The Annals of Statistics

Likelihood based inference for current status data on a grid: A boundary phenomenon and an adaptive inference procedure

Runlong Tang, Moulinath Banerjee, and Michael R. Kosorok

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In this paper, we study the nonparametric maximum likelihood estimator for an event time distribution function at a point in the current status model with observation times supported on a grid of potentially unknown sparsity and with multiple subjects sharing the same observation time. This is of interest since observation time ties occur frequently with current status data. The grid resolution is specified as cnγ with c > 0 being a scaling constant and γ > 0 regulating the sparsity of the grid relative to n, the number of subjects. The asymptotic behavior falls into three cases depending on γ: regular Gaussian-type asymptotics obtain for γ < 1/3, nonstandard cube-root asymptotics prevail when γ > 1/3 and γ = 1/3 serves as a boundary at which the transition happens. The limit distribution at the boundary is different from either of the previous cases and converges weakly to those obtained with γ ∈ (0, 1/3) and γ ∈ (1/3, ∞) as c goes to ∞ and 0, respectively. This weak convergence allows us to develop an adaptive procedure to construct confidence intervals for the value of the event time distribution at a point of interest without needing to know or estimate γ, which is of enormous advantage from the perspective of inference. A simulation study of the adaptive procedure is presented.

Article information

Ann. Statist., Volume 40, Number 1 (2012), 45-72.

First available in Project Euclid: 15 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G09: Resampling methods 62G20: Asymptotic properties
Secondary: 62G07: Density estimation

Adaptive procedure boundary phenomenon current status model isotonic regression


Tang, Runlong; Banerjee, Moulinath; Kosorok, Michael R. Likelihood based inference for current status data on a grid: A boundary phenomenon and an adaptive inference procedure. Ann. Statist. 40 (2012), no. 1, 45--72. doi:10.1214/11-AOS942.

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Supplemental materials

  • Supplementary material: More proofs for the current paper “Likelihood based inference for current status data on a grid: A boundary phenomenon and an adaptive inference procedure”. The supplementary material contains the details of the proofs of several theorems and lemmas in Sections 3.1 and 3.3 of this paper.