Extension of a Two-Stage Conditionally Unbiased Estimator of the Selected Population to the Bivariate Normal Case

Abstract

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.

Keywords:

• Biased
• Bivariate normal distribution
• Clinical trial
• Conditional estimation
• Correlated observations
• Estimation after selection
• Maximum
• Ranking and selection
• Surrogate
• UMVUE

Mathematics Subject Classification:

• Primary 62F10
• Secondary 62H12

Acknowledgments

The authors wish to thank the reviewer for his/her attention to detail and helpful comments which improved the manuscript. We would also like to thank Mark F. Brady for his discussions about the clinical trial design used in GOG 0182.

The research of the first author was supported in part by National Science Foundation Grant No. DMS9704400 and a National Cancer Institute grant to the Gynecologic Oncology Group Statistical and Data Center (CA37517). The research of the second author was supported by National Science Foundation Grant No. DMS-0072207.