Fixed-α and fixed-β efficiencies

¹ Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand
² School of Mathematics, University of Manchester, M13 9 PL Manchester, UK
saralees.nadarajah@manchester.ac.uk

Received: 6 April 2011

Abstract

Consider testing H₀ : F ∈ ω₀ against H₁ : F ∈ ω₁ for a random sample X₁, ..., X_n from F, where ω₀ and ω₁ are two disjoint sets of cdfs on ℝ = (−∞, ∞). Two non-local types of efficiencies, referred to as the fixed-α and fixed-β efficiencies, are introduced for this two-hypothesis testing situation. Theoretical tools are developed to evaluate these efficiencies for some of the most usual goodness of fit tests (including the Kolmogorov–Smirnov tests). Numerical comparisons are provided using several examples.