Characterization and estimation of some probability distributions by minimum χ2−divergence principle

Abstract EN

A probability distribution can be characterized through various methods, given a prior probability distribution g(x) and some available information on moments of the random variable X, the original probability distribution f(x) is determined such that the χ2- divergence measure of the distance between f(x) and g(x) is a minimum.

The minimum chi-squared density function f(x) is determined.

The expressions of the non-central moments as well as the cumulative distribution function F(x), survival function S(x), and hazard function h(x) are also determined under different available information on moments.

In this paper we discussed the characterization of the exponential and exponentiated exponential (EExp) distribution.

The available information on moments included: first moment, second moment and the first two moments.

Some illustrative examples are included for special values of the parameters.

Moreover, we have considered the principle of minimizing chi square divergence and used it for characterizing the probability distributions given a prior distribution as the exponential and the partial information in terms of averages and variance.

It is observed that the probability distributions which minimize the χ2 -distance also minimize the Kullback's measure of the directed divergence.

It is shown that by applying the minimum chi square divergence principle, new probability distributions are obtained.

Hence, the probability models can be revised to find the best estimated probability models given the new information on moments