The problem is to estimate the mean of the selected population. The selection rule is to choose the population with the largest sample mean when such sample means are calculated from the first stage sample. An estimator of the selected mean is unbiased if its expected value equals the expected value of the selected mean. We seek conditionally unbiased estimators of the selected mean given the ordering of the set of sample means based on the first stage sample. Conditionally unbiased estimators are of course unconditionally unbiased. For several distributions such as the normal, with unknown mean, and binomial, no conditionally unbiased estimators exist based on a one stage sample. We propose a two stage sample where observations at stage two are taken from the selected population only. Such a procedure has the advantage of yielding conditionally unbiased estimators and enables, possibly a better allocation of available sample points. We find the uniformly minimum variance conditionally unbiased estimators (UMVCUE) for the normal case when the variance is known or when a common unknown variance is present. We also find the UMVCUE for the gamma case and indicate that the method is suitable for many other cases as well.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
- selected mean
- two stage sample
- unbiased estimators
- uniformly minimum variance conditionally unbiased estimator