Tests for the Chacteristic Exponent and the Scale Parameter of Symmetric Stable Distributions

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Abstract
Let S (α, β, a, c) be a stable distribution, where α∈ (0, 2), −1 ≤ β ≤ 1, a ∈ IR and c > 0 are respectively, the characteristic exponent, the skewness, the location and the scale parameters. In this paper we shall develop tests for the parameters α and c of a symmetric stable random variable S (α, 0,0, c). The test statistics for the hypthesis H0: α = α0 vs. H1: α > α0 and H0:c = c0 vs. H1 : c > c0 are functions of the empirical characteristic function , where X1, …, Xn is a random sample of size n from a stable population S (α, 0,0, c). We shall study the asymptotic properties of the power functions of the tests above and their behaviour in the neighborhood of α0 and c0.