On Ban Estimators for Chi Squared Test Criteria

Abstract

Wijsman (1959a) developed the theory of BAN estimators of a parameter $\beta$ under some fairly general conditions assuming that $n^{1/2}(y_n - g(\beta)) \rightarrow_L N_s(0, \sigma(\beta))$. The present article considers the complementary, but somewhat more general, approach under the constraint equation model that restricts the parameter $\mu$ so that $f(\mu) = 0$ under general conditions requiring $n^{1/2}(y_n - \mu) \rightarrow_L N_s(0, \sigma^\ast(\mu))$. At the same time, this article weakens Wijsman's differentiability requirement by introducing a $p$-differentiability condition for regular estimators. Next the theory of BAN estimation is developed for a model combining features of both of these approaches. As a special case of the model above, weighted least squares estimators for a general linear model are shown to be BAN.