Robust estimation and testing of location for symmetric stable distributions

Along the line of growing interest in studying stable distributions, we propose a robust estimation technique for the location parameter δ of these distributions. It will be based on using some selector statistics Δ = $x1-q1-xq1x1-q2-xq2$ where x_q is the qth sample quantile and 0 < q₁ < q ₂ < 1/2, for selecting the index parameter α of a symmetric stable distribution where 0 < α ≤ 2. We then use a class of robust estimators, T = $T2l$(F_n), where $T2l$(F_n) is 2$l$-trimmed mean and 0 < $l$ ≤ 1/2, for estimating the location parameter δ. A new class of tests is then introduced and investigated V = $T2lFnx1-q1-xq1$ for testing H₀ : δ = 0, against H_A : δ ≠ 0. For two-sample problem, we introduce the class of tests V ={09}$T2l1Fnx-T2l2Fny12x1-q1-xq1+y1-q2-yq2${09}for testing H₀ : δ₁ − δ ₂ = 0 vs. H_A : δ₁ − δ ₂ ≠ 0. V statistic is similar to Student's t statistic. For each α, the asymptotic distributions of Δ and V will be derived and general formulas of means and variances of Δ and V will be given. Normal (α = 2) and Cauchy (α = 1) distributions will be used as special cases from the family of symmetric stable distributions to show the derivation of the asymptotic; distributions, the means and the variances. The performance of our robust procedures for estimating and testing is investigated using both computer simulation and real examples. Also, the power of these tests is studied. Our robust tests can be adapted to investigate other symmetric distributions.