Statistica Sinica 16(2006), 165-181

ESTIMATION FOLLOWING A GROUP SEQUENTIAL TEST

FOR DISTRIBUTIONS IN THE

ONE-PARAMETER EXPONENTIAL FAMILY

Aiyi Liu$^{1}$, W. J. Hall$^{2}$, Kai F. Yu$^{1}$ and Chengqing Wu$^{1, 3}$

$^{1}$National Institute of Child Health and Human Development,

$^{2}$University of Rochester and $^{3}$University of Science and Technology of China

Abstract: We consider unbiased estimation following a group sequential test for distributions in a one-parameter exponential family. We show that, for an estimable parameter function, there exists uniquely an unbiased estimator depending on the sufficient statistic and based on the truncation-adaptation criterion (Liu and Hall (1999)); moreover, this estimator is identical to one based on the Rao-Blackwell method. When completeness fails, we show that the uniformly minimum-variance unbiased estimator may not exist or might possess undesirable performance. A Phase-II clinical trial application with exponentially distributed responses is included.

Key words and phrases: Clinical trials, completeness, Laplace transform, minimum variance, truncation-adaptation, unbiased estimation.