Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 2, 282--292

AN INEQUALITY FOR THE PITMAN ESTIMATORS RELATED TO THE STAM INEQUALITY

By

ABRAM KAGAN, University of Maryland, College Park, USA

SUMMARY. An inequality is proved for the variances of the Pitman estimators of a location parameter $\theta$ based on samples of a fixed size $n\geq 2$ from populations\\ $F_{1}(x-\theta), F_{2}(x-\theta)$ and $F(x-\theta), F=F_{1}\ast F_{2}$. The inequality is a natural small sample version of the Stam inequality\\ $1/I\geq 1/I_{1} +1/I_{2}$ where $I_{1}, I_{2}$ and $I$ are, respectively, the Fisher information on $\theta$ contained in an observation $X_{1}\sim F_{1}(x-\theta)$, $X_{2}\sim F_{2}(x-\theta)$ and $X\sim F(x-\theta)$. Some related inequalities are proved.

*AMS (1991) subject classification}. *Primary 62F11; secondary 62B10.

*Key words and phrases: *Pitman estimator, Fisher information, Stam inequality.