Affiliations: [a] Department of Econometrics and Business Statistics, Monash University, Victoria, Australia | [b] School of Economics, Finance and Marketing, RMIT University, Melbourne, Australia
Correspondence:
[*]
Corresponding author: Maxwell L. King, Department of Econometrics and Business Statistics, Monash University, Victoria, Australia. E-mail:max.king@monash.edu
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.
Keywords: Local to unity asymptotics, Neyman-Pearson lemma, nuisance parameters, power envelope, unit root testing
