The strong law under semiparametric random censorship models (Q1969132)

Let \((Z_i, \delta_i), 1\leq i\leq n\), be independent data, where \(Z_i = \min(X_i, Y_i)\) and \(\delta_i\) indicates whether \(Z_i = X_i\) (uncensored) or \(Z_i = Y_i\) (censored). The goal then is to construct estimates of the distribution function \(F\) of \(X\), say, from the available \((Z, \delta)\)-sequence. An important role is played by the function \[ m(z) =\text{I\negthinspace P}(\delta=1 |Z = z). \] In a completely nonparametric framework the function \(m\) remains unspecified. From the author's previous work it is known that under some parametric modeling of \(m\), estimators \(F_n\) of \(F\) may be deduced which are more efficient than the classical Kaplan-Meier estimator. In the present paper it is shown that for a large class of \(\varphi\)'s, the empirical integrals \(\int \varphi d F_n\) converge with probability one to \(\int^{\tau}\varphi d F\), where \(\tau\) is the least upper bound for the support of \(Z\). Proofs rest on proper (reverse super-) martingale representations of \(\int \varphi d F_n\) in \(n\).