An Empirical Process P-Value Test When a Nuisance Parameter is Present Under Either or Both Hypotheses

Abstract

We present a new test when there is a nuisance parameter λ under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the subset of λ on which a p-value test based on a test statistic Tn(λ) rejects the null hypothesis. The PVOT has only been explored in Hill and Aguilar (2013) and Hill (2012) as a way to smooth over a trimming parameter for heavy tail robust test statistics. Our key contributions are: (i) we show that a weighted average local power of a test based on Tn(λ) is identically a weighted average mean PVOT, and the PVOT used for our test is therefore a point estimate of the weighted average probability of PV test rejection, under the null; (ii) an asymptotic critical value upper bound for our test is the significance level itself, making inference easy (as opposed to supremum and average test statistic transforms which typically require a bootstrap method for p-value computation); (iii) we only require Tn(λ) to have a known or bootstrappable limit distribution, hence we do not require √n-Gaussian asymptotics as is nearly always assumed, and we allow for some parameters to be weakly or non-identified; and (iv) a numerical experiment, in which local asymptotic power is computed for a test of omitted nonlinearity, reveals the asymptotic critical value is exactly the significance level, and the PVOT test is virtually equivalent to a test with the greatest weighted average power in the sense of Andrews and Ploberger (1994)We give examples of PVOT tests of omitted nonlinearity, GARCH effects and a one time structural break. A simulation study demonstrates the merits of PVOT test of omitted nonlinearity and GARCH effects, and demonstrates the asymptotic critical value is exactly the significance level.