## 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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