数理解析研究所講究録 1273 巻 2002 年 100-114

100
Model checking for semiparemetric linear
transformation model

北里大学大学院臨床統計専攻

聡 (Satoshi Hattori)

服部

Kitasato University Graduate School, Div. of Biostatistics

1

Introduction

In clinical trials, regression models are frequently used and play important roles. For example,
in cancer clinical trials which are desined to evaluate the survival benefit of anti-cancer drugs,
regression modek for time-to-event data are used to evaluate the treatment effect adjusting theeffect of covariates and to estimate(predict) survival functions of various kind of patients. The
proportional hazard model is frequenty and routinely used in the recent clinical trials with
time-to-event data as endpoints. Of course theproportional hazard model is quite useful andimportant because theproportional hazard model is a semiparametric model, which seems to
be more robust than the parametric models, and can be fifitted easily by many commercial software
packages. On the other hand, the Cox proportional hazard model may not fifit well since the proportional hazard assumption, the Cox proportional hazard model requires, isn’t neccesarily aweak one.
Recently the practical inference procedure of some kind of semiparametric models have been aetablished with the theoretical justififications. For example,, Ying and, Tsiatis(1990),
Ying(1993) and other authers have developed the inference procedure for the semiparametric accelerated failure time model andand Ying(1994) has established the simple inference procedure forthe semiparametric additive hazard model. The linear transformation model is another attractive
alternative semiparametric model. The linear transformation model is defined as$\mathrm{C}\mathrm{o}\mathrm{x}$$\mathrm{C}\mathrm{o}\mathrm{x}$$\mathrm{C}\mathrm{o}\mathrm{x}$$\mathrm{W}\mathrm{e}\mathrm{i}$$\mathrm{L}\mathrm{i}\mathrm{n}(1990)$$\mathrm{L}\mathrm{i}\mathrm{n}$$S_{Z}(t)=g\{h(t)+Z^{T}\beta\}$,(1)

where $Sz(t)$ is the survival function of the patient with the covariate $Z$ , $h(t)$ is the unknown nondecreasing function and is the known continuous and strictly increasing function. Here we assume
that $Z$ is bounded and, without loss of generality, $|Z|\leq 1$ is assumed. When $g^{-1}(t)=log(-log(t))$ ,(1) reduces to$g$$log(-log(S_{Z}(t)))=h(t)+Z^{T}\beta$ .

This is the Cox proportional hazard model. And when $g^{-1}(t)=$ -logii(t), (1) reduces to-logit(Sz (t)) $=h(t)+Z^{T}\beta$ .

101
proportinal
This is the proportional odds model which is the important alternative to thehazard model (Bennett (1982)). So changing the link function, , the linear transformation model
proportional hazard model andprovides a large class of semiparametric models containing theand Ying(1995),
the proportional odds regression model as special cases. Recently Cheng,
Cheng, Wei, Ying(1997) and Fine, Wei and Ying(1998) proposed the inference procedure for the
linear transformation model with the univariate, possibly right-censored data. And the extensions,and Wilcox(2000), Cheng and Wang(2001),
to more complicated data were done (and Ying(1997) proposed an graphical
Wei and Ying(2001) . For the univariate data, Cheng,plot. Though their model checking procedure is very usefulmodel checking procedure based on
to check and select the model, it is desirable to establish the formal model checking procedure,
proportional hazard model,
since the graphical procedure may be subjective. For the
Wei and Ying(1993) proposed an quite useful model checking procedure based on the cumulative
martingale-based residuals. Their method provides the formal omnibus test and , in addition, thegraphical model checking technique which is very useful to investigate what kind of misspecification
occure. In this article, we develop the model checking technique for the linear transformation modelwith the univariate, possible right-censored data based on the cumlative sum of martingale-type$\mathrm{C}\mathrm{o}\mathrm{x}$$g$$\mathrm{C}\mathrm{o}\mathrm{x}$$\mathrm{W}\mathrm{e}\mathrm{i}$$\mathrm{L}\mathrm{i}\mathrm{n}$$\mathrm{C}\mathrm{a}\mathrm{i},$$\mathrm{W}\mathrm{e}\mathrm{i}$$\mathrm{W}\mathrm{e}\mathrm{i}$$)$$\mathrm{p}\mathrm{p}$$\mathrm{L}\mathrm{i}\mathrm{n}$$\mathrm{C}\mathrm{o}\mathrm{x}$residuals.

2The inference procedure for the linear transformation model
In this section, we summarize the inference procedure for the linear transformation model
posed by Cheng, Wei and Ying $(1995, 1997)$ . The linear transformation model is defifined equivalently
to (1) as$\mathrm{p}\mathrm{r}\mathrm{e}\succ$$h(T)=-Z^{T}\beta+\epsilon$,

is the random variable whose distribution function $F=1-g$
is the failure time,
whereis completely known. The specification of the distribution of corresponds to that of the linkfunction . The distribution of is the standard extreme value distribution and the standart logistic
distribution for the Cox proportional hazard model and the proportional odds model respectively.
To estimate the unknown parameter and , the survival function $G$ of the censoring time $C$ isassumed not to depend on the covariate $Z$ . To estimate the regression coeffiffifficient , Cheng,
and Ying(1995) proposed the unbiased estimating equation,$\mathrm{T}$$\epsilon$$\epsilon$$g$$\epsilon$$\beta$$h$$\beta$$U( \beta)=\sum_{i=1j}^{n}\sum_{=1}^{n}w(Z_{ij}^{T}\beta)Z_{ij}\{\frac{\Delta_{j}I(X_{i}\geq X_{j})}{\hat{G}^{2}(t)}-\xi(Z_{ij}^{T}\beta)\}$,$\mathrm{W}\mathrm{e}\mathrm{i}$(2)where $\Delta_{i}=I(Tj\geq C_{i})$ , $\xi(s)=\int_{-\infty}^{\infty}\{1-F(t+s)\}dF(t)$ , $w(.)$ is a weight function, $Zij=Z_{\dot{l}}-Zj$ andis the Kaplan-Meier estimator for $G$ . Cheng, Wei and Ying(1995) showed that , which is thesolution of the equation $U(\beta)=0$ , converges to the true value almost surely and the distribution
converges asymptotically to normal distribution with mean zero whose asymptotic
ofvariance-covariance matrix can be consistently estimated by the sandwich-type variance estimator.
In addition, Cheng, Wei and Ying(1997) proposed the inference procedure for the survival function$\hat{\beta}$$\hat{G}$$\beta 0$$n^{\frac{1}{2}}(\hat{\beta}-\beta 0)$102$S_{z}(t)$. To this end, they proposed the unbiased estimating equation for $h(t)$ ,$V(h(t))= \dot{.}\sum_{=1}^{n}\{\frac{I(X_{i}\geq t)}{\hat{G}(t)}-g(Z^{T}.\cdot\hat{\beta})\}$

where
that

$\tau$$\hat{h}(t)$2.1

is a constant satisping $P(X>\tau)>0$ and, the solution of $V(h(t))=0$ , converges to$\hat{\beta}$,$t\in[0,\tau]$(3)

is the estimator derived by (2), and proved
uniformly inalmost surely.
$[0, \tau]$$h_{0}(t)$Model cheching method based on the cumulative sum of martingale residuals

At fifirst, we define the martingale-type residual for the linear transformation model in the similar matter for the Cox proportional hazard model. For the Cox proportional hazard model, themartingale residual is defined as$\tilde{M}_{\dot{1}}(t)=N_{\dot{1}}(t)-\int_{0}^{t}\mathrm{Y}.\cdot(u)e^{Z^{T}\overline{\beta}}\dot{.}d\tilde{\Lambda}(u)$where $N\dot{.}(t)=I(X\dot{.}\leq t, \Delta:=1)$ ,, is the maximum partial likelihood estimatorandis the Breslow estimator of the baseline hazard function (Barlow and Prentice 1988,
Thernea, Grambsch and Fleming 1990). Similarly we define the martingale residual for the linear
trmsformation model, using the relationand (1) as$\mathrm{Y}_{}(t)=I(X.\cdot\geq t)$$\tilde{\beta}$$\tilde{\Lambda}(t)$$\Lambda z\dot{.}(t)=-log(S_{Z}\dot{.}(t))$$\hat{M}_{\dot{1}}(t)=N_{\dot{1}}(t)+\int_{0}^{t}\mathrm{Y}_{\dot{1}}(u)dlog(g\{\hat{h}(t+)+Z_{\dot{1}}^{T}\hat{\beta}\})$.

Using this martingale residual, we defifine the goodness-of-fit statistics similar to that proposed,byand Ying(1993) for the
proportional hazard model. Defifine the multi-parameterstochastic process,$\mathrm{L}\mathrm{i}\mathrm{n}$$\mathrm{W}\mathrm{e}\mathrm{i}$$\mathrm{C}\mathrm{o}\mathrm{x}$$H(t, z)= \sum_{\dot{\iota}=1}^{n}I(Z.\cdot\leq z)\hat{M}_{}(t)$.

If the model is correctly specified, by the Taylor series expansion,equivalent to$n^{-\frac{1}{2}}\tilde{H}(t, z)$$=$$n^{-\frac{1}{2}} \sum_{l=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)dM_{l}(u)$$+$$n^{-1} \sum_{\mathrm{t}=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)$$\cross$$d[\tilde{g}\{h_{0}(u)+\beta_{0}^{T}Z_{l}\}$$n^{-\frac{1}{2}}H(t,z)$

is asymptoticffiy$n^{\frac{1}{2}}\{\hat{h}(u)-h_{0}(u)+(\hat{\beta}-\beta 0)^{T}Z_{l}]$where.
Cheng,
Wei and Ying(1997),
Prom$\tilde{g}(x)=\frac{dg(x)}{dx}/g(x)$$W_{Z_{l}}(t)$$n^{\frac{1}{2}}\{\hat{h}(t)-h_{0}(t)+(\hat{\beta}-\beta 0)^{T}Z_{l}]$$=$$\frac{1}{a(t)}[\{b(t)+a(t)Z_{l}\}^{T}D$is asymptotically equivalent to103$\cross$$\{n^{-\frac{3}{2}}\sum_{i=1j}^{n}\sum_{=1}^{n}w(Z_{ij}^{T}\beta_{0})Z_{ij}e_{ij}(\beta_{0})$$+$$2n^{\frac{-1}{2}} \sum_{i=1}^{n}\int_{0}^{\infty}\frac{q(u)}{\pi(u)}dM_{i}^{c}(u)\}$$+$$n^{\frac{-1}{2}} \sum_{i=1}^{n}r_{i}(t)$$+$$n^{\frac{-1}{2}} \sum_{i=1}^{n}\frac{\pi(t)}{G(t)}\int_{0}^{\infty}\frac{1}{\pi(u)}dM_{i}^{c}(u)]$,(4)

where varioius kind of quantities in (4) are as follows,$a(t)$\^a (t)$=$$\lim_{narrow\infty}$$=$$\lim_{narrow\infty}\frac{-1}{n}\sum_{i=1}^{n}f(\hat{h}_{0}(t)+Z_{i}^{T}\hat{\beta})$$=$$\lim_{narrow\infty}\hat{b}(t)$$=$$\lim_{narrow\infty}\frac{-1}{n}\sum_{i=1}^{n}f(\hat{h}_{0}(t)+Z_{i}^{T}\hat{\beta})Z_{i}$$=$$\lim_{narrow\infty}\hat{D}^{-1}$$=$$\lim_{narrow\infty}\frac{1}{n^{2}}\sum_{i=1j}^{n}\sum_{=1}^{n}w(Z_{ij}^{T}\beta_{0})\xi^{l}(Z_{ij}^{T}\beta_{0})Z_{ij}Z_{ij}^{T}$$=$$\lim_{narrow\infty}\hat{\pi}(t)$$=$$\lim_{narrow\infty}\frac{1}{n}\sum_{i=1}^{n}I(X_{i}\geq t)$$=$$\lim_{narrow\infty}\hat{q}(t)$$=$$\lim_{narrow\infty}\frac{1}{n^{2}}\sum_{i=1j}^{n}\sum_{=1}^{n}w(Z_{ij}^{T}\beta_{0})Z_{ij}\frac{\delta_{j}I(X_{i}\geq X_{j})}{\hat{G}^{2}(t)}I(X_{j}\geq t)$$e_{ij}(\beta_{0})$$=$$\frac{\delta_{j}I(X_{i}\geq X_{j})}{\hat{G}^{2}(t)}-\xi(Z_{ij}^{T}\beta_{0})$$r_{i}(t)$$=$$\frac{I(X_{i}\geq t)}{G(t)}-S_{Z}.(t)$$M_{i}^{c}(t)$$=$$I(X_{i} \leq t, \Delta_{i}=0)-\int_{0}^{\infty}I(X_{i}\geq u)d\Lambda_{G}(u)$$b(t)$$D^{-1}$$\pi(t)$$q(t)$

,

,

,

,

,

,

,

,

is the cumulative hazard
is the density function of $F$ , the distribution function of andis the martingale corresponding to thefunction of the common censoring time. Note thatcounting process for the censoring time C. Then$f$$\epsilon$$\Lambda_{G}$$M_{i}^{c}(t)$$n^{-\frac{1}{2}}H(t, z)$$\simeq$$n^{-\frac{1}{2}} \sum_{l=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)dM_{l}(u)$$+$$n^{-1} \sum_{l=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)$104$\cross$$d[\tilde{g}\{h_{0}(u)+Z_{l}^{T}\beta_{0}\}W_{Z_{l}}(t)]$.

(5)

In appendix, the fifirst and second term of the r.h.s of (5) is proved to be asmptotically equivalent to
the zer0-mean Gaussian process. To evaluate the goodness-of-fifit objectively, the null distribution of
the goodness-of-fifit multi-parameter stochastic process is needed. Although, under the correct model
specification, the goodness-0f-fit stochastic process is asymptotically equivalent to the zer0-mean
Gaussian process, it is difficult to know the covariance structure of the Gaussian process analytically.
So we use the simulation techniques to approximate the null Gaussian process following to the idea
originally proposed by Lin, Wei and Ying(1993).
To approximate the null process, we defifine another stochastic process,$n^{-\frac{1}{2}}\hat{H}(t, z)$$=$$n^{-\frac{1}{2}}.

$+$$n^{-\frac{5}{2}} \sum_{l=1}^{n}\sum_{\dot{|}=1j}^{n}\sum_{=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)Z_{j}\dot{.}e_{j}^{\wedge}.\cdot(\hat{\beta})$\cdot\sum_{=1}^{n}I(Z\dot{.}\leq z)\hat{M}.\cdot(t)c_{:}$$\cross d[\tilde{g}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\{\hat{b}(u+)+\hat{a}(u+)Z_{l}\}^{T}\hat{D}]$$\cross \mathcal{L}_{l}$$+$$2n^{-\frac{3}{2}} \sum_{l=1}^{n}.\cdot\sum_{=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)\int_{0}^{\infty}\frac{\hat{q}(u)}{\hat{\pi}(u)}d\hat{M}_{\dot{1}}^{c}(u)$$\cross d[\tilde{g}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\{\hat{b}(u+)+\hat{a}(u+)Z_{l}\}^{T}\hat{D}]$$\cross \mathcal{L}_{l}$$+$$n^{-\frac{3}{2}} \sum_{l=1}^{n}.\cdot\sum_{=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)$$\cross d[\tilde{g}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\hat{r}\dot{.}(u+)]$$\cross \mathcal{L}_{l}$$+$$n^{-\frac{3}{2}} \sum_{l=1}^{n}.\cdot\sum_{=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)$$\cross d[\tilde{g}\{\hat{h}(u)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\frac{\hat{\pi}(u+)}{\hat{G}(u+)}\int_{0}^{u}\frac{1}{\hat{\pi}(x)}d\hat{M}^{c}.\cdot(x)]$(6)$\cross \mathcal{L}_{l}$where, $..n$ are the sequence of the standard nomal random variable independent of
the data, {( , , Z. )}. In appendix, it is shown that conditional on the data,is thezero-mean Gaussian process and converges to the limiting process ofasymptotically.
Note that conditional on the data, , $i=1,2$ , $..n$ are the only random components in.
In practice, arbitary numbers of realizations of the null process
can be easily simulated bygeneratingin computer. Comparing the observed cumlative martingale residuals
to the simulated null process, we can evaluate the goodness-0f-fit$\mathcal{L}_{:},i=1,2$$X\dot{.}$$\Delta_{:}$$n^{-\frac{1}{2}}\hat{H}(z,t)$$\cdot$$n^{-\frac{1}{2}}H(z,t)$$\hat{H}(z,t)$$\mathcal{L}_{:}$$\hat{H}(z,t)$$\{\mathcal{L}:,i=1,2, ..n\}\mathrm{s}$

105
2.2

An omnibus test

Similar to Lin, Wei and Ying(1993), an omnibus goodness-0f-fit test is defined as$H_{omn}$$\sup$$=$$|H(t, z)|$$z\in[-1,1]^{p},0\leq t\leq\tau$$=$$\sup$$| \sum I(Z_{i}\leq z)\hat{M}_{i}(t)|n$$z\in[-1,1]^{\mathrm{p}},0\leq t\leq\tau i=1$

Since the null distribution of $H(t, z)$ is zer0-mean Gaussian process, an remarkably large value ofcan be approximated by$H_{mn}$ suggests model misspecification. As the distribution of, $H_{mn}$ can be approximated by $\hat{H}_{mn}=\sup_{z,t}|\hat{H}(t, z)|$ where the supremum is takenover $z\in[-1,1]^{p}$ , $0\leq t\leq\tau$ . So the -value, $P\{H_{mn}\geq h_{mn}\}$ where $h_{mn}$ is a realization of $H_{mn}$ ,can be easily generated bycan be approximated by $P\{\hat{H}_{omn}\geq h_{omn}\}$ . The realizations ofsimulation.
$n^{-\frac{1}{2}}H(t, z)$$n^{-\frac{1}{2}}\hat{H}(t, z)$$\mathrm{p}$$\hat{H}_{mn}$2.3

Checking the misspecification of functional form of covariates

An omnibus goodness-0f-fit test may be a powerful guide to judge whether the fifitted model isappropriate or not. When the fitted model, however, seem to be not appropreate, it is desirable,
and Ying(1993), weto know what kind of model misspecification was made. Similar todefine one parameter stochastic process as$\mathrm{L}\mathrm{i}\mathrm{n}$$H^{(k)}(\tau, z)$$=$$\mathrm{W}\mathrm{e}\mathrm{i}$$\sum_{i=1}^{n}I(Z_{i}^{(k)}\leq z^{(k)})\hat{M}_{i}(\tau)$is the -th element of . This stochastic process is aspecial case ofwhere $z\in[-1,1]$ and$H(t, z)$ . So the sample path of $H^{(k)}(\tau, z)$ under null hypothesis(i.e. the fifitted model is correct) canbe easily obtained though simulation and be displayed graphically because the sample path is one20
dimensional function. Plotting the realization of $H^{(k)}(\tau, z)$ with some simulated realization
realizations) of null distribution, it may be evaluated graphicall how strange the obtained realization
is. Furthermore -value based on $\sup|H^{(k)}(\tau, z)|$ can be evaluated in the same way as the omnibus$Z_{i}^{(k)}$$\mathrm{k}$$Z_{i}$$\mathrm{s}$$\mathrm{p}$

test.

2.4

Checking the misspecification of link function

To test the link misspecification, we can use the special case of $H(t, z)$ setting$H(t, 1)= \sum_{i=1}^{n}\hat{M}_{i}(t)$$\mathrm{z}=1$.

.

can evaluate the goodness-of-fifit of the fifitted model graphically by ploting the
Similar to
obtained realization and the some(say 20) simulated realizations of the null process and subjectively
by evaluating the -value using the simulation technique$H_{f}^{(k)}(z),\mathrm{w}\mathrm{e}$$\mathrm{P}$106
3

Appendix

3.1

Weak convergence of$n^{-\frac{1}{2}}H(t,$z)

To show the weak convergence of, it is suffiffifficient to prove the weak convergence ofthe 1st and 2nd terms of
converge to the zer0-mean Gaussian process, where$n^{-\frac{1}{2}}H(t, z)$$n^{-\frac{1}{2}}\tilde{H}(t, z)$$n^{-\frac{1}{2}}\tilde{H}(t, z)$$\simeq$$+$$\cross$since$n^{-\frac{1}{2}}H(t, z)$and$n^{-\frac{1}{2}}\tilde{H}(t, z)$$n^{-\frac{1}{2}} \sum_{l=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)dM_{l}(u)$$n^{-1} \sum_{l=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)$$d[\tilde{g}\{h_{0}(u)+Z_{l}^{T}h\}W_{Z_{l}}(t)]$,(7)

are asymptotically equivalent.

Weak convergence of the 1st term of (7)thecentral limit theorem, the arbitary fifinite dimensional projection converge
to the Gaussim distribution. So to show the weak convergence to the zero-mean Gaussian procaesof, it is sufficient to show its tightness in $D([-1,1]^{p}\cross[0,\tau])$ . Using
the same argument as appendix 1of Spiekerman and, the covariates can be restricted tothe continuous ones without loss of generality. And for simplicity, the dimension of the covariates
is assumed unity. To show the tightness, it is sufficient to show the moment inequalities,$\mathrm{R}\mathrm{o}\mathrm{m}$$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{r}\cdot \mathrm{a}\mathrm{t}\mathrm{e}$$n^{-\frac{1}{2}} \sum_{=1}^{n}.\cdot\int_{0}^{t}I(Z.\cdot\leq z)dM.\cdot(u)$$\mathrm{L}\mathrm{i}\mathrm{n}(1996)$$E[\psi^{2}(z_{1}, z_{2;}t_{1},t)\psi^{2}(z_{1}, z_{2;}t, t_{2})]$$\leq$$K(t-t_{1})(t_{2}-t)Pr^{2}\{Z\in[z_{1}, z_{2})\}$$E[\psi^{2}(z_{1}, z;t_{1}, t_{2})\psi^{2}(z, z_{2};t_{1},t_{2})]$$\leq$$K(t_{2}-t_{1})^{2}$$\cross$for $\forall z_{1}<z<z_{2}$ , $t_{1}<t<t2$ , where$\mathrm{K}$$Pr\{Z\in[z_{1}, z)\}Pr\{Z\in[z, z_{2})\}$

(8)

(9)

is some constant and$\psi(z_{1}, z_{2};t_{1}, t_{2})=n^{-\frac{1}{2}}\sum_{\dot{|}=1}^{n}\int_{t_{1}}^{t_{2}}dM_{}(u)I(Z\dot{.}\in[z_{1}, z_{2}))$.

It is suffiffifficient to the moment inequarities (8) and (9) under the assumption that $Pr \{Z\in[z_{1}, z)\}\geq\frac{1}{n}$and $Pr \{Z\in[z, z_{2})\}\geq\frac{1}{n}$ (Bickel and Wichura(1971),, Wei and Ying(1993)).
$\mathrm{L}\mathrm{i}\mathrm{n}$

l.s.h

of

(8)

$=$

$E[\psi^{2}(z_{1}, z_{2}; t_{1}, t)E[\psi^{2}(z_{1}, z_{2};t, t_{2})|Z_{\dot{1}}$$=$$E[\psi^{2}(z_{1}, z_{2}; t_{1}, t)\cross$, $i=1,2\ldots n$ ,$\psi^{2}(z_{1}, z_{2;}t_{1},t)]]$$\frac{1}{n}\sum_{\dot{|}=1}^{n}E[\int_{t}^{t_{2}}\mathrm{Y}.\cdot(u)dlog(g\{h_{0}(u)+Z^{T}.\cdot h\}I(Z\dot{.}\in[z_{1}, z))|Z.\cdot,$ $\psi^{2}(z_{1}, z_{2;}t_{1},t)]]$$=$$E[ \psi^{2}(z_{1}, z_{2;}t_{1}, t)\frac{1}{n}\dot{.}\sum_{=1}^{n}\int_{t}^{t_{2}}\mathrm{Y}\dot{.}(u)dlog(g\{h_{0}(u)+Z_{\dot{1}}^{T}\beta_{0}\}I(Z_{\dot{1}} \in[z_{1}, z))]$107
The 2nd equality is obtained by the independent increament property of the martingale and thestandard moment caluculation of the counting process martingale. Furthermore$\leq$$KE[ \psi^{2}(z_{1}, z_{2}; t_{1}, t)\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\in[z_{1}, z))(t2-t)]$$=$$KE[ \frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\in[z_{1}, z))\frac{1}{n}\sum_{j=1}^{n}\int_{t_{1}}^{t}\mathrm{Y}_{i}(u)dlog(g\{h_{0}(u)+Z_{j}^{T}\beta_{0}\}I(Z_{j}\in[z_{1}, z)](t_{2}-t)$$\leq$$K(t2-t)(t-t1) \{\frac{n(n-1)}{n^{2}}Pr^{2}(Z\in[z_{1}, z))+\frac{1}{n}Pr^{2}(Z\in[z_{1}, z))\}$$\leq$$K(t2-t)(t-t1) \{\frac{n(n-1)}{n^{2}}Pr^{2}(Z\in[z_{1}, z))+Pr^{2}(Z\in[z_{1}, z))\}$$\leq$$2K(t2-t)(t-t1)Pr^{2}(Z\in[z_{1}, z))$So the moment inequility (8) is obtained. Next we show thel.s.h$2\mathrm{n}\mathrm{d}$moment inequility (9).

of

(9)

$=$

$E[E[\psi^{2}(z_{1}, z;t_{1}, t2)\psi^{2}(z, z_{1}; t_{12}, t)|Z_{i}, i=1,2, \ldots n]]$$=$$\frac{1}{n^{2}}\sum_{i\neq j}E[\{\int_{t_{1}}^{t_{2}}dM_{i}(u)\}^{2}\{\int_{t_{1}}^{t_{2}}dM_{j}(u)\}^{2}I(Z_{i}\in[z_{1}, z),$$\leq$$K \frac{1}{n^{2}}n(n-1)(t2-t_{1})^{2}Pr\{Z_{i}\in[z_{1}, z), Zj\in[z, z_{2})\}$$\leq$$K(t_{2}-t_{1})^{2}Pr\{Z_{i}\in[z_{1}, z)\}Pr\{Zj\in[z, z_{2})\}$$Z_{j}\in[z, z_{2}))]$

So the moment inequility (9) is also obtained.

Weak convergence of the 2nd term of (7)Using the integration by part,the $2nd$ term

Here $Wz_{i}(t)$ is$\mathrm{r}$

of

(7)

$=$

$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\int_{0}^{t}\mathrm{Y}_{i}(u)\tilde{g}\{h_{0}(u)+Z_{i}^{T}\beta\}dW_{Z}\dot{.}(u)$

(10)

$+$

$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\int_{0}^{t}\mathrm{Y}_{i}(u)Wz.\cdot(u)d\tilde{g}\{h_{0}(u)+Z_{i}^{T}\beta\}$(11)-expressed as,$Wz \dot{.}(t)=\frac{1}{a(t)}[(b(t)+a(t)Z_{i})^{T}D(W_{1}+W_{2})+W_{3}(t)+W_{4}(t)]$where$W_{1}$$=$$n^{-\frac{3}{2}} \sum_{i=1j}^{n}\sum_{=1}^{n}w(Z_{ij}^{T}\beta_{0})Z_{ij}e_{ij}(\beta_{0})$$W_{2}$$=$$2n^{-\frac{1}{2}} \sum_{i=1}^{n}\int_{0}^{\infty}\frac{q(u)}{\pi(u)}dM_{i}^{c}(u)$108$W_{3}(t)$$=$$n^{-\frac{1}{2}}.

$W_{4}(t)$$=$$n^{-\frac{1}{2}} \dot{.}\sum_{=1}^{n}\frac{\pi(t)}{G(t)}\int_{0}^{t}\frac{1}{\pi(u)}dM^{c}\dot{.}(u)$\cdot\sum_{=1}^{n}r:(t)$,,Cheng, Wei and Ying(1997) proved thatandare asymptotically equivalent
to the zero-mean Gaussian process(appendix of Cheng, Wei and Ying(1997)).
To show the weak convergence of (10) to the zero-mean Gaussian
prepare alemma.
$W_{1}$$W_{3}(t)$$W_{2}$$W_{4}(t)$$\mathrm{B}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s},\mathrm{w}\mathrm{e}$Lemma 3.1Let
be asequences of bounded variation functions on [0, and $\{g_{n}(z,t)\}$ be a sequence ofbounded variateion with respect to t in [0, for each z in [-1,1] such that$\{f_{n}(t)\}$$\tau]$$\tau]$

1. $\sup_{0\leq t\leq\tau}|f_{n}(t)-f_{\infty}(t)|arrow 0$, where,is continuous on
2. $\sup_{0\leq t\leq\tau,-1\leq z\leq 1}|g_{n}(z,t)-g_{\infty}(z,t)|arrow \mathrm{O},where$
is bounded on [-1, 1]
bounded variation with respect to for each .
$f_{\infty}(t)$$[0, \tau]$$g_{\infty}(z,t)$$t$$\cross[0,\tau]$with$\mathrm{z}$Then$\sup_{0\leq t\leq\tau,-1\leq z\leq 1}|\int_{0}^{t}f_{n}(u)dg_{n}(z,u)-\int_{0}^{t}f_{\infty}(u)dg_{\infty}(z,u)|arrow \mathrm{O}$

.

(12)

Proof
(12) is asimple extension of lemma A.3 of Bilias, Gu and Ying (1997). (Q.E.D)
Here we show the weak convergence of (10). Using the integration by part,
(10)

$=$

$\frac{1}{n}\sum_{\dot{|}=1}^{n}I(Z_{\dot{1}} \leq z)\int_{0}^{t}\mathrm{Y}.\cdot(u)\tilde{g}\{h_{0}(u)+Z_{\dot{1}}^{T}\beta\}Z_{\dot{1}}^{T}$$+$$\frac{1}{n}\sum_{\dot{|}=1}^{n}I(Z\dot{.}\leq z)\int_{0}^{t}\mathrm{Y}.\cdot(u)\tilde{g}\{h_{0}(u)+Z^{T}.\cdot\beta\}$$\cross$$D(W_{1}+W_{2})d \frac{a(u)}{a(u)}$$d[ \frac{b^{T}(u)}{a(u)}D(W_{1}+W_{2})+\frac{1}{a(u)}(W_{3}(u)+W_{4}(u))]$$=$$\frac{1}{n}\sum_{i=1}^{n}I(Z\dot{.}\leq z)\int_{0}^{t}\mathrm{Y}\dot{.}(u)\tilde{g}\{h\mathrm{o}(u)+Z_{\dot{1}}^{T}\beta\}Z^{T}.\cdot D(W_{1}+W_{2})d\frac{a(u)}{a(u)}$$+$$\int_{0}^{t}d[\frac{1}{n}\dot{.}\sum_{=1}^{n}I(Z.\cdot\leq z)\mathrm{Y}.\cdot(u)\tilde{g}\{h_{0}(u)+Z^{T}\dot{.}\beta\}\frac{b^{T}(u)}{a(u)}D(W_{1}+W_{2})+\frac{1}{a(u)}(W_{3}(u)+W_{4}(u))]$$\int_{0}^{t}\frac{b^{T}(u)}{a(u)}D(W_{1}+W_{2})+\frac{1}{a(u)}(W_{3}(u)+W_{4}(u))$$\cross$(13)$d[ \frac{1}{n}\dot{.}\sum_{=1}^{n}I(Z_{\dot{1}} \leq z)\mathrm{Y}_{\dot{1}}(u)\tilde{g}\{h_{0}(u)+Z_{\dot{1}}^{T}\beta\}]$,,Note that the 1st term of (13) is zero. Letandbe the limiting zer0-mean,,
Gaussian process ofrespectively whose weak convergences were provedandin the appendix of Cheng, Wei and Ying (1997). Note that there exist nondecreasing functions$\mathcal{W}_{1}$$W_{1}$$\mathrm{B}$$W_{2}$$W_{3}(t)$$W_{4}(t)$$\mathcal{W}_{2}$$\mathcal{W}_{3}(t)$$\mathcal{W}_{4}(t)$109
and $I(Z_{i}\leq z)$. Sincesuch thatandare monotone functions on , their psedo dimensions are unity. From lemma 5.3 and theorem 4.8
is manageable (Pollard 1990, . 38). Similarly, $I(Z_{i}\leq$of Pollard(1990),
is also manageable(theis also manageable. Thenlemma A2 of Bilias, Gu and Ying(1997) . So by the uniform law of large numbers(Pollard(1990),
converges uniformly to some nonrandom functionp.41),, which is proved inandofalmost surely. Since this fact and the continuity onterm of (13) isthe later part of this appendix, ensure the conditions of the lemma 3.1, theasmptotically equivalent to the zero-mean Gaussian process. The 2nd term of (13) converges to aW4(u) duconverges uniformly to somemean Gaussian process sincenonrandom function almost surely.

$\tilde{g}^{+}(u)$$\tilde{g}\{h_{0}(u)+Z_{i}^{T}\beta\}=\tilde{g}^{+}(u)-\tilde{g}^{-}(u)$$\tilde{g}^{-}(u)$$\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}-\mathrm{Y}_{i}(u)\tilde{g}^{+}(u)$$\mathrm{R}$$I(Z_{i}\leq z)\mathrm{Y}_{i}(u)\tilde{g}^{+}(u)$$\mathrm{p}$$I(Z_{i}\leq z)\mathrm{Y}_{i}(u)\tilde{g}\{h_{0}(u)+Z_{i}^{T}\beta\}$$z)\mathrm{Y}_{i}(u)\tilde{g}^{-}(u)$$)$$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\mathrm{Y}_{i}(u)\tilde{g}\{h_{0}(u)+Z_{i}^{T}\beta\}$$[0, \tau]$$\mathcal{W}_{3}(t)$$\mathcal{W}_{4}(t)$$3\mathrm{r}\mathrm{d}$$\int_{0}^{t}[\frac{b^{T}(u)}{a(u)}D(\mathcal{W}_{1}+\mathcal{W}_{2})+\frac{1}{a(u)}(\mathcal{W}_{3}(u)+$$\mathrm{z}\mathrm{e}\mathrm{r}(\succ$$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq \mathrm{z})\mathrm{Y}\mathrm{i}(\mathrm{u})\mathrm{g}\{\mathrm{h}\mathrm{o}\{\mathrm{X}\mathrm{i})+Z_{i}^{T}\beta\}$$)$$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\mathrm{Y}_{i}(u)\tilde{g}\{h_{0}(u)+Z_{i}^{T}\beta\}$

Next we show the weak convergence of (11).
(11)

$=$

$\cross$$+$$\cross$$\frac{1}{n}\sum_{\dot{0}=1}^{n}I(Z_{i}\leq z)\int_{0}^{t}\mathrm{Y}_{i}(u)\frac{1}{a(u)}\{b(u)+a(u)Z_{i}\}^{T}h_{\acute{0}}(u)\tilde{g}’\{h_{0}(u)+Z_{\dot{l}}^{T}\beta\}du$(14)$D(W_{1}+W_{2})$$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\int_{0}^{t}\mathrm{Y}_{i}(u)\frac{1}{a(u)}\{W_{3}(u)+W_{4}(u)\}$(15)$h_{\acute{0}}(u)\tilde{g}’\{h_{0}(u)+Z_{i}^{T}\beta\}du$Using the similar argument to that for the 1st term of (10), by the uniform law of large number(Pollard(1990), . 41),
converges uniformly to some non-random function almost surely. And $D(W_{1}+W_{2})$ converge
to Gaussian random variable. So (14) is tight and converges weakly to some Gauusian process. Furthermore, it can be shown by the similar argument to that for the 1st term (10) thatconverges uniformaly to some non-random function almost surely by the uniform law of large number. Because of the weak convergence of$W_{3}(u)+W_{4}(u)$ to
and the almost sure representation theorem(P011ard(1990), .
uniformly and almost surely in some proba45), $W_{3}(u)+W_{4}(u)$ can converges tobility space. So$\mathrm{p}$$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\int_{0}^{t}\mathrm{Y}_{i}(u)\frac{1}{a(u)}\{b(u)+a(u)Z_{i}\}^{T}h_{\acute{0}}(u)\tilde{g}’\{h_{0}(u)+Z_{i}^{T}\beta\}du$$\frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\mathrm{Y}_{i}(u)\frac{1}{a(u)}h_{\acute{0}}(u)\tilde{g}’\{h_{0}(u)+Z_{i}^{T}\beta\}$$\mathcal{W}_{3}(t)+\mathcal{W}_{4}(t)$$\mathrm{p}$$\mathcal{W}_{3}(u)+\mathcal{W}_{4}(u)$$\sup_{0\leq t\leq\tau,-1\leq z\leq 1}|\int_{0}^{t}f_{n}(z,u)(W_{3}(u)+W_{4}(u))du-\int_{0}^{t}f_{\infty}(z,u)(\mathcal{W}_{3}(u)+\mathcal{W}_{4}(u))du|$$\leq$$\sup_{0\leq t\leq\tau,-1\leq z\leq 1}|f_{n}(z, u)(W_{3}(u)+W_{4}(u))-f_{\infty}(z, u)(\mathcal{W}_{3}(u)+\mathcal{W}_{4}(u))|\tau$,

and $f_{\infty}= \lim f_{n}(u)$ . Again
where
applying the almost sure presentation theorem, this means that (13) converges weakly to thein the original probability space.
Gaussian process$f_{n}(z, u)= \frac{1}{n}\sum_{i=1}^{n}I(Z_{i}\leq z)\mathrm{Y}_{i}(u)\frac{1}{a(u)}h_{0}’(u)\tilde{g}’\{h_{0}(u)+Z_{i}^{T}\beta\}$$\int f_{\infty}(z, u)(\mathcal{W}_{3}(u)+\mathcal{W}_{4}(u))du$110
Continuity ofand$\mathcal{W}_{3}(t)$It is suffiffifficient to show
simple algebra,$\mathcal{W}_{4}(t)$$E|\mathcal{W}_{3}(t)-\mathcal{W}_{3}(s)|^{4}\leq K|t-s|^{2}$$E|W_{3}(t)-W_{3}(s)|^{4}$by the Kolmogorov-Centsov theorem.By$=$$E[ \frac{1}{n^{2}}|\sum_{\dot{|}=1}^{n}(r:(t)-r:(s))|^{4}]$$=$$\frac{1}{n}E|r_{1}(t)-r_{1}(s)|^{4}$$+$$\frac{n^{2}-n}{n^{2}}E|r_{1}(t)-r_{1}(s)|^{2}E|r_{2}(t)-r_{2}(s)|^{2}$

(16)

, the
Because of the uniform boundedness ofterm of (16) is dominated by
where$E|r_{1}(t)-r_{1}(s)|^{2}\leq K^{\frac{1}{2}}|t-s|$
is some constant. From the simple algebra, one can show
. So
$E|W_{3}(t)-W_{3}(s)|^{4} \leq\frac{K}{n}+(1-\frac{1}{n})K|t-s|^{2}$ is obtained.Using Fatou’s lemma, the aimed inequality,, is obtained.
, it is sufficient to prove
Similar toto show the continuityof.
$r_{1}(t)$$\mathrm{l}\mathrm{s}\mathrm{t}$$\frac{K}{n}$$\mathrm{K}$$E|\mathcal{W}_{3}(t)-\mathcal{W}_{3}(s)|^{4}\leq K|t-s|^{2}$$E|\mathcal{W}_{4}(t)-\mathcal{W}_{3}(4)|^{4}\leq K|t-s|^{2}$$\mathcal{W}_{3}(t)$$\mathcal{W}_{4}(t)$$E|W_{4}(t)-W_{4}(s)|^{4}$$=$$E[ \frac{1}{n^{2}}|\dot{.}\sum_{=1}^{n}(\frac{\pi(t)}{G(t)}\int_{0}^{t}\frac{1}{\pi(u)}dM_{}^{c}(u)-\frac{\pi(s)}{G(s)}\int_{0}^{s}\frac{1}{\pi(u)}dM_{}^{c}(u))|^{4}$$=$$E[ \frac{1}{n^{2}}\sum_{=1}^{n}(\frac{\pi(t)}{G(t)})^{4}|\int_{s}^{t}\frac{1}{\pi(u)}dM^{c}\dot{.}(u)|^{4}$$+$$\frac{2}{n^{2}}\dot{.}\sum_{\neq j}(\frac{\pi(t)}{G(t)})^{2}(\frac{\pi(t)}{G(t)}-\frac{\pi(s)}{G(s)})^{2}$$\cross$(16)$| \int_{s}^{t}\frac{1}{\pi(u)}dM_{\dot{1}}(u)|^{2}|\int_{0}^{s}\frac{1}{\pi(u)}dM_{\dot{1}}(u)|^{2}]$The 1st term and 2nd terms are bounded by and $(1- \frac{1}{n})K|t-s|^{2}$ respectively. Rom the Fatou’s
lemma,
is obtained.
$\frac{K}{n}$$E|\mathcal{W}_{4}(t)-\mathcal{W}_{4}(s)|^{4}\leq K|t-s|^{2}$

3.2

Weak convergence of$n^{-\frac{1}{2}}\hat{H}(t, z)$conditional on the data,
Here we show that conditional on the data
converges weakly to theunconditional limiting Gaussian process ofwhen the fifitted model is correctly specified.
It is straightforward to show the covariance function of
converges almost surely to thatof the limiting Gaussian process of. So the finite dimensional conditional distributionof theconverges to that of he limiting Gaussian process of. To show the
weak convergence, it is sufficient to show the tightness of the each term of (6). To this end, it issufficient to show the moment inequalities(Bickel and Wichura(1971)). Define$\{X_{\dot{1}}, \Delta:, Z\dot{.}\}$$n^{-\frac{1}{2}}\hat{H}(t, z)$$n^{-\frac{1}{2}}\tilde{H}(t, z)$$n^{-\frac{1}{2}}\hat{H}(t,z)$$n^{-\frac{1}{2}}\tilde{H}(t, z)$$n^{-\frac{1}{2}}\tilde{H}(t, z)$$n^{-\frac{1}{2}}\hat{H}(t, z)$$\Psi_{l}^{(1)}(z_{1}, z_{2} :t_{1},t_{2})$$=$$\Psi_{+,l}^{(2)}(z_{1}, z_{2} :t_{1},t_{2})$$=$$\{\hat{M}_{l}(t_{2})-\hat{M}_{l}(t_{1})\}I(z_{1}\leq Z_{l}\leq z_{2})$$. \cdot\sum_{=1j}^{n}\sum_{=1}^{n}\int_{t_{1}}^{t_{2}}\mathrm{Y}_{l}(u)Z_{j}\dot{.}\hat{e}_{j}\dot{.}(\hat{\beta})$

111$\mathrm{x}d[\tilde{g}^{+}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\{\hat{b}(u+)+\hat{a}(u+)Z_{l}\}^{T}\hat{D}]$$\cross I(z_{1}\leq Z_{l}\leq z_{2})$$\Psi_{+,l}^{(3)}(z_{1}, z_{2} :t_{1},t_{2})$$=$$2n^{-\frac{3}{2}} \sum_{i=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)\int_{0}^{\infty}\frac{\hat{q}(u)}{\hat{\pi}(u)}d\hat{M}_{i}^{c}(u)$$\cross d[\tilde{g}^{+}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\{\hat{b}(u+)+\hat{a}(u+)Z_{l}\}^{T}\hat{D}]$$\cross I(z_{1}\leq Z_{l}\leq z_{2})$$\Psi_{+,l}^{(4)}(z_{1}, z_{2} :t_{1},t_{2})$$=$$n^{-\frac{3}{2}} \sum_{i=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)$$\cross d[\tilde{g}^{+}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\hat{r}_{i}(u+)]$$\cross I(z_{1}\leq Z_{l}\leq z_{2})$$\Psi_{+,l}^{(5)}(z_{1}, z_{2} :t_{1},t_{2})$$=$$n^{-\frac{3}{2}} \sum_{i=1}^{n}\int_{0}^{t}I(Z_{l}\leq z)\mathrm{Y}_{l}(u)$$\cross d[\tilde{g}^{+}\{\hat{h}(u)+Z_{l}^{T}\hat{\beta}\}\frac{1}{\hat{a}(u+)}\frac{\hat{\pi}(u+)}{\hat{G}(u+)}\int_{0}^{u}\frac{1}{\hat{\pi}(x)}d\hat{M}_{i}^{c}(x)]$$\cross I(z_{1}\leq Z_{l}\leq z_{2})$

(18)

where

$\Psi_{-,l}^{(k)}(z_{1}, z_{2} :$Z_{l}^{T}\hat{\beta}\}$.
are nondecreasing functions such that
and$g\{\hat{h}(u+)+$andare defined in the similar matter. By the boundedness of$\tilde{g}^{+}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}$ont_{1},t2)$$[0, \tau]$$\tilde{g}=\tilde{g}^{+}-\tilde{g}^{-}$$\tilde{g}^{-}\{\hat{h}(u+)+Z_{l}^{T}\hat{\beta}\}$$M_{l}(t)$,$\{\Psi_{i}^{(1)}(z_{1}, z_{2} :t_{1}, t)\}^{2}\{\Psi_{j}^{(1)}(z_{1}, z_{2} :t,t_{2})\}^{2}$$\leq$$\cross$$+$$(M_{i}(t_{2})-M_{i}(t))^{2}(M_{j}(t)-M_{j}(t_{1}))^{2}$$I(z_{1}\leq Z_{i}\leq z_{2})I(z_{1}\leq Z_{j}\leq z_{2})$.

$L_{11}|t_{2}-t||t-t_{1}|I(z_{1}\leq Z_{i}\leq z_{2})I(z_{1}\leq Z_{j}\leq z_{2})$(19)
is some constant. Noting that conditional on the data,whereof each terms of (6), for $\forall z_{1}<z<z_{2},t_{1}<t<t_{2}$ ,$L_{11}$$E|n^{-\frac{1}{2}} \sum_{l=1}^{n}\Psi_{l}^{(1)}(z_{1}, z_{2} :

$=$

$+$

$\cross$$\leq$

t_{1},t)\mathcal{L}_{l}|^{2}|n^{-\frac{1}{2}}\sum_{l=1}^{n}\Psi_{l}^{(1)}(z_{1}, z_{2} :$\frac{1}{n^{2}}\sum_{i=1j}^{n}\sum_{=1}^{n}\Psi_{i}^{(1)}(z_{1}, z_{2} :t_{1}, t)^{2}\Psi_{j}^{(1)}(z_{1}, z_{2} :$\frac{4}{n^{2}}\sum_{i_{1}<i_{2},i_{3}<i_{4}}\Psi_{i_{1}}^{(1)}(z_{1}, z_{2} :$\Psi_{i_{3}}^{(1)}(z_{1}, z_{2} :t, t_{2})\Psi_{i_{4}}^{(1)}(z_{1}, z_{2} :$\{\mathcal{L}_{l}\}$is the only random elementst,t_{2})\mathcal{L}_{l}|^{2}$t, t_{2})^{2}E[\mathcal{L}_{i}^{2}\mathcal{L}_{j}^{2}]$t_{1},t)\Psi_{i_{2}}^{(1)}(z_{1}, z_{2} :t_{1},t)$t, t_{2})E[\mathcal{L}_{i_{1}}\mathcal{L}_{i_{2}}\mathcal{L}_{j_{1}}\mathcal{L}_{j_{2}}]$$\frac{3}{n^{2}}\sum_{i=1j}^{n}\sum_{=1}^{n}(M_{i}(t_{2})-M_{i}(t))^{2}(M_{j}(t)-M_{j}(t_{1}))^{2}I(z_{1}\leq Z_{i}\leq z_{2})I(z_{1}\leq Z_{j}\leq z_{2})$

112
$+$

,

$3L_{11}|t_{2}-t||t-t_{1}| \frac{1}{n}\sum_{i=1}^{n}I(z_{1}\leq Z\dot{.}\leq z_{2})\frac{1}{n}\sum_{j=1}^{n}I(z_{1}\leq Z_{j}\leq z_{2})$(20)

where the last inequality holds because of (19) and
if $i_{1}<i_{2},j_{1}<j_{2}$ . Since
the 1st term vanishes as tends toby the independent incremant property of martingale, themoment inequality was obtained (theorem 3 and p.1666 of Bickel and Wichura (1971)). Similarly$E[c_{:_{1}}c_{:_{2}}\mathcal{L}_{j_{1}}\mathcal{L}_{j_{2}}]=0$$\infty$$\mathrm{n}$$E|n^{-\frac{1}{2}} \sum_{l=1}^{n}\Psi_{+,l}^{(1)}(z_{1}, z :$\leq$$+$

t_{1},t_{2})\mathcal{L}_{l}|^{2}|n^{-\frac{1}{2}}\sum_{l=1}^{n}\Psi_{+,l}^{(1)}(z, z_{2} :t,t_{2})\mathcal{L}_{l1^{2}}$$\frac{1}{n^{2}}.\cdot\sum_{=1j}^{n}\sum_{=1}^{n}(M\dot{.}(t_{2})-M\dot{.}(t_{1}))^{2}(M_{j}(t_{2})-M_{j}(t_{1}))^{2}I(z_{1}\leq Z_{\dot{1}} \leq z)I(z\leq Z_{j}\leq z_{2})$$3L_{11}|t_{2}-t_{1}|^{2} \frac{1}{n}\sum_{\dot{|}=1}^{n}I(z_{1}\leq Z.\cdot \leq z)\frac{1}{n}.\cdot\sum_{=1}^{n}I(z\leq Z_{j}\leq z_{2})$

,

(21)

can be obtained. Since the 1st term converges to$\{E[(\int_{t_{1}}^{t_{2}}dM_{1}(u))^{2}I(z_{1}\leq Z_{1}\leq z)]\}^{2}$$= \{E[I(z_{1}\leq Z_{1}\leq z)E[(\int_{t_{1}}^{t_{2}}dM_{1}(u))^{2}|Z_{1}]]\}^{2}$$\leq-\{E[I(z_{1}\leq Z_{1}\leq z)\int_{t_{1}}^{t_{2}}\mathrm{Y}_{1}(u)d\log(g\{h_{0}(u)+Z_{1}^{T}ffi\})]\}^{2}$

Then the 1st term of (21) convergaae to smoe meaeure whose marginal is continuous and thereby
tightness is obtained (Bickel and Wichura 1971 the0rem3 and p.1666).
Since
,
,
and
are bounded, for,$\mathrm{Y}_{l}(u)$$Z_{j}.\cdot,\hat{e}_{j}.\cdot(\hat{\beta})$$\frac{1}{\hat{a}(u+)}\{\hat{b}(u+)+\hat{a}(u+)Z_{l}\}^{T}\hat{D},\hat{q}(u),\hat{\pi}(u),\hat{M}^{c}.\cdot(u)$$\frac{1}{G(u+)}$$\mathrm{k}=2,3,4,5$$|\Psi_{+,l}^{(k)}(z_{1}, z_{2} :t_{1},t_{2})|$$\leq$$\cross$$\leq$$K_{k}|\tilde{g}^{+}\{\hat{h}(t_{2}+)+Z_{l}^{T}\hat{\beta}\}-\tilde{g}^{+}\{\hat{h}(t_{1}+)+Z_{l}^{T}\hat{\beta}\}|$$I(z_{1}\leq Z_{l}\leq z_{2})$

,

(22)

$\tilde{K}_{k}|\hat{h}(t2)-\hat{h}(t_{1})|I(z_{1}\leq Z_{l}\leq z_{2})$

where
and
are some constants. Noting that conditional on the data,
elements of each terms of 6, for $\forall z_{1}<z<z_{2}$ , $t_{1}<t<t_{2}$ ,$K_{k}$$\tilde{K}_{k}$$E|n^{-\frac{1}{2}} \sum_{l=1}^{n}\Psi_{+,l}^{(k)}(z_{1}, z_{2} :$\leq$$+$$\cross$$\leq$

t_{1},t)\mathcal{L}_{l}|^{2}|n^{-\frac{1}{2}}\sum_{l=1}^{n}\Psi_{+,l}^{(k)}(z_{1},z_{2} :$\frac{1}{n^{2}}\sum_{\dot{|}=1j}^{n}\sum_{=1}^{n}\Psi_{+,}^{(k)}(z_{1}, z_{2} :.

t_{1},t)^{2}\Psi_{+i}^{(k)}(z_{1}, z_{2} :$\frac{14}{n^{2}n^{2}}\dot{.}\sum_{4},\cdot.\Psi_{+,11}^{(k)}(z_{1}, z_{2} :$\Psi_{+_{3}}^{(k)},.\cdot(z_{1}, z_{2} :is the only randomt,t_{2})\mathcal{L}_{l}|^{2}$t,t_{2})^{2}E[\mathcal{L}_{\dot{1}}^{2}\mathcal{L}_{j}^{2}]$t_{1},t)\Psi_{+,_{2}}^{(k)}(z_{1}, z_{2} :t,t_{2})\Psi_{+,\dot{1}4}^{(k)}(z_{1}, z_{2} :$\{\mathcal{L}_{l}\}$(23)t_{1},t):_{1}<_{213}<$t,t_{2})E[c_{:_{1}}c_{:_{2}}\mathcal{L}_{j_{1}}\mathcal{L}_{j_{2}}]$$L_{k1}| \hat{h}(t)-\hat{h}(t1)|^{2}|\hat{h}(t2)-\hat{h}(t)|^{2}\frac{1}{n}\dot{.}\sum_{=1}^{n}I(z_{1}\leq Z\dot{.}\leq z_{2})$

,

(20)

113
where
if

is some constant and the last inequality holds because of (22) and. Similarly,
and$L_{k1}$$E[\mathcal{L}_{i_{1}}\mathcal{L}_{i_{2}}\mathcal{L}j_{1}\mathcal{L}j_{2}]=0$$j_{1}\neq j_{2}$$i_{1}\neq i_{2}$$E|n^{-\frac{1}{2}} \sum_{l=1}^{n}\Psi_{+,l}^{(k)}(z_{1}, z :t_{1}, t_{2})\mathcal{L}_{l}|^{2}|n^{-\frac{1}{2}}\sum_{l=1}^{n}\Psi_{+,l}^{(k)}(z, z_{2} :t, t_{2})\mathcal{L}_{l}|^{2}$$L_{k2}| \hat{h}(t_{2})-\hat{h}(t_{1})|^{2}\frac{1}{n}\sum_{i=1}^{n}I(z_{1}\leq Z_{i}\leq z)\frac{1}{n}\sum_{i=1}^{n}I(z\leq Z_{i}\leq z_{2})$$\leq$

From (24) and (25), the tightness of the 2nd-5th term of (6) is obtained (Bickel and Wichura(1971) . The moment inequalities forsame matter.
$\Psi_{-,l}^{(k)}(z_{1}, z_{2} :

$)$

.

(25)

and p.1666 of
are obtained in the$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}3$t_{1},t2)$

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