Likelihood ratio tests for monotone functions. (English) Zbl 1043.62037

Banerjee, Moulinath; Wellner, Jon A.

Ann. Stat. 29, No. 6, 1699-1731 (2001).

Summary: 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.

MSC:

62G10	Nonparametric hypothesis testing
62G05	Nonparametric estimation
60J65	Brownian motion
62E20	Asymptotic distribution theory in statistics

Keywords:

constrained estimation; local alternatives; Gaussian process; Kullback-Leibler discrepancy; monotone functions; slope processes

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