A Unified Framework for Characterizing Wrapped Distributions Using Truncated Moments and Reverse Hazard Function

Abstract

Bell and Nadarajah [1] provided an exceptional review of the existing "Wrapped Distributions" in an alphabetic order. We suggest every interested reader carefully read the "Introduction" given in their paper. The current paper deals with various characterizations of these distributions in two main sections, one for the continuous case and the other for the discrete case. The characterizations for the continuous wrapped case are based on a simple relationship between two truncated moments. It should be mentioned that for this characterization the cumulative distribution function need not have a closed form and depends on the solution of a first order differential equation, which provides a bridge between probability and differential equation. The fact that our characterizations in the continuous case does not require that the distribution function have closed\ form, set\ them apart from other existing characterizations results. We like to point out that our paper is a theoretical research in a challenging and mathematically elegant field, called the "Characterizations of Distributions". As we mentioned in our previous works, sometimes in real life cases, it is very difficult to obtain samples from a continuous distribution. The observed values are generally discrete due to the fact that they are not measured in continuum. In some cases, it may be possible to measure the observations via a continuous scale, however, they may be recorded in a manner in which a discrete model seems more suitable. Consequently, the discrete models are appearing quite frequently in applied fields and have attracted the attention of many researchers. The characterizations for the discrete wrapped case will be based on certain function of the random variable whose form depends on the nature of the distribution of the random variable.