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Consistency of the GMLE with mixed case interval-censored data. (English) Zbl 0938.62109

The case \(K\) model of interval censoring is considered. I.e., it is supposed that a (random) number \(K\) of inspection times \(Y_1<Y_2<\dots\) are defined and for the random quality of interest \(X\) only the interval \((L,R]=(Y_k,Y_{k+1}]\ni X\) is observed. \(X\) is independent of \(Y_i\). The aim is to estimate the PDF \(F\) of \(X\) by observations of \(n\) independent copies \((L_i,R_i]\). The generalized likelihood function for this problem is \(\Lambda_n(F)=\prod_{j=1}^n [F(R_j)-F(L_j)]\). The GML estimator for \(F\) is the minimizer of \(\Lambda_n\) by all picewise constant d.f.’s with possible jumps only in \(L_i\) and \(R_i\), \(i=1,\dots,n\).
Theorem 1. Let \(E(K)<\infty\) and let \[ \mu(B)=\sum_{k=1}^\infty \Pr(K=k)\sum_{j=1}^K\Pr(Y_j\in B |K=k). \] Then \( \int|\hat F_n(x)-F(x)|d\mu\to 0 \) a.s.

MSC:

62N01 Censored data models
62N05 Reliability and life testing
62G07 Density estimation
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