Estimating the survival functions in a censored semi-competing risks model

Abstract

Rivest and Wells (2001) proposed estimators of the marginal survival functions in a right-censored model that assumes an Archimedean copula between the survival time and the censoring time. We study the extension of these estimators to the context of right-censored semi-competing risks data with an independent second level censoring time. We intensively use martingale techniques to derive their large sample properties under mild assumptions on the true distribution of the data. As compared to the simpler context of right-censored data, a primary difference is the need to enlarge the filtrations with respect to which we use the Doob-Meyer decompositions of counting processes.

A   Delta method in the Skorohod space

The goal of this appendix is to prove a version of the Delta method in the Skorohod space D[0,1] when the limit process is continuous, because we do not find it in the literature. The proof we give is essentially the same as the proof of the Delta method for real random variables. An alternative proof consists in using the powerful functional Delta method (van der Vaart and Wellner, 1996) after having said that in this context, the weak convergence in D[0,1] is equivalent to the weak convergence in the uniform sense, as in the modern theory of empirical processes; this fact results from theorem 1.7.2 in van der Vaart and Wellner (1996) and theorem 6.6 in Billingsley (1999). All our statements are proved for stochastic processes in the Skorohod space $D\bigl([0,1], {\mathbb{R}}^d\bigr)$ which we shortly denote by D[0,1]. Similarly we shortly denote by C[0,1] the space of continuous functions $C\bigl([0,1], {\mathbb{R}}^d\bigr)$

Lemma A.1

Let (X _n ) be a sequence of random variables in D[0,1] which converges weakly to a random variable X in C[0,1]. Then the sequence $\bigl(\sup_{0 \leq s \leq 1}\Vert X_n(s)\Vert\bigr)$ converges weakly to $\sup_{0 \leq s \leq 1}{\Vert X(s)\Vert}$ .

Proof

This follows from the continuity of $x \mapsto \sup_{0 \leq s \leq 1}\Vert x(s)\Vert$ outside a set with null measure with respect to the law of X.

Lemma A.2

Let (Z _n ) be a sequence of random variables in D[0,1] such that $\sup_{0 \leq t \leq 1}\bigl\Vert Z_n(t) \bigr\Vert = o_P(1)$ and the sequence $(\sqrt{n}Z_n)$ converges weakly to a random variable W in C[0,1]. Let $R \colon {\mathbb{R}}^d \to {\mathbb{R}}$ be a function such that $R(z)=o\bigl(\Vert z\Vert \bigr)$ when z→0. Then $\sup_{0 \leq t \leq 1}\left|\sqrt{n}R\bigl(Z_n(t)\bigr)\right| =o_P(1)$ .

Proof

One has

$$ \left|\sqrt{n}R\bigl(Z_n(t)\bigr)\right| = \bigl\Vert \sqrt{n} Z_n(t) \bigr\Vert h\bigl(Z_n(t)\bigr) \leq \sup\limits_{0 \leq s \leq 1}\bigl\Vert \sqrt{n}Z_n(s) \bigr\Vert h\bigl(Z_n(t)\bigr) $$

where h is the function defined by $h(z)=\left|R(z)\right|/\Vert z\Vert$ if z ≠ 0 and h(0) = 0, which is continuous at 0. The sequence $\bigl(\sup_{0 \leq s \leq 1}\Vert \sqrt{n}Z_n(s)\Vert\bigr)$ converges weakly to $\sup_{0 \leq s \leq 1}\Vert W(s)\Vert$ by Lemma A.1, hence, owing to Slutsky’s theorem, it suffices to show that $\sup_{0 \leq t \leq 1} h\bigl(Z_n(t)\bigr)=o_P(1)$, but this follows from the continuity of h at 0.

Proposition A.1

(Delta method) Let (X _n ) be a sequence of random variables in D[0,1] and θ a function belonging to D[0,1]. Assume that $\sup_{0 \leq t \leq 1}\bigl\Vert X_n(t)-\theta(t) \bigr\Vert = o_P(1)$ and that the sequence $\sqrt{n}\bigl(X_n-\theta\bigr)$ converges weakly in D[0,1] to a random variable G in C[0,1]. Let $g\colon {\mathbb{R}}^d \to {\mathbb{R}}^k$ be a differentiable function. Denote by R _μ the remainder in the first order Taylor expansion of g at μ , that is, R _μ (y) = g(μ + y) − g(μ) − g′(μ)(y). If $\sup_{0 \leq t \leq 1}\bigl\Vert R_{\theta(t)}(y) \bigr\Vert=o\bigl(\Vert y\Vert\bigr)$ , then $\sqrt{n}\bigl(g(X_n)-g(\theta)\bigr) = g'(\theta)\sqrt{n}\bigl(X_n-\theta\bigr) + o_P(1)$ uniformly on [0,1], therefore $\sqrt{n}\bigl(g(X_n)-g(\theta)\bigr)$ converges weakly to g′(θ)G in D[0,1].

Proof

One has

$$ \left\Vert \sqrt{n}\bigl(g(X_n)-g(\theta)\bigr) - g'(\theta)\sqrt{n}\bigl(X_n-\theta\bigr) \right\Vert \leq \sqrt{n}R(X_n-\theta) $$

where $R(y) = \sup_{0 \leq t \leq 1} \bigl\Vert R_{\theta(t)}(y) \bigr\Vert$, hence the result follows from the preceding lemma.

Denoting by H the limiting process in the above proposition, its variance function is ${\mathbb{E}} H(s){H(t)}' = D_{\theta(s)} C(s,t) D'_{\theta(t)}$ where D _z is the matrix of g′(z) and $C(s,t) = {\mathbb{E}} G(s){G(t)}'$ is the variance function of G.