Admissibility of the likelihood ratio test when a nuisance parameter is present only under the alternative
type: Item
instance of: scholarly article
title: Admissibility of the likelihood ratio test when a nuisance parameter is present only under the alternative
zbMATH Open document ID: 0843.62023
author: Donald W. K. Andrews
author: Werner Ploberger
published in: The Annals of Statistics
publication date: 1996-08-21T00:00:00Z
review text: This paper considers hypothesis tests when a nuisance parameter is present only under the alternative hypothesis. Such tests are nonstandard and the classical likelihood ratio (LR) test does not possess its usual chi-square asymptotic null distribution in this context. It also does not possess its usual asymptotic optimality properties. We show that the LR test and two asymptotically equivalent tests, namely, the sup Wald and sup Lagrange multiplier (LM) tests, are asymptotically admissible. In fact, we show that these tests are best tests, in a certain sense, against alternatives that are sufficiently distant from the null hypothesis. We establish these results first under a set of high-level assumptions. Then we provide primitive sufficient conditions for a number of examples. The examples considered include tests of (i) cross-sectional constancy in nonlinear models, (ii) threshold effects in autoregressive models and (iii) variable relevance in nonlinear models, such as Box-Cox transformed regressor models. Two examples that are covered by the high-level results, but for which primitive conditions are not provided, are tests of (i) white noise versus first-order autoregressive moving average structure and (ii) white noise versus first order generalized autoregressive conditional heteroskedasticity. Next, we consider finite sample admissibility of the LR test for the Gaussian linear regression model with known variance. Minor modifications to the proof of the asymptotic admissibility result yield finite sample admissibility. The types of hypotheses covered by this result include tests of (i) single and multiple changepoints, (ii) variable relevance for Box-Cox transformed regressors and (iii) cross-sectional constancy, among others. The remainder of this paper is organized as follows: Section 2 presents the main asymptotic admissibility result under a set of high-level assumptions. Section 3 presents examples and provides primitive sufficient conditions for the hi…
Mathematics Subject Classification ID: 62F05
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 885112
zbMATH Keywords: sup Lagrange multiplier tests
zbMATH Keywords: changepoint test
zbMATH Keywords: empirical process
zbMATH Keywords: likelihood ratio test
zbMATH Keywords: nuisance parameter under alternative
zbMATH Keywords: sup Wald test
zbMATH Keywords: cross-sectional constancy
zbMATH Keywords: nonlinear models
zbMATH Keywords: threshold effects
zbMATH Keywords: autoregressive models
zbMATH Keywords: variable relevance
zbMATH Keywords: Box-Cox transformed regressor models
zbMATH Keywords: white noise versus first-order autoregressive moving average structure
zbMATH Keywords: first order generalized autoregressive conditional heteroskedasticity
zbMATH Keywords: finite sample admissibility
zbMATH Keywords: Gaussian linear regression model
zbMATH Keywords: asymptotic admissibility
zbMATH Keywords: examples
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176324316
OpenAlex ID: W2081789018
DOI: 10.1214/AOS/1176324316
description: scientific article; zbMATH DE number 885112