Modern Stochastics: Theory and Applications

Survival analysis models time to an event of interest (e.g., lifetime). It is a powerful tool in biometrics, epidemiology, engineering, and credit risk assessment in financial institutions. The proportional hazards model proposed in Cox (1972) [3] is a widely used technique to characterize a relation between survival time and covariates.

Our model is presented in Augustin (2004) [1] where the baseline hazard function $\lambda (\cdot )$ is assumed to belong to a parametric space, while we consider $\lambda (\cdot )$ belonging to a closed convex subset of $C[0,\tau ]$. In practice covariates are often contaminated by errors, so we deal with errors-in-variables model. Kukush et al. (2011) [5] derive a simultaneous estimator of the baseline hazard rate $\lambda (\cdot )$ and the regression parameter β and prove the consistency of the estimator. At that, the parameter set $\varTheta _{\lambda }$ for the baseline hazard rate is assumed to be bounded and separated away from zero. The asymptotic normality of the estimator is shown in Chimisov and Kukush (2014) [2]. In [7, 6] we construct an estimator $({\hat{\lambda }_{n}^{(1)}}(\cdot ),\hspace{2.5pt}{\hat{\beta }_{n}^{(1)}})$ of $\lambda (\cdot )$ and β over the parameter set $\varTheta =\varTheta _{\lambda }\times \varTheta _{\beta }\hspace{2.5pt}$, where n is the sample size and $\varTheta _{\lambda }$ is a subset of $C[0,\tau ]$, which is unbounded from above and not separated away from zero. The estimator is consistent and can be modified to be asymptotically normal.

The goal of present paper is to construct confidence intervals for integral functionals of $\lambda (\cdot )$ and a confidence region for β based on the estimators from [7, 6]. We impose certain restrictions on the error distribution. Actually we handle three cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.

The paper is organized as follows. Section 2 describes the observation model, gives main assumptions, defines an estimator under an unbounded parameter set, and states the asymptotic normality result from [7, 6]. Sections 3 and 4 present the main results: a confidence region for the regression parameter and confidence intervals for integral functionals of the baseline hazard rate. Section 5 provides a method to compute auxiliary consistent estimates, and Section 6 concludes.

Throughout the paper, all vectors are column ones, $\mathsf{E}$ stands for the expectation, $\mathsf{Var}$ stands for the variance, and $\mathsf{Cov}$ for the covariance matrix. A relation holds eventually if it is valid for all sample sizes n starting from some random number, almost surely.

Let T denote the lifetime and have the intensity function A covariate X is a time-independent random vector distributed in ${\mathbb{R}}^{m}$, β is a parameter belonging to $\varTheta _{\beta }\subset {\mathbb{R}}^{m}$, and $\lambda (\cdot )\in \varTheta _{\lambda }\subset C[0,\tau ]$ is a baseline hazard function.

We observe censored data, i.e., instead of T only a censored lifetime $Y:=\min \{T,C\}$ and the censorship indicator $\varDelta :=I_{\{T\le C\}}$ are available, where the censor C is distributed on a given interval $[0,\tau ]$. The survival function of censor $G_{C}(u):=1-F_{C}(u)$ is unknown. The conditional pdf of T given X is The conditional survival function of T given X equals

We deal with an additive error model, which means that instead of X, a surrogate variable is observed. We suppose that a random error U has known moment generating function $M_{U}(z):=\mathsf{E}{e}^{{z}^{\top }U}$, where $||z||$ is bounded according to assumptions stated below. A couple $(T,X)$, censor C, and measurement error U are stochastically independent.

Introduce assumptions from [7, 6].

Consider independent copies of the model $(X_{i},T_{i},C_{i},Y_{i},\varDelta _{i},U_{i},W_{i})$, $i=1,\dots ,n$. Based on triples $(Y_{i},\varDelta _{i},W_{i})$, $i=1,\dots ,n$, we estimate true parameters $\beta _{0}$ and $\lambda _{0}(t)$, $t\in [0,\tau ]$. Following Augustin (2004) [1], we use the corrected partial log-likelihood function with

The estimator [7, 6] of the baseline hazard rate $\lambda (\cdot )$ and parameter β is defined as follows.

Theorem 3 from [7, 6] proves that under conditions (i) to (vii) the corrected estimator $({\hat{\lambda }_{n}^{(1)}},{\hat{\beta }_{n}^{(1)}})$ is a strongly consistent estimator of the true parameters $(\lambda _{0},\beta _{0})$. In the proof of Theorem 3 from [7, 6], it is shown that eventually and for R large enough, the upper bound on the right-hand side of (2) can be taken over the set ${\varTheta }^{R}:={\varTheta _{\lambda }^{R}}\times \varTheta _{\beta }$, with where $\bar{B}(0,R)$ denotes the closed ball in $C[0,\tau ]$ with center in the origin and radius R. Thus, we assume that for all $n\ge 1$, and $({\hat{\lambda }_{n}^{(1)}},{\hat{\beta }_{n}^{(1)}})\in {\varTheta }^{R}$. Notice that ${\varTheta }^{R}$ is a compact set in $C[0,\tau ]$.

Definition 2 from [7, 6] provides, based on $({\hat{\lambda }_{n}^{(1)}},{\hat{\beta }_{n}^{(1)}})$, a modified estimator $({\hat{\lambda }_{n}^{(2)}},{\hat{\beta }_{n}^{(2)}})$ which is consistent and asymptotically normal.

Below we use notations from [2]. Let For $i=1,2,\dots $, introduce random variables with Let with where $\varphi =(\varphi _{\lambda },\varphi _{\beta })\in C[0,\tau ]\times {\mathbb{R}}^{m}$ and ${q^{\prime }}$ denotes the Fréchet derivative.

Denote as $\mathsf{E}_{X}[\cdot ]$ the conditional expectation given a random variable X. Remember that $M_{U}(z)=\mathsf{E}{e}^{{z}^{\top }U}$. For simplicity of notation, we write $M_{k,\beta }$ instead of $M_{U}((k+1)\beta )$. Using differentiation in z one can easily prove the following.

Now, we state conditions on measurement error U under which one can construct unbiased estimators for $a(t)$, $b(t)$ and $p(t)$, $t\in [0,\tau ]$.

Now, we can construct estimators of $a(t)$, $b(t)$ and $p(t)$ for $t\in [0,\tau ]$. Take $\hat{\varLambda }(t):={\int _{0}^{t}}{\hat{\lambda }_{n}^{(2)}}(s)ds$ as a consistent estimator of $\varLambda (t)$, $t\in [0,\tau ]$. Indeed, the consistency of ${\hat{\lambda }_{n}^{(2)}}(\cdot )$ implies a.s. as $n\to \infty $.

For any fixed $(\lambda ,\beta )\in {\varTheta }^{R}\hspace{2.5pt}$ and for all $t\in [0,\tau ]$, a sequence converges to $b(t;\lambda ,\beta )$ a.s. due to SLLN. The sequence is equicontinuous a.s. on the compact set ${\varTheta }^{R}$, and the limiting function is continuous on ${\varTheta }^{R}$. The latter three statements ensure that the sequence converges to b uniformly on ${\varTheta }^{R}$. Thus, a.s. as $n\to \infty $.

In a similar way for all $t\in [0,\tau ]$, a.s. and a.s. Then is a consistent estimator of $T(t)K(t)$, $t\in [0,\tau ]$.

We state the convergence of the Kaplan–Meier estimator. Remember that $Y=\min \{T,C\}$. Let $G_{Y}(t)$ be the survival function of Y.

In our model, the lifetime T has a continuous survival function, and if we assume that the same holds true for the censor C, then the first condition of Theorem 4 is satisfied. Next, it holds $G_{Y}(t)=G_{T}(t)G_{C}(t)$ and due to condition (v) for all small enough positive ε there exists $0<\delta <\frac{1}{2}$ such that Therefore, the second condition holds as well, with $S=\tau -\varepsilon $.

Relation (9) is equivalent to the following: there exists a random variable $C_{S}(\omega )$ such that a.s. for all $n\ge 2$,

Let We have Due to the above-stated consistency of $\hat{T}(\cdot )\hat{K}(\cdot )$ and since $\hat{G}_{C}$ is bounded by 1, the first summand in (10) converges to zero a.s. as $n\to \infty $.

Consider the second summand. Let $S=\tau -\varepsilon $ for some fixed $\varepsilon >0$. There are two possibilities: $Y_{(n)}\le S$ and $S<Y_{(n)}\le \tau $. In the first case, In the second case,

It holds that $Y_{(n)}\to \tau \hspace{1em}\text{a.s.}\hspace{2.5pt}$ Utilizing Theorem 4, we first tend $n\to \infty $ and then $\varepsilon \to 0$ and obtain convergence of the second summand of (10) to 0 a.s. as $n\to \infty $.

The convergence of $Y_{(n)}$ yields the convergence of the third summand. Finally,

Because $\mathsf{E}\zeta _{i}=0$, it holds $\varSigma _{\beta }=4\cdot \mathsf{E}\zeta _{1}{\zeta _{1}^{\top }}$. Therefore, we take as an estimator of $\varSigma _{\beta }$. We have Then and eventually ${\hat{M}}^{-1}\hat{\varSigma }_{\beta }{\hat{M}}^{-1}>0$. Convergences (5) and (11) yield Thus,

Given a confidence probability $1-\alpha $, the asymptotic confidence ellipsoid for β is the set Here $({\chi _{m}^{2}})_{\alpha }$ is the upper quantile of ${\chi _{m}^{2}}$ distribution.

Theorem 1 implies the following statement.

We show that asymptotic variance ${\sigma _{\varphi }^{2}}$ is positive and construct its consistent estimator.

Now, we can construct an estimator for the asymptotic variance ${\sigma _{\varphi }^{2}}\hspace{2.5pt}$. Rewrite Let Results of Section 3 yield that $\hat{A}$ is a consistent estimator of A. Denote and define $\hat{\varphi }_{\lambda }$ as a solution in $L_{2}[0,\tau ]$ to the Fredholm integral equation with a degenerate kernel Eventually, a solution is unique because the limiting equation (12) has a unique solution. The function $\hat{\varphi }_{\lambda }$ can be assumed right-continuous and it converges a.s. to $\varphi _{\lambda }$ from (12) in the supremum norm. Therefore, is a consistent estimator of $\varphi _{\beta }$.

Finally, we construct an estimator of ${\sigma _{\varphi }^{2}}$. Put with and

Lemma 2 and the consistency of auxiliary estimators yield the following consistency result.

For fixed $\varepsilon >0$, consider an integral functional of the baseline hazard rate, $I_{f}(\lambda _{0})={\int _{0}^{\tau -\varepsilon }}\lambda _{0}(u)f(u)du$. Corollary 1 gives which together with (14) yields Let where $z_{\alpha /2}$ is the upper quantile of normal law. Then $I_{n}$ is the asymptotic confidence interval for $I_{f}(\lambda _{0})$.

In Section 3, we constructed estimators in a form of absolutely convergent series expansions. E.g., in Theorem 2 (a) we derived an expansion of such kind for $t\in [0,\tau ]$: and a.s. as $n\to \infty $. Now, we show that we can truncate the series.

Let $\{N_{n}:n\ge 1\}$ be a strictly increasing sequence of nonrandom positive integers. Fix t for the moment and omit this argument t. Consider the head of series $B(W_{i})$, Fix $j\ge 1$, then for $n\ge j$ it holds: The latter expression tends to zero as $j\to \infty $. Therefore, almost surely We conclude that a.s. as $n\to \infty $. Moreover, with probability one the convergence is uniform in $(\lambda ,\beta )$ belonging to a compact set. Therefore, it is enough to truncate the series $B(W,t)$ by some large numbers, which makes feasible the computation of estimators from Section 3.

At the end of Section 3, we constructed asymptotic confidence intervals for integral functionals of the baseline hazard rate $\lambda _{0}(\cdot )$, and at the end of Section 4, we constructed an asymptotic confidence region for the regression parameter β. We imposed some restrictions on the error distribution. In particular, we handled the following cases: (a) the measurement error is bounded, (b) it is normally distributed, and (c) it has independent components which are shifted Poisson random variables. Based on truncated series, we showed a way to compute auxiliary estimates which are used in construction of the confidence sets.

In future we intend to elaborate a method to construct confidence regions in case of heavy-tailed measurement errors.