Extension of a two-stage conditionally unbiased estimator of the selected population to the bivariate normal case

Cohen and Sackrowitz (1989) derive a uniformly minimum variance conditionally unbiased estimator (UMVCUE) for the mean of a selected population for the univariate normal case with variance known and unknown as well as for the gamma case in a two-stage design. We extend this methodology to the bivariate normal case where the covariance matrix is assumed to be known. The population with the largest sample mean of the first dimension is selected for additional observations in a second stage. The goal of the analysis is to find an unbiased estimate of the mean of the second dimension with all of the data.