Efficient robust estimation of parameter in the random censorship model

For an open set Θ of ^k, let s{P_θ: θ Θs} be a parametric family of probabilities modeling the distribution of i.i.d. random variables X₁,…, X_n. Suppose X_i's are subject to right censoring and one is only able to observe the pairs (min(X_i, Y_i), [X_i Y_i]), i = 1,…, n, where [A] denotes the indicator function of the event A, Y₁,…, Y_n are independent of X₁,…, X_n and i.i.d. with unknown distribution Q₀. This paper investigates estimation of the value θ that gives a fitted member of the parametric family when the distributions of X₁ and Y₁ are subject to contamination. The constructed estimators are adaptive under the semi-parametric model and robust against small contaminations: they achieve a lower bound for the local asymptotic minimax risk over Hellinger neighborhoods, in the Hájel—Le Cam sense. The work relies on Beran (1981). The construction employs some results on product-limit estimators.