Given two independent simple random samples of equal size from two normal distributions, N(μ, σ2i) (i = 1, 2), the problem is to estimate the common mean μ, — ∞ < μ, < ∞. The variance ratio ρ = σ22/σ21 is unknown. Two classes of randomized unbiased estimators are studied. Let ( , S1, S2) denote the sample means and sums of squares of deviations. For both classes of estimators a two-sided F-test is performed. If F = S2/S1 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 S2/S1 > ρ and by if S2/S1 < 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.