Generalized copula-graphic estimator

Abstract

In this paper, a copula-graphic estimator is proposed for censored survival data. It is assumed that there is some dependent censoring acting on the variable of interest that may come from an existing competing risk. Furthermore, the full process is independently censored by some administrative censoring time. The dependent censoring is modeled through an Archimedean copula function, which is supposed to be known. An asymptotic representation of the estimator as a sum of independent and identically distributed random variables is obtained, and, consequently, a central limit theorem is established. We investigate the finite sample performance of the estimator through simulations. A real data illustration is included.

Appendix 1: Technical lemmas

In this section, we give the technical lemmas needed in the proof to Theorem 1.

Lemma 1

Under the conditions in Theorem 1, we have

$$ \sup_{0\leq t\leq T}\bigl \vert R_{n1}(t)\bigr \vert =O \bigl(n^{-1}\log \log n\bigr)\quad \text{\textit{a.s.}} $$

Proof

It is easy to see that, with probability 1,

The first term is O(n ^−1/2(loglogn)^1/2) a.s., cf. Földes and Rejtő (1981). The same order bound is proved to hold for the second term in Lemma 4 below, and the proof is complete. □

Lemma 2

Under the conditions in Theorem 1, we have

$$ \sup_{0\leq t\leq T}\bigl \vert R_{n2}(t)\bigr \vert =O \bigl(n^{-1}\log \log n\bigr)\quad \text{\textit{a.s.}} $$

Proof

With probability 1 we have

Then, the assertion of the lemma follows from a result of Földes and Rejtő (1981). □

Lemma 3

Under the conditions in Theorem 1, we have

$$ \sup_{0\leq t\leq T}\bigl \vert R_{n3}(t)\bigr \vert =O \bigl(n^{-3/4}(\log n)^{3/4}\bigr)\quad \text{\textit{a.s.}} $$

Proof

Divide [0,T] into k _n =O(n ^1/2(logn)^1/2) subintervals [t _i ,t _i+1] of length O(n ^−1/2(logn)^1/2). Then, as in the proof of Lo and Singh (1986), we have

For I, we have by Taylor expansion and the fact that $\sup_{0\leq t\leq T}\vert \overline{H}_{n}(t)-\overline{H}(t)\vert =O(n^{-1/2}(\log \log n)^{1/2})$ a.s. (Földes and Rejtő 1981):

Now further subdivide each interval [t _i ,t _i+1] into a _n =O(n ^1/4(logn)^−1/4) subintervals of length O(n ^−3/4(logn)^3/4). By using Bernstein’s inequality we can show that this term is bounded a.s. by

$$ c\max_{1\leq i\leq k_{n}}\max_{0\leq j\leq a_{n}-1}\bigl \vert H_{n}(t_{i,j+1})-H(t_{i,j+1})-H_{n}(t_{i})+H(t_{i}) \bigr \vert +O\bigl(n^{-3/4}(\log n)^{3/4}\bigr) $$

for some constant c>0. By the modulus of continuity result for the Kaplan–Meier estimator (see Schäfer 1986) we obtain that I=O(n ^−3/4(logn)^3/4) a.s. The II term is treated similarly and leads to the same order. It requires the almost sure behavior of the modulus of continuity of the $H_{n}^{1}$ estimator, and this follows from Lemma 5 below. In that Lemma take a _n =n ^−1/2(logn)^1/2. □

Lemmas 4 and 5 below are needed for the proofs of Lemmas 1 and 3, respectively. They have some independent interest since they provide the almost sure rate of convergence and the almost sure behavior of the modulus of continuity for the estimator of the cumulative incidence function of Z subject to δ=1 ($H_{n}^{1}$).

Lemma 4

For $T<\min (T_{F},T_{G},T_{\widetilde{G}})$, we have

$$ \sup_{0\leq t\leq T}\bigl \vert H_{n}^{1}(t)-H^{1}(t) \bigr \vert =O\bigl(n^{-1/2}(\log \log n)^{1/2}\bigr)\quad \text{\textit{a.s.}} $$

Proof

Define the following empirical estimators for the distribution function $\widetilde{H}(t)=P(U\leq t)$ and for the subdistribution functions $\widetilde{H}^{0}(t)=P(U\leq t,\rho =0)$ and $\widetilde{H}^{11}(t)=P(U\leq t,\rho =1,\delta =1)$:

Then, H ¹(t) can be expressed in terms of $\widetilde{H}$, $\widetilde{H}^{0}$, and $\widetilde{H}^{11}$, and $H_{n}^{1}(t)$ can be expressed in terms of the corresponding empiricals. Similar as in Stute (1995), we obtain

$$ H_{n}^{1}(t)=\int_{0}^{t}\exp \biggl\{ n\int_{0}^{u}\log \biggl(1+ \frac{1}{n(1-\widetilde{H}_{n}(z))}\biggr)\,d\widetilde{H}_{n}^{0}(z) \biggr\} \,d \widetilde{H}_{n}^{11}(u) $$

and

$$ H^{1}(t)=\int_{0}^{t}\exp \biggl\{ \int _{0}^{u}\frac{d\widetilde{H}^{0}(z)}{1-\widetilde{H}(z)} \biggr\} \,d \widetilde{H}^{11}(u). $$

It follows that $\sup_{0\leq t\leq T}\vert H_{n}^{1}(t)-H^{1}(t)\vert$ is smaller than

(4)

The second term in (4) is O(n ^−1/2(loglogn)^1/2) a.s. For the first term in (4), we use (with obvious abbreviations) that

with θ between 0 and a−b. Note that exp(b) is uniformly bounded in [0,T]. Looking at (a−b), we have

(5)

The second term in (5) is O(n ^−1/2(loglogn)^1/2) a.s. For the first term in (5), we use that, for x≥0,

$$ x-\frac{1}{2}x^{2}\leq \log (1+x)\leq x. $$

It follows that the first term in (5) is bounded above by

$$ \sup_{0\leq z\leq T}\biggl \vert \frac{1}{1-\widetilde{H}_{n}(z)}-\frac{1}{1-\widetilde{H}(z)}\biggr \vert +\frac{1}{2n}\sup_{0\leq z\leq T}\frac{1}{(1-\widetilde{H}_{n}(z))^{2}}. $$

This is O(n ^−1/2(loglogn)^1/2) a.s. since $\sup_{0\leq z\leq T}\vert \widetilde{H}_{n}(z)-\widetilde{H}(z)\vert$ has the same order and since $\widetilde{H}(T)<1$. □

Lemma 5

Suppose that $T<\min (T_{F},T_{G},T_{\widetilde{G}})$. Suppose that H(t)=P(Z≤t) and H ¹(t)=P(Z≤t,δ=1) have bounded first derivatives in [0,T]. Let {a _n } be a sequence of positive constants tending to zero with a _n n(logn)⁻⁵>Δ>0 for all n sufficiently large. Then

$$ \sup_{0\leq t,s\leq T,\vert t-s\vert \leq a_{n}}\bigl \vert H_{n}^{1}(t)-H_{n}^{1}(s)-H^{1}(t)+H^{1}(s) \bigr \vert =O\bigl(a_{n}^{1/2}n^{-1/2}(\log n)^{1/2}\bigr)\quad \text{\textit{a.s.}} $$

Proof

We make the same partition of the interval [0,T] as in Lemma A.5 of Van Keilegom and Veraverbeke (1997). Exploiting the monotonicity of H ¹(t) and $H_{n}^{1}(t)$ and also the Lipschitz continuity of H ¹(t), we obtain that it suffices to prove that

where {t _ij }, i=1,…,m, j=−b _n ,…,b _n is a grid of points with $m= [ \frac{T}{a_{n}} ]$ ([⋅] denoting the integer part) and $b_{n}\sim a_{n}^{1/2}n^{1/2}(\log n)^{-1/2}$. At this point, we use the almost sure asymptotic representation for $H_{n}^{1}(t)$ as it can be derived as a special case of the more general result of Sánchez-Sellero et al. (2005):

$$ H_{n}^{1}(t)-H^{1}(t)=\frac{1}{n}\sum _{i=1}^{n}\widetilde{\widetilde{\psi }}_{i}(t)+R_{n}(t), $$

where

with

the function $\widetilde{C}(t)$ being that in the remark of Sect. 2; and sup_0≤t≤T |R _n (t)|=O(n ⁻¹(logn)³) a.s. It follows that it suffices to show that

$$ \max_{1\leq i\leq m-1}\max_{-b_{n}<j,k<b_{n}}\Biggl \vert \frac{1}{n}\sum_{r=1}^{n}\bigl(\widetilde{\widetilde{ \psi }}_{r}(t_{ik})-\widetilde{\widetilde{\psi }}_{r}(t_{ij})\bigr)\Biggr \vert =O\bigl(a_{n}^{1/2}n^{-1/2}( \log n)^{1/2}\bigr). $$

To achieve this, we use Bernstein’s inequality as in Van Keilegom and Veraverbeke (1997). The random variables $\widetilde{\widetilde{\psi }}_{r}(t_{ik})-\widetilde{\widetilde{\psi }}_{r}(t_{ij})$ are bounded, and $\operatorname {Var}(\widetilde{\widetilde{\psi }}_{r}(t_{ik})-\widetilde{\widetilde{\psi }}_{r}(t_{ij}))$ is bounded by a constant times a _n . The latter fact is shown by checking six appropriate groups of terms in

For example, by direct calculation,

for some constant c>0 by the Lipschitz continuity of H. The other groups of terms are treated similarly. □