Let x be distributed as a normal variable with mean [mu] and variance [image]. Let y be distributed as a normal variable with mean [mu] and variance [image]. Let [image] be distributed as a chi-square variable with m21 degrees of freedom, and let [image] be sim 1 22 ilary distributed with m2 degrees of freedom. Suppose all random variables are independent. If [image] are known, then the minimum variance unbiased estimator of [mu]. is [image] We will replace [image] s. in the above combined estimator and the following is proved: Under the above conditions on the random variables x, y, [image], and [image], and [image], a necessary and sufficient condition that the quantify [image] is an unbiased estimator of [mu] which has uniformly smaller variance than either x or y is that m1 and m2 are both larger than nine.