A COMPARISON OF RESTRICTED AND UNRESTRICTED ESTIMATORS IN ESTIMATING LINEAR FUNCTIONS OF ORDERED SCALE

PARAMETERS OF TWO GAMMA DISTRIBUTIONS

YUAN-TSUNG CHANG^{1} AND NOBUO SHINOZAKI^{2}

1Department of Studies on Contemporary Society, Mejiro University, Iwatsuki, Saitama 339-8501, Japan

2Department of Administration Engineering, Faculty of Science and Technology, Keio University, Yokohama, Kanagawa 223-8522, Japan

(Received May 15, 2000; revised September 25, 2001)

Abstract. The problem of estimating linear functions of ordered scale parameters of two Gamma distributions is considered. A necessary and sufficient condition on the ratio of two coefficients is given for the maximum likelihood estimator (MLE) to dominate the crude unbiased estimator (UE) in terms of mean square error. A modified MLE which satisfies the restriction is also suggested, and a necessary and sufficient condition is also given for it to dominate the admissible estimator based solely on one sample. The estimation of linear functions of variances in two sample problem and also of variance components in a one-way random effect model is mentioned.

Key words and phrases: MLE, unbiased estimator, admissible estimator, variance estimation.