Expected Value \def \Zero{{\displaystyle\bf 0}} %0 bold (\displaystyle for \frac) \def \Un{{\displaystyle\bf 1}} %1 bold (\displaystyle for \frac) %___italic___________________________________________ \newcommand{\imin}{\mathop{\it min\/}\limits} \newcommand{\imax}{\mathop{\it max\/}\limits} \newcommand{\iinf}{\mathop{\it inf\/}\limits} \newcommand{\isup}{\mathop{\it sup\/}\limits} \newcommand{\ilim}{\mathop{\it lim\/}\limits} %___italic + bold ___________________________________ \def\xx{\mbox{\boldmath$x$}} \def\aa{\mbox{\boldmath$a$}} \def\XX{\mbox{\boldmath$X$}} \def\AA{\mbox{\boldmath$A$}} \def\yy{\mbox{\boldmath$y$}} \def\bb{\mbox{\boldmath$b$}} \def\YY{\mbox{\boldmath$Y$}} \def\BB{\mbox{\boldmath$B$}} \def\zz{\mbox{\boldmath$z$}} \def\cc{\mbox{\boldmath$c$}} \def\ZZ{\mbox{\boldmath$Z$}} \def\CC{\mbox{\boldmath$C$}} \def\ii{\mbox{\boldmath$i$}} \def\dd{\mbox{\boldmath$d$}} \def\II{\mbox{\boldmath$I$}} \def\DD{\mbox{\boldmath$D$}} \def\vv{\mbox{\boldmath$v$}} \def\uu{\mbox{\boldmath$u$}} \def\VV{\mbox{\boldmath$V$}} \def\UU{\mbox{\boldmath$U$}} \def\hh{\mbox{\boldmath$h$}} \def\rr{\mbox{\boldmath$r$}} \def\HH{\mbox{\boldmath$H$}} \def\RR{\mbox{\boldmath$R$}} \def\jj{\mbox{\boldmath$j$}} \def\ff{\mbox{\boldmath$f$}} \def\JJ{\mbox{\boldmath$J$}} \def\FF{\mbox{\boldmath$F$}} \def\TT{\mbox{\boldmath$T$}} \def\WW{\mbox{\boldmath$W$}} \def\ww{\mbox{\boldmath$w$}} \def\QQ{\mbox{\boldmath$Q$}} \def\qq{\mbox{\boldmath$q$}} \def\GG{\mbox{\boldmath$G$}} \def\KK{\mbox{\boldmath$K$}} \def\LL{\mbox{\boldmath$L$}} \def\MM{\mbox{\boldmath$M$}} \def\NN{\mbox{\boldmath$N$}} \def\SS{\mbox{\boldmath$S$}} \def\pp{\mbox{\boldmath$p$}} \def\ttt{\mbox{\boldmath$t$}} %\tt for TeleType font \def\sss{\mbox{\boldmath$s$}} %\ss for gotic beta \def\g{\mbox{\boldmath$g$}} %\gg for >> \def\ee{\mbox{\boldmath$e$}} %\ee_1=(1,0,0,...,0) %___math_____________________________________________ \def \beq{\begin{equation}} \def \eeq{\end{equation}} \def \lf{\left\{} %{ \def \rf{\right\}} %} \def \lq{\left[} %[ \def \rq{\right]} %] \def \lc{\left(} %( \def \rc{\right)} %) \def \tran{^{\rm T}} \def \suml{\sum\limits} \def \prodl{\prod\limits} \def \intl{\int\limits} \def\barXX{\bar{\XX}} \def\barYY{\bar{\YY}} %___Greek______________________________________________ \def \SIGMA{{\bf\Sigma}} \def \ssigma{\mbox{\boldmath$\sigma$}} \def \TH{{\bf\Theta}} \def \th{\mbox{\boldmath$\theta$}} \def \barth{\bar{\theta}} \def \OO{{\bf\Omega}} \def \LAM{{\bf\Lambda}} \def \lam{\mbox{\boldmath$\lambda$}} \def \PSI{{\bf\Psi}} \def \PHI{{\bf\Phi}} \def \mm{\mbox{\boldmath$\mu$}} \def \nn{\mbox{\boldmath$\nu$}} \def \ba{\mbox{\boldmath$\alpha$}} \def \be{\mbox{\boldmath$\beta$}} \def \btau{\mbox{\boldmath$\tau$}} \def \DELTA{{\bf\Delta}} \def \del{\mbox{\boldmath$\delta$}} \def \eeps{\mbox{\boldmath$\varepsilon$}} %__Expressions______________________________________________ %__"md_*.tex" \def\pdf{probability density function } \def\pdfs{probability density functions } %__"test_hyp.tex","app3.tex" \def\prob{probability } \def\probs{probabilities } \def\dist{distinguishing } \def\suf{sufficient } \def\stats{statistics} \def\hyp{hypothesis} \def\hyps{hypotheses} %__"tables.tex" \def\kn{\;{\rm known}} \def\wh{{\rm where}\quad} \def\cmpn{\;{\rm are}\;{\rm components}\;{\rm of}\;} %__"norm_ue.tex" \def\csf{complete sufficient for } \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) $$ \beq = {\intl_{g(t,a)\in B,g(s,a)\in B_0}S_T(t\mid a)d F_A (a\mid t)}/{\intl_{g(s,a)\in B_0}S_T(t\mid a)dF_A (a\mid t)}. \eeq Denote by $$ T^0(z)=h(z,A) $$ the time needed to attain the level $z$ $(z\leq z_0)$ of the real degradation. The conditional survival function and the conditional mean of $T^0(z)$ given the survival until the degradation attains the level $z$ and given survival until the moment $T^0(z)$ are \beq P(T^0(z)>t\mid T>T^0(z))= \frac{\intl_{g(t,a)T^0(z))= \frac{\intl_{{\cal A}}h(z_0,a)S_T(h(z,a)\mid a) dF_A (a)}{\intl_{{\cal A}}S_T(h(z,a)\mid a) dF_A (a)}, \eeq respectively. \vskip 0.5cm \centerline{\bf 4. Modeling covariate effects on degradation and survival} \vskip 0.5cm Suppose units are observed under the covariate $x=x(t)=(x_1(t),\dots,x_m(t)),t\geq 0$, and the values of the real degradation process are measured at time moments $t_1,\dots,t_m$. In this case, following Bagdonavi\v{c}ius and Nikulin (2001), the real degradation process is modeled by \beq Z_r(t\mid x)=g(\varphi(t,\beta,x),A), \eeq where \beq \varphi(t,\beta,x)=\int_0^{t}e^{\beta x^T(u)}du, \eeq is the link function, $\beta=(\beta_1,\dots,\beta_m)$ is the vector of the regression parameters. If the covariate is constant over time then \beq \varphi(t,\beta,x)=e^{\beta x^T}t. \eeq \vskip 0.3cm \noindent {\bf Model 4} (Degradation with measurement errors). {\it The observed degradation values are \beq Z(t_j\mid x)=Z_r(t_j\mid x)\;U(t_j), \eeq where $e_j=\ln U(t_j)$ are i.i.d. random variables, $e_j\sim N(0,\sigma^2)$.} \vskip 0.3cm \noindent {\bf Model 5} (Degradation with the noise). {\it The observed degradation process is $$ 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}. $$ The given model implies that the survival function and the mean of the random variable $\tau$ given the covariate $x$ are \beq S_{\tau}(t\mid x)=P(\tau>t\mid x(s),0\leq s \leq t) = \intl_{g(\varphi(t,\beta,x),a)t) $$ $$ =\frac1{S_\tau(t\mid x)}\intl_{g(\varphi(t,\beta,x),a)\leq z}S_T(t\mid a,x)dF_A (a), \eeq and the mean real degradation attained at the moment $t$ given the survival until this moment is \beq m_{r}(t,x)=E Z_r(t\mid x)=\frac1{S_T(t\mid x)} \intl_{\cal A} (g(\varphi(t,\beta,x),a)\wedge z_0-z_*)S_T(t\mid a,x) dF_A (a). \eeq For any $B\subset {\bf R},B_0\subset {\bf R}$, the probability that at the moment $t$ the value of the real degradation will be in the set $B$ given that at the moment $s$ it is in the set $B_0$ and given $x(u),0\leq u\leq t$, is $$ 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 $$ \beq \hskip 6cm = \frac{\intl_{g(\varphi(t,\beta,x),a)\in B,g(\varphi(s,\beta,x),a)\in B_0}S_T(t\mid a,x) d F_A (a)}{\intl_{g(\varphi(s,\beta,x),a)\in B_0}S_T(t\mid a,x)dF_A (a)}. \eeq Denote by \beq T^0(z,x)=\psi(h(z,A),\beta,x) \eeq the time needed to attain the level $z$ of the real degradation under the covariate $x$. The conditional survival function and the conditional mean of $T^0(z,x)$ given the survival until $T^0(z,x)$, and given $x(u),0\leq u\leq t$, are \beq P(T^0(z,x)>t\mid \tau>T^0(z,x), x(u),0\leq u\leq t)=\hskip 5cm $$ $$ \hskip 5cm \frac{\intl_{g(\varphi(t,\beta,x),a)T^0(z), x(u),0\leq u\leq t)=\hskip 5cm $$ $$ \hskip 5cm \frac{\intl_{\cal A}\psi(h(z,a),\beta,x) S_T(\psi(h(z,a),\beta,x)\mid a,x)dF_A (a)}{\intl_{\cal A}S_T(\psi(h(z,a),\beta,x)\mid a,x) dF_A (a)}, \eeq respectively. \vskip 0.5cm \centerline{\bf 5. Semiparametric estimation of degradation and survival characteristics:} \centerline{\bf models without covariates} \vskip 0.5cm Assume that the c.d.f. $F_A$ and the intensities $\lambda$ are completely unknown. Fix the degradation measurement moments $t_{i,1},\dots,t_{i,m_i}$ of the $i$th unit $(i=1\dots,n)$. If the failure time $\tau_i$ of this unit occurs in the interval $[t_{i,j_i},t_{i,j_i+1})$ $(j_i=1,\dots,m_i;t_{m_{i}+1}=\infty)$ then the values $Z_{i1},\dots,Z_{i,j_i}$ of the degradation process $Z_i$ of the $i$th unit are observed at the time moments $t_{i1},\dots,t_{i,j_i}$. Set $$ 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_10$. \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 $$ \hat S_T(t\mid \hat A_i)=\exp\Bigl\{-\int_{g(0,\hat A_i)}^{g(t,\hat A_i)}h'(z,\hat A_i)\,d\hat\Lambda(z)\Bigr\}= $$ $$ \exp\{-\suml_{j=1}^n\Un_{\{\delta_j=1,g(0,\hat A_i)\leq\hat Z_jt) $$ $$ = {\intl_{g(t,a)\in B,g(s,a)\in B_0}\hat S_T(t\mid a)d \hat F_A (a)}/{\intl_{g(s,a)\in B_0}\hat S_T(t\mid a)d\hat F_A (a)}= $$ $$ \frac{\suml_{i:g(t,\hat A_i)\in B,g(s,\hat A_i)\in B_0} \hat S_T(t\mid \hat A_i) }{\suml_{i:g(s,\hat A_i)\in B_0} \hat S_T(t\mid \hat A_i)}. $$ 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 $$ \hat P(T^0(z)>t\mid \tau>T^0(z))= \frac{\intl_{g(t,a)T^0(z))= \frac{\intl_{{\cal A}}h(z,a)\hat S_T(h(z,a)\mid a) d\hat F_A (a)}{\intl_{{\cal A}}\hat S_T(h(z,a)\mid a) d\hat F_A (a)}= \frac{\suml_{i=1}^nh(z,\hat A_i) \hat S_T(h(z,\hat A_i)\mid \hat A_i) }{\suml_{i=1}^n\hat S_T(h(z,\hat A_i)\mid \hat A_i)}, $$ 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)t, x(u),0\leq u\leq t) $$ \beq = \frac{\suml_{i:g(\varphi(t,\hat\beta,x),\hat A_i)\in B,g(\varphi(s,\hat\beta,x),\hat A_i)\in B_0} \hat S_T(t\mid \hat A_i,x) }{\suml_{i:g(\varphi(s,\hat\beta,x),\hat A_i)\in B_0}\hat S_T(t\mid \hat A_i,x)}. \eeq The conditional survival function of $T^0(z,x)$ given the survival until the degradation attains the level $z$ and the surviving until the moment $t_1$, and given $x(u),0\leq u\leq t$, is estimated by \beq \hat P(T^0(z)>t\mid \tau>T^0(z),x(u),0\leq u\leq t)= $$ $$ \frac{\suml_{i:g(\varphi(t,\hat\beta,x),\hat A_i)T(z_0),x(u),0\leq u\leq t)= $$ $$ \frac{\suml_{i=1}^n\psi(h(z,\hat A_i),\hat\beta,x) \hat S_T(\psi(h(z,\hat A_i),\hat \beta,x)\mid \hat A_i,x) }{\suml_{i=1}^n\hat S_T(\psi(h(z,\hat A_i),\hat \beta,x)\mid \hat A_i,x)}, $$ respectively. \vskip 1cm \centerline{\bf References} \vskip 0.5cm \noindent Andersen,P.K., Borgan,O., Gill,R. and Keiding,N. 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