## The Annals of Statistics

### Likelihood Ratio Tests for Monotone Functions

#### Abstract

We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function.The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data)and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion $+t^2$ and greatest convex minorants thereof. Inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function.We also study the behavior of the statistic under local and fixed alternatives.

#### Article information

Source
Ann. Statist., Volume 29, Number 6 (2001), 1699-1731.

Dates
First available in Project Euclid: 5 March 2002

https://projecteuclid.org/euclid.aos/1015345959

Digital Object Identifier
doi:10.1214/aos/1015345959

Mathematical Reviews number (MathSciNet)
MR1891743

Zentralblatt MATH identifier
1043.62037

Subjects
Primary: 62G05: Estimation
Secondary: 60G15: Gaussian processes 62E20: Asymptotic distribution theory

#### Citation

Banerjee, Moulinath; Wellner, Jon A. Likelihood Ratio Tests for Monotone Functions. Ann. Statist. 29 (2001), no. 6, 1699--1731. doi:10.1214/aos/1015345959. https://projecteuclid.org/euclid.aos/1015345959

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