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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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