SECOND ORDER ASYMPTOTIC COMPARISON OF ESTIMATORS UNDER UNIVERSAL DOMINATION CRITERION

The main purpose of this paper is to formulate theories of universal optimality, in the sense that some criteria for performances of estimators are considered over a class of loss functions. It is shown that the difference of the second order terms between two estimators in any risk functions is expressed as a form which is characterized by a peculiar value associated with the loss functions, which is referred to as the loss coefficient. This means that the second order optimal problem is completely characterized by the value of the loss coefficient. Furthermore, from the viewpoint of change of the loss coefficient, the relationship between two estimators is classified into six types. On the basis of this classification, the concept of universal second order admissibility is introduced. Some sufficient conditions are given to determine whether any estimators are universally admissible or not.     

Universal admissibility in second order asymptotics

Some sufficient conditions for an estimator to be universally second order admissible are derived. Those sufficient conditions consist of the elementary integrals with respect to the Fisher information and the limits of some functions characterized by the dealt statistical model, and thus can be checked with comparative ease. In location model and scale model, the sufficient condition for the linear estimator with respect to the maximum likelihood estimator (MLE) to be universally second order admissible is given. Furthermore, a guide for classifying any estimator into either the universal admissibility or the non-universal admissibility is proposed.     

Asymptotics of Likelihood Ratio Tests for General One-sided Hypotheses in the Two-sample Normal Model

We consider the general one-sided hypotheses testing problem expressed as H₀: θ₁ ? h(θ₂) versus H₁: θ₁ < h(θ₂), where h( · ) is not necessary differentiable. The values of the right and the left differential coefficients, h′_?( · ) and h′₊( · ), at nondifferentiable points play an essential role in constructing the appropriate testing procedures with asymptotic size α on the basis of the likelihood ratio principle. The likelihood ratio testing procedure is related to an intersection–union testing procedure when h′_?(θ₂) ? h′₊(θ₂) for all θ₂, and to a union–intersection testing procedure when there exists a θ₂ such that h′_?(θ₂) < h′₊(θ₂).     

On Second Order Admissibilities in Two-Parameter Logistic Regression Model

In two-parameter family of distribution, conditions for a modified maximum likelihood estimator to be second-order admissible are given. Applying these results to two-parameter logistic regression model, it is shown that the maximum likelihood estimator is always second-order inadmissible and the Rao-Blackwellized minimum logit chi-squared estimator is second-order admissible if and only if the number of the doses is greater than or equal to 6.