Sharp minimax distribution estimation for current status censoring with or without missing

Abstract

Nonparametric estimation of the cumulative distribution function and the probability density of a lifetime X modified by a current status censoring (CSC), including cases of right and left missing data, is a classical ill-posed problem with biased data. The biased nature of CSC data may preclude us from consistent estimation unless the biasing function is known or may be estimated, and its ill-posed nature slows down rates of convergence. Under a traditionally studied CSC, we observe a sample from (Z, ∆) where a continuous monitoring time Z is independent of X, ∆:= I (X ≤ Z) is the status, and the bias of observations is created by the density of Z which is estimable. In presence of right or left missing, we observe corresponding samples from (∆Z, ∆) or ((1 − ∆)Z, ∆); the data are again biased but now the density of Z cannot be estimated from the data. As a result, to solve the estimation problem, either the density of Z must be known (like in a controlled study) or an extra cross-sectional sampling of Z, which is typically simpler than an underlying CSC study, be conducted. The main aim of the paper is to develop for this biased and ill-posed problem the theory of efficient (sharp-minimax) estimation which is inspired by known results for the case of directly observed X. Among interesting aspects of the developed theory: (i) While sharp-minimax analysis of missing CSC may follow the classical Pinsker's methodology, analysis of CSC requires a more complicated estimation procedure based on a special smoothing in both frequency and time domains; (ii) Efficient estimation requires solving an old-standing problem of approximating aperiodic Sobolev functions; (iii) If smoothness of the cdf of X is known, then its rate-minimax estimation is possible even if the density of Z is rougher. Real and simulated examples, as well as extensions of the core models to dependent X and Z and case-control CSC, are presented.