On Bahadur Efficiency of Power Divergence Statistics

TL;DR

This paper analyzes the Bahadur efficiency of power divergence statistics, showing the information divergence statistic's superior efficiency under certain conditions and comparing efficiencies across different orders.

Contribution

It extends previous results by relaxing assumptions and evaluates the Bahadur efficiency of power divergence statistics for all positive orders, providing a comprehensive comparison.

Findings

Information divergence statistic is infinitely more Bahadur efficient than  power divergence statistics.

Power divergence statistics of orders 0<  are mutually Bahadur comparable.

All  power divergence statistics are more efficient than those with  > 1.

Abstract

It is proved that the information divergence statistic is infinitely more Bahadur efficient than the power divergence statistics of the orders $\alpha >1$ as long as the sequence of alternatives is contiguous with respect to the sequence of null-hypotheses and the the number of observations per bin increases to infinity is not very slow. This improves the former result in Harremo\"es and Vajda (2008) where the the sequence of null-hypotheses was assumed to be uniform and the restrictions on on the numbers of observations per bin were sharper. Moreover, this paper evaluates also the Bahadur efficiency of the power divergence statistics of the remaining positive orders $0< \alpha \leq 1.$ The statistics of these orders are mutually Bahadur-comparable and all of them are more Bahadur efficient than the statistics of the orders $\alpha > 1.$ A detailed discussion of the technical definitions and conditions is given, some unclear points are resolved, and the results are illustrated by examples.