Point optimal testing: A survey of the post 1987 literature

Maxwell L King,Sivagowry Sriananthakumar

doi:10.3233/mas-150323

Maxwell L King, Sivagowry Sriananthakumar

Open Access

https://doi.org/10.3233/mas-150323

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Abstract

In the absence of uniformly most powerful (UMP) tests or uniformly most powerful invariant (UMPI) tests, King [80] suggested the use of Point Optimal (PO) tests, which are most powerful at a chosen point under the alternative hypothesis. This paper surveys the literature and major developments on point optimal testing since 1987 and suggests some areas for future research. Topics include tests for which all nuisance parameters have been eliminated and dealing with nuisance parameters via (i) a weighted average of p values, (ii) approximate point optimal tests, (iii) plugging in estimated parameter values, (iv) using asymptotics and (v) integration. Progress on using point-optimal testing principles for two-sided testing and multi-dimensional alternatives is also reviewed. The paper concludes with thoughts on how best to deal with nuisance parameters under both the null and alternative hypotheses, as well as the development of a new class of point optimal tests for multi-dimensional testing.

Highlights

Constructing hypothesis tests or choosing which test to use in econometrics can be difficult
A related paper by Elliott and Pesavento (2009) considered the problem of testing the null hypothesis of no cointegration when the cointegrating variables are known to have a unit root. They traced out the power envelope using point optimal tests that maximize the weighted average power for different weightings over the unknown cointegrating vector parameter space
A high proportion of this new literature has been in the very highly researched area of unit root testing. This has proved to be an extremely difficult testing problem that point optimal testing and its application in a local-to-unity asymptotic setting by Elliott et al (1996) and more recently Müller (2011) have helped solve, we continue to see innovations that result in power improvements

Summary

Introduction

Constructing hypothesis tests or choosing which test to use in econometrics can be difficult. Based on a range of early applications largely involving testing the covariance matrix of the linear regression model (see Spjotvoll (1967), Davies (1969), Berenblut and Webb (1973), Fraser, Guttman and Styan (1976), Bhargava, Franzini and Narendranathan (1982), King (1981b, 1983a, 1983b, 1984, 1985a, 1985b, 1986, 1987a), Franzini and Harvey (1983), Sargan and Bhargava (1983), Evans and King (1985a, 1985b, 1988), King and Smith (1986), Shively (1986, 1988a, 1988b), Nyblom (1986) and Dufour and King (1991)), King (1987c) argued the case for the use of point optimal testing He observed they best suit problems in which the parameter space under the alternative hypothesis can be restricted in scope by theoretical and technical (such as variances being positive) considerations. It involves finding a probability density function over the ∆ space, h∆ (δ ) , constructing

It also requires that a critical value c exists such that

Θb ω