Admissibility of the likelihood ratio test when a nuisance parameter is present only under the alternative. (English) Zbl 0843.62023

Andrews, Donald W. K.; Ploberger, Werner

Ann. Stat. 23, No. 5, 1609-1629 (1995).

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 high-level assumptions. Section 4 states the finite sample admissibility results for tests concerning a Gaussian linear regression model. Section 5 gives proofs of the results stated in earlier sections.

MSC:

62F05	Asymptotic properties of parametric tests
62C15	Admissibility in statistical decision theory
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

sup Lagrange multiplier tests; changepoint test; empirical process; likelihood ratio test; nuisance parameter under alternative; sup Wald test; cross-sectional constancy; nonlinear models; threshold effects; autoregressive models; variable relevance; Box-Cox transformed regressor models; white noise versus first-order autoregressive moving average structure; first order generalized autoregressive conditional heteroskedasticity; finite sample admissibility; Gaussian linear regression model; asymptotic admissibility; examples