Confidence intervals for ranked means

Abstract

Suppose that observations from populations π1, …, πk (k ≥ 1) are normally distributed with unknown means μ1., μk, respectively, and a common known variance σ2. Let μ[1] μ … ≤ μ[k] denote the ranked means. We take n independent observations from each population, denote the sample mean of the n observation from π1 by Xi (i = 1, …, k), and define the ranked sample means X[1] ≤ … ≤ X[k]. The problem of confidence interval estimation of μ(1), …,μ[k] is stated and related to previous work (Section 1). The following results are obtained (Section 2). For i = 1, …, k and any γ(0 < γ < 1) an upper confidence interval for μ[i] with minimal probability of coverage γ is (− ∞, X[i]+ h) with h = (σ/n1/2) Φ−1(γ1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ[i] with minimal probability of coverage γ is (Xi[i] – g, + ∞) with g = (σ/n1/2) Φ−1(γ1/i). For the upper confidence interval on μ[i] the maximal probability of coverage is 1– [1 – γ1/k-i+1]i, while for the lower confidence interval on μ[i] the maximal probability of coverage is 1–[1– γ1/i] k-i+1. Thus the maximal overprotection can always be calculated. The overprotection is tabled for k = 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence interval h(X1, …, Xk) for μ[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ[1], …,μ[k]), does not exist.