Testing Homogeneity of Uniform Scale Distributions against Two-Sided and One-Sided Alternatives

Abstract

Consider the model where X_ij, i = 1, 2, …, k; j = 1, …, n _i are independent uniform random variables with scale parameter θ _i > 0. Test H ₀: θ_l = ··· = θ _k versus H ₂: not H ₀. Also consider the alternative H ₁: θ_l ≤ ··· ≤ θ _k . For H ₀ versus H ₂ and n _i = n, we obtain a complete class of constant-size permutation invariant tests and show that each test in the class is unbiased. The likelihood ratio test and others are in this class. For H ₀ versus H ₁ we obtain a complete class of constant-size tests based on a set of variables that is a transformation of the sufficient statistics. Again we show that each test in the class is unbiased. The likelihood ratio test is in this class. We derive the exact and asymptotic distribution of the likelihood ratio test. All results also hold for testing homogeneity of location parameters of exponential distributions. This latter case is of considerable practice importance in reliability.

Key Words:

• Association
• Complete class
• Conditionally increasing in sequence
• Exponential distributions with unknown location parameters
• Likelihood ratio test
• Permutation invariant
• Simple order alternative
• Unbiased test