Second-order power comparisons for a class of nonparametric likelihood-based tests

Asymptotic Results on a General Class of Empirical Statistics: Power and Confidence Interval Properties

“…4) satisfy (26). Thus, the second-order local maximinity of the ELR statistic, that was so far known in the Cressie-Reid and ED subclasses (Bravo, 2003), now stands extended to our general class. There are two key ingredients in the phenomenon just noted: (i) P (1) (γ ) = O(γ 3 ) for the ELR statistic, (ii)P (1) (γ ) is an odd function for every statistic in our class, even though unlike with ED statistics, U(g 3 , y) can involve odd powers of y.…”

“…As will be seen later, a(.) can be nonzero for our general class of statistics, a fact that may be contrasted with what happens for ED statistics (Bravo, 2003). Under contiguous alternatives, the power function corresponding to (6) is given by…”

“…, X n be independent scalar-valued random variables from an unknown common distribution with mean θ . We work under the same conditions as in Bravo (2003); these conditions justify the Edgeworth expansions considered later. Let…”

“…Remark 2 In contrast with the stochastic expansions considered in Baggerly (1998), Bravo (2003) or Mukerjee (2005), the one in (1) does not involve any unknown population moment other than θ. This helps not only in finding an asymptotic representation for the associated confidence interval (cf.…”

“…Let σ be the unknown population standard deviation under θ 0 , and we consider contiguous alternatives of the form H n : θ = θ n , where θ n = θ 0 + n −1/2 γ σ and γ is free from n. As in Bravo (2003), we note that in the present context of nonparametric inference, the population distribution functions under the null and contiguous alternative hypotheses are related in the sense that they are both assumed to belong to the same class of distributions, indexed by the mean θ . Thus the distribution of X i − θ n , under θ n , is the same as that of X i − θ 0 , under θ 0 (the difference with a parametric location model is that the form of the distribution is now unknown).…”