Efficiency of a Certain Modification of the Studentized Range of Symmetric Stable Random Variables

Abstract

We study the properties of tests constructed by a simple modification of the studentized range of the sample from a symmetric stable population in the problem of testing the hypothesis \( \mathrm{\mathscr{H}} \)_α (the stability index equals α, α ∈ (1, 2)) against the alternative \( \mathrm{\mathscr{H}} \)₂. We obtain approximate formulas for the calculation of critical values and estimation of the test power and develop a method for the estimation of the accuracy of these approximations. The major part of the paper deals with the construction of the approximations to the distribution function of the normalized sum of squares of symmetric stable random variables and estimation of the accuracy of approximations.