Optimal Tests when a Nuisance Parameter is Present Only Under the Alternative

TLDR

The study addresses nonstandard testing problems, such as structural change with unknown changepoint, where classical asymptotic optimality results for standard tests do not apply. The paper derives asymptotically optimal tests for problems where a nuisance parameter is present only under the alternative hypothesis. Optimal tests are generated using a weighted average power criterion. © 1994 The Econometric Society.

Abstract

This paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null. For example, the results apply to tests of structural change with unknown changepoint. The testing problem considered is nonstandard and the classical asymptotic optimality results for the Lagrange multiplier, Wald, and likelihood ratio do not apply. A weighted average power criterion is used here to generate optimal tests. This criterion is similar to that used by A. Wald (1943) to obtain the classical asymptotic optimality properties of Wald tests in 'regular' testing problems. Copyright 1994 by The Econometric Society.