Annals of Statistics

Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data

Yuan Wu and Ying Zhang

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Abstract

The analysis of the joint cumulative distribution function (CDF) with bivariate event time data is a challenging problem both theoretically and numerically. This paper develops a tensor spline-based sieve maximum likelihood estimation method to estimate the joint CDF with bivariate current status data. The $I$-splines are used to approximate the joint CDF in order to simplify the numerical computation of a constrained maximum likelihood estimation problem. The generalized gradient projection algorithm is used to compute the constrained optimization problem. Based on the properties of $B$-spline basis functions it is shown that the proposed tensor spline-based nonparametric sieve maximum likelihood estimator is consistent with a rate of convergence potentially better than $n^{1/3}$ under some mild regularity conditions. The simulation studies with moderate sample sizes are carried out to demonstrate that the finite sample performance of the proposed estimator is generally satisfactory.

Article information

Source
Ann. Statist., Volume 40, Number 3 (2012), 1609-1636.

Dates
First available in Project Euclid: 5 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1346850067

Digital Object Identifier
doi:10.1214/12-AOS1016

Mathematical Reviews number (MathSciNet)
MR3015037

Zentralblatt MATH identifier
1254.62046

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 60G05: Foundations of stochastic processes

Keywords
Bivariate current status data constrained maximum likelihood estimation empirical process sieve maximum likelihood estimation tensor spline basis functions

Citation

Wu, Yuan; Zhang, Ying. Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data. Ann. Statist. 40 (2012), no. 3, 1609--1636. doi:10.1214/12-AOS1016. https://projecteuclid.org/euclid.aos/1346850067


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Supplemental materials

  • Supplementary material: Technical lemmas. This supplemental material contains some technical lemmas including their proofs that are imperative for the proofs of Theorems 3.1 and 3.2.
    Digital Object Identifier: doi:10.1214/12-AOS1016SUPP
    Supplemental PDF (103 KB)

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