## The Annals of Statistics

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

#### 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

https://projecteuclid.org/euclid.aos/1346850067

Digital Object Identifier
doi:10.1214/12-AOS1016

Mathematical Reviews number (MathSciNet)
MR3015037

Zentralblatt MATH identifier
1254.62046

#### 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.