Confidence intervals for ranked means

Suppose that observations from populations π₁, …, π_k (k ≥ 1) are normally distributed with unknown means μ₁., μ_k, 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 X _i (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/n^1/2) Φ^?1(γ^1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ_i] with minimal probability of coverage γ is (X _ii] – g, + ∞) with g = (σ/n^1/2) Φ^?1(γ^1/i). For the upper confidence interval on μ_i] the maximal probability of coverage is 1– 1 – γ^1/k-i+1]ⁱ, while for the lower confidence interval on μ_i] the maximal probability of coverage is 1–1– γ^1/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(X ₁, …, X _k) for μ_i] with probability of coverage γ(0 < γ < 1) for all μ = (μ_1], …,μ_k]), does not exist.