## Inconsistency of the MLE for the joint distribution of interval-censored survival times and continuous marks.(English)Zbl 1164.62076

An interval censoring model is considered for a survival time $$X$$ and a continuous mark $$Y$$. I.e., the observed data are $$W=(T,\Delta,Z)$$, where $$T=(T_1,\dots,T_k)$$ is a random vector of observation times independent of $$(X,Y)$$, $$\Delta=(\Delta_1,\dots,\Delta_{k+1})$$, $$\Delta_j=1_{\{T_{j-1}<X\leq T_j\}}$$ ($$T_0=0$$, $$T_{k+1}=+\infty$$), and $$Z=Y\cdot 1_{\{X\leq T_k\}}$$ (i.e., $$Y$$ is observed only if $$X\leq T_k$$). A nonparametric maximum likelihood estimator (NMLE) is constructed for the joint CDF $$F$$ of $$(X,Y)$$ by $$n$$ i.i.d. copies of $$W$$. Its limit as $$n\to\infty$$ does not generally coincide with the true CDF $$F$$, so the NMLE is inconsistent. To derive a consistent estimate the authors propose to discretize the mark $$Y$$. Results of simulations are presented.

### MSC:

 62N02 Estimation in survival analysis and censored data 62N01 Censored data models 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference
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### References:

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