The Large-Sample Power of Rank Order Tests in the Two-Sample Problem

This paper studies the large-sample power of certain rank order tests against one-parameter alternatives in the two-sample problem. The first m of N independent random variables are supposed identically distributed, each with a density function f₁(x, θ), the remaining $N - m$ with a density function f₂(x, θ). When θ = 0 both density functions are the same. Let aN1, ⋯, aNN be a set of constants defined by (3.2) below; let bN1, ⋯, bNN be another set of constants; and let R₁, ⋯ RN be the ranks of the N random variables. A statistic of the type ∑Ni = 1 aNibNRᵢ is called an L statistic. Part I of this paper characterizes the locally best rank order statistic for testing H₀:θ = 0 against the alternative that θ is positive and "close" to zero. This turns out to be any one of an equivalent class of L statistics. Under certain regularity conditions it is possible to determine the large-sample power of L statistics. Of particular interest is the large-sample power of the locally best L statistic. For arbitrary bN1, ⋯, bNN it is usually difficult to determine whether the regularity conditions hold. Hence, in Part II a special class of L statistics, the Lₕ statistics, are studied. For these, the regularity conditions are easier to verify and the large-sample power is determined. The best L statistic can, in a certain sense, be approximated by Lₕ statistics.