ON TESTING AGAINST RESTRICTED ALTERNATIVES FOR THE VARIANCES OF GAUSSIAN MODELS

Let π₁,…,π_p be p independent normal populations with means μ₁…, μ_p and variances σ²₁,…, σ²_p respectively. Let X(n_i) be a simple random sample of size n_i from π_i, i = 1,…,p. Given the simple random samples X(n₁),…, X(n_p) from π₁,…,π_p respectively, a test has been proposed for testing the homogeneity of variances H₀: σ²₁=…σ²_p, against the restricted alternative, H₁: σ²₁≥…≥σ²_p, with at least one strict inequality. Some properties of the test are discussed and critical values are tabulated.     

A Classification Rule Whose Probability of Misclassification is Independent of the Variance

An observation ×_o is to be classified into one of two normal populations φ₁ and φ₂. A classification rule, the Two-stage sample Rule, R(TS), whose probability of misclassification, P[MC], is independent of the common but unknown variance is proposed. Some optimal properties of R(TS) are also discussed and some values of P[MC | R(TS)], the probability of misclassification given the rule R(TS), are tabulated.     

THE OPTIMAL CLASSIFICATION RULE FOR EXPONENTIAL POPULATIONS

Let π₁ and π₂ be two exponential populations with location parameters α and β respectively. It is shown that the probability of misclassification for the optimal rule, R_E, depends on α and β only through their ratio α/β=r, and that the smaller the ratio r (i.e. the smaller α is compared to β), the greater the superiority of the optimal rule R_E over the commonly used rule R_N. When α and β are unknown, the sample-based version R_E(s) of R_E exhibits the same pattern of superiority over the sample-based R_N(s) of R_N.