Form-Invariance of the Non-Regular Exponential Family of Distributions

Abstract

Abstract The weighted distributions are used when the sampling mechanism records observations according to a nonnegative weight function. Sometimes the form of the weighted distribution is the same as the original distribution except possibly for a change in the parameters that are called the form-invariant weighted distribution. In this paper, by identifying a general class of weight functions, we introduce an extended class of form-invariant weighted distributions belonging to the non-regular exponential family which included two common families of distribution: exponential family and non-regular family as special cases. Some properties of this class of distributions such as the su-cient and minimal su-cient statistics, maximum likelihood estimation and the Fisher information matrix are studied.

Keywords

• Fisher information matrix
• Form-invariance
• Non-regular exponential family
• Maximum likelihood estimation
• Weighted distribution