Bhattacharrya and Kshirsagar Lower Bounds for the Natural Exponential Family (NEF)

Abstract

The natural exponential families (NEF) play an important role in many fields of statistics. Two subclasses of NEF have interesting mathematical properties and have therefore been investigated by many researchers. These two subfamilies contain the distributions with quadratic variance function (NEF-QVF) and with cubic variance function (NEF-CVF), respectively.

In estimation theory, the variance of an unbiased estimator is of eminent importance whenever it is necessary to evaluate its accuracy. In many cases, however, it is difficult to calculate the variance because of its complicated form. In such cases one can use a lower bound of the variance as approximation. The best known lower bounds for unbiased estimators are the Cramér–Rao bound (by Rao, 1945 and Cramér, 1946), Bhattacharyya bound (by Bhattacharyya, 1946, 1947), Hammersley–Chapman–Robbins bound (by Hammersley, 1950 and Chapman and Robbins, 1951) and Kshirsagar bound (by Kshirsagar, 2000). The Bhattacharyya and Kshirsagar bounds, respectively, are generalizations of the Cramér–Rao and Hammersley–Chapman–Robbins bounds. Several authors such as Shanbhag (1972, 1979), Pommeret (1996), Mohtashami Borzadaran (2001), Tanka and Akahira (2003) and Tanaka (2006) derived some useful properties and characterizations of the Bhattacharrya bounds for NEF, NEF-QVF and NEF-CVF. This paper presents explicitly the Bhattacharyya and Kshirsagar bounds for some examples (normal, gamma, exponential, negative binomial, Abel and Takacs distributions) from NEF-QVF and NEF-CVF. The aim is twofold, first we want to illustrate by the examples that the bounds may well be used as approximation of unknown variances and secondly we want to compare the two bounds to show that sometimes the calculation of the Bhattacharyya bound is sufficient since the Kshirsagar bound is only marginally larger but much more difficult to compute.