Weighted multimodal family of distributions with sine and cosine weight functions

Ayman Alzaatreh,Jaber Kazempoor,Adel Ahmadi Nadi

doi:10.1016/j.heliyon.2020.e04757

Ayman Alzaatreh, Jaber Kazempoor + Show 1 more

Open Access

https://doi.org/10.1016/j.heliyon.2020.e04757

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Abstract

In this paper, the moment of various types of sine and cosine functions are derived for any random variable. For an arbitrary even probability density function, the sine and cosine moments are used to define new families of univariate multimodal probability density and their corresponding characteristic functions. For illustration, two weighted multimodal generalizations of the t distribution are investigated. Furthermore, a method of calculating some interesting improper integrals is also presented. Finally, an explicit expression of the probability density function of the sum of independent t-distributed random variables with odd degrees of freedom is derived.

Highlights

A multimodal distribution is a probability distribution with at least two modes
Multimodal distributions are popular in modelling various types of data
The first method is by applying weighted distribution technique with weight functions involving cosine and sine

Summary

Introduction

A multimodal distribution is a probability distribution with at least two modes (local maxima). The reader is referred to [1, 2] and the references cited therein Another way of modelling univariate multimodal data is by using weighted distributions. Some new family of multimodal distributions are proposed and discussed. These families are constructed in two ways. The first method is by applying weighted distribution technique with weight functions involving cosine and sine This technique requires calculating the moments of sine and cosine functions which is discussed in this paper. The second method is defining new family of distributions based on sine and cosine functions.

Main results

Applications

Conclusion