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February 2021 Sharp minimax distribution estimation for current status censoring with or without missing
Sam Efromovich
Ann. Statist. 49(1): 568-589 (February 2021). DOI: 10.1214/20-AOS1970

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 $(Z,\Delta )$ where a continuous monitoring time $Z$ is independent of $X$, $\Delta :=I(X\leq Z)$ 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 $(\Delta Z,\Delta )$ or $((1-\Delta )Z,\Delta )$; 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.

Citation

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Sam Efromovich. "Sharp minimax distribution estimation for current status censoring with or without missing." Ann. Statist. 49 (1) 568 - 589, February 2021. https://doi.org/10.1214/20-AOS1970

Information

Received: 1 September 2019; Revised: 1 January 2020; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/20-AOS1970

Subjects:
Primary: 62G07, 62G07
Secondary: 60J10

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 1 • February 2021
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