Efficient robust estimation of parameter in the random censorship model

Abstract

For an open set Θ of Rk, let \\s{Pθ: θ ∈ Θ\\s} be a parametric family of probabilities modeling the distribution of i.i.d. random variables X1,…, Xn. Suppose Xi's are subject to right censoring and one is only able to observe the pairs (min(Xi, Yi), [Xi ⩽ Yi]), i = 1,…, n, where [A] denotes the indicator function of the event A, Y1,…, Yn are independent of X1,…, Xn 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.