Point optimal testing: A survey of the post 1987 literature

Abstract

In the absence of uniformly most powerful (UMP) tests or uniformly most powerful invariant (UMPI) tests, King (1987c) 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 multidimensional 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.

Point Optimal Testing with Roots That Are Functionally Local to Unity

“…The primary need for this development came from the desire to assess the effect of local departures on the functional limit theory for unit root processes and its many applications to regression and unit root testing (Chan and Wei (1987), Phillips (1987), Phillips (1988)). The methodology assisted asymptotic power analysis and the construction of point optimal unit root tests (Elliott et al (1996) and for a recent overview King and Sriananthakumar (2016)). Most recently, the methods have been used to study the uniform properties of various methods of confidence interval construction for autoregressive coefficients (Mikusheva (2007), Mikusheva (2012), Phillips (2014)).…”

Distributions You Can Count On …But What’s the Point?

“…Our findings are somewhat disappointing and we are unable to recommend that practitioners change their current practices as the performance of the point optimal tests is, at best, mixed. It may be possible to improve the performance of the point optimal tests by a more refined analysis of (i) the problem of nuisance parameters and (ii) the construction of p-values, along the lines suggested by King and Sriananthakumar (2015). This we leave for further work.…”

“…Point optimal tests have had a long and varied career in econometrics; see King (1987) and King and Sriananthakumar (2015) for an overview. These tests optimize power at a particular parameter value under the alternative, the idea being to have good power at a point where incorrectly accepting the null really matters.…”

Point optimal testing with roots that are functionally local to unity

“…The primary need for this development came from the desire to assess the effect of local departures on the functional limit theory for unit root processes and its many applications to regression and unit root testing (Chan and Wei (1987), Phillips (1987), Phillips (1988)). The methodology assisted asymptotic power analysis and the construction of point optimal unit root tests (Elliott et al (1996) and for a recent overview King and Sriananthakumar (2016)). Most recently, the methods have been used to study the uniform properties Date: April 18, 2017.…”