Empirical copulas for consecutive survival data

Abstract

In the analysis of medical data the whole lifetime is often split into pieces characterizing the various stages in the development of a chronical disease. In this paper we provide a nonparametric copula function estimator for two consecutive survival data which are subject to truncation and right censorship. We also discuss an extension of Spearman’s Rho and Kendall’s Tau to the present situation.

Appendix

The following result will be needed in the proof of Theorem 2 and applied to the marginals F _1n and F _2n of F _n . It has been formulated for general distribution function estimators and their quantiles. For the special case of a classical univariate empirical d.f., Lemma A allows for remarkable extensions since then the oscillation behavior of empirical processes yields exact almost sure bounds for the remainder terms leading to the famous Bahadur (1966) representation of empirical quantiles for i.i.d. data. See also Stute (1982). Together with oscillation results for multivariate empirical processes, see Stute (1984), Gaenssler and Stute (1987) discussed bounds for the remainders of the empirical copula process. For the estimators discussed in this paper only tightness but no sharp oscillation bounds are available so far. Therefore, for the sake of completeness, we included Lemma A in a form which summarizes the conditions needed for distributional convergence and an in-probability representation without bounds.

Lemma A

Let $\tilde{F}_{n}$ be univariate nondecreasing estimators of a distribution function $\tilde{F}$ such that the processes

$$\tilde{\alpha}_n = n^{1/2} [\tilde{F}_n - \tilde{F}] $$

are asymptotically C-tight on compact subsets of the real line. Assume furthermore that $\tilde{F}$ is continuously differentiable at $\tilde{F}^{-1}(u)$ with

$$\tilde{F}'\bigl(\tilde{F}^{-1}(u)\bigr) > 0 . $$

Then we have

$$ \tilde{F}'\bigl(\tilde{F}^{-1}(u)\bigr) n^{1/2} \bigl[ \tilde{F}_n^{-1}(u) - \tilde{F}^{-1}(u) \bigr] = - \tilde{\alpha}_n \bigl(\tilde{F}^{-1}(u)\bigr) + o_{\mathbb{P}}(1) . $$

The last representation holds uniformly in u whenever u varies in compact subsets of (0,1).

Proof

Since $\tilde{\alpha}_{n}$ is asymptotically C-tight, we have $\tilde{F}_{n} \to\tilde{F}$ on compacta, in probability. Under the assumed regularity conditions at $\tilde{F}^{-1}(u)$, we get

$$\tilde{F}_n^{-1}(u) \to\tilde{F}^{-1}(u)\quad \mbox{in probability}. $$

By the assumed tightness,

$$\tilde{\alpha}_n \bigl( \tilde{F}_n^{-1}(u) \bigr) - \tilde{\alpha}_n \bigl(\tilde{F}^{-1}(u)\bigr) \to0\quad \mbox{in probability} . $$

The last difference, however, equals

$$n^{1/2} \bigl[ \tilde{F}_n\bigl(\tilde{F}_n^{-1}(u) \bigr) - \tilde{F}\bigl(\tilde{F}_n^{-1}(u)\bigr) - \tilde{F}_n\bigl(\tilde{F}^{-1}(u)\bigr) + \tilde{F}\bigl(\tilde{F}^{-1}(u)\bigr) \bigr] . $$

We also have

$$\tilde{F}_n\bigl(\tilde{F}_n^{-1}(u)\bigr) = u\quad \mbox{within}\ o_{\mathbb{P}}\bigl(n^{-1/2}\bigr) $$

and

$$\tilde{F}\bigl(\tilde{F}^{-1}(u)\bigr) = u . $$

Conclude that

$$n^{1/2} \bigl[ \tilde{F}\bigl(\tilde{F}^{-1}(u)\bigr) - \tilde{F}\bigl(\tilde{F}_n^{-1}(u)\bigr) \bigr] = \tilde{\alpha}_n \bigl(\tilde{F}^{-1}(u)\bigr) + o_{\mathbb{P}}(1) . $$

By the mean value theorem, the left-hand sided is asymptotically equal to

$$\tilde{F}'\bigl(\tilde{F}^{-1}(u)\bigr) n^{1/2} \bigl[\tilde{F}^{-1}(u) - \tilde{F}_n^{-1}(u)\bigr] . $$

This completes the proof. □