Unbiased Estimation of the Common Mean of Two Normal Distributions Based on Small Samples of Equal Size

Abstract

Abstract Given two independent simple random samples of equal size from two normal distributions, N(μ, σ2 i ) (i = 1, 2), the problem is to estimate the common mean μ, — ∞ < μ, < ∞. The variance ratio ρ = σ2 2/σ2 1 is unknown. Two classes of randomized unbiased estimators are studied. Let (, S 1, S 2) denote the sample means and sums of squares of deviations. For both classes of estimators a two-sided F-test is performed. If F = S 2/S 1 falls in the interval (1/ρ, ρ), where ρ is a parameter, the estimator used is . Otherwise the estimator is for the first class whereas for the second class μ is estimated by if S 2/S 1 > ρ and by if S 2/S 1 < 1/ρ*. The variances and the efficiency functions of these estimators are studied. Explicit formulae are given for the case of samples of size n = 3. Recommendations are made for a robust estimation procedure.