A new class of distribution free procedures for testing homogeneity of scale parameters against ordered alternatives

Abstract

Let X_{i 1},X_{i 2}…X_{i n},i = 1,2,…,K independent random samples of size n_i from absolutely continuous distributions with distribution functions F_i. We assume that these distribution functions have zero as the common quantile of order a(0≤α≤1), i.e., F_i(0) = α for i = 1,2, …,k. It is also assumed that F_i's are identical in all respects except possibly their scale parameters i.e., We wish to test the null hypothesis H₀:σ₁ = σ₂ = …=σ_k against the ordered alternative H₁ : σ1≤σ2≤…≤σ_k with at least one strict inequality. Define for i< j, i, j = 1,2,…,k, Ø_ij(X_ic, X_jd) = 1 if 0 ≤ x_ic ≤ X_jd or X_jd ≤ x_ic ≤0, and -1 if 0 ≤ x_jd ≤ X_jd or X_ic≤ x_jd ≤0, and 0 other-wise. Let U_ij be the corresponding two-sample U-statistic. For testing H₀ against H₁, with F_i(0) = α i = 1,2,…k, we propose a class of distribution free tests based on the statistics. Large values of T_k are significant for testing H₀ against H₁. The optimum values of a_is are obtained. The tests are quite efficient.

Keywords:

• Asymptotic relative efficiency
• U-statistics
• Districbution-free tests