A New Family of Continuous Distributions: Properties and Estimation

Mohamed Hussein,Howaida Elsayed,Gauss M Cordeiro

doi:10.3390/sym14020276

Abstract

We introduce a new flexible modified alpha power (MAP) family of distributions by adding two parameters to a baseline model. Some of its mathematical properties are addressed. We show empirically that the new family is a good competitor to the Beta-F and Kumaraswamy-F classes, which have been widely applied in several areas. A new extension of the exponential distribution, called the modified alpha power exponential (MAPE) distribution, is defined by applying the MAP transformation to the exponential distribution. Some properties and maximum likelihood estimates are provided for this distribution. We analyze three real datasets to compare the flexibility of the MAPE distribution to the exponential, Weibull, Marshall–Olkin exponential and alpha power exponential distributions.

Highlights

Introduction and EstimationSymmetry 2022, 14, The characteristics of classical distributions are limited and cannot represent all situations found in applications or may not provide a good fit in many practical situations.Adding parameters to a parent distribution is an effective way to make it more flexible and to improve the goodness of fit to real data
A simulation is performed to study the behavior of the maximum-likelihood estimates (MLEs) of the parameters α, β and λ of the modified alpha power exponential (MAPE) distribution in terms of their averages, absolute biases (ABs) and mean square errors (MSEs) of the estimates obtained from 1000 samples under different scenarios for sample sizes n = 50, 100, 150, 200, 250
Marshall–Olkin exponential (MOE), using a random sample of size 100 generated from the MAPE distribution with α = 2, β = 2 and λ = 0.5

Summary

Introduction

Adding parameters to a parent distribution is an effective way to make it more flexible and to improve the goodness of fit to real data. There are different methods to include parameters to a distribution. [1] seems to have been the first to raise an exponential cumulative distribution function (cdf) to a positive power. [6] added two parameters (r, p) to the survival function by considering a countable mixture of positive integer powers of general survival functions, where the mixing proportions are Pascal (r,p). [7] included two parameters to any probability density function (pdf), based on the form of a mixture, to construct a new family. [8] defined a method of adding a parameter to a parent distribution called the alpha power (AP) transformation. The cdf of the AP family (for x ∈ R) has the form

Methods

Results

Conclusion