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Volume 19, Issue 4, December 2020, Pages 518 - 525

A Class of Beta Second Kind Mixture Distributions

Authors
Mian Arif Shams Adnan1, *, ORCID, Humayun Kiser2
1Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio, 43402, USA
2School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, England
*Corresponding author. Email: maadnan@bgsu.edu
Corresponding Author
Mian Arif Shams Adnan
Received 21 December 2019, Accepted 20 March 2020, Available Online 4 January 2021.
DOI
10.2991/jsta.d.201229.001How to use a DOI?
Keywords
Methods of moments; Mixing distribution; Mixtured distribution
Abstract

A class of mixture distributions have been derived which we call beta second kind mixtures of distributions. Various integral representations of beta functions can be obtained using these mixture beta distributions. Estimation of unknown parameters along with some characteristics of these distributions are also been found.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Mixture distribution [1,2] was first coined in 1894. A number of authors like Pearson [3], Rider [4], Blichke [5–8], Cohen [9], Chahine [10], Hasselblad [11], Day [12], Jewell [13], Roy et al. [14–16], Adnan et al. [17–26] defined mixtures of two distributions and studied various mixtured distributions which they called poisson mixture, binomial mixture, negative binomial mixture, chi-square mixture, erlang mixture, laplace mixture, pareto mixture, F mixture, weibull mixture and Maxwell mixture of distributions. Adnan et al. [27,28] also studied several properties of triple and folded Gamma mixture distributions.

2. PRELIMINARIES

A mixture distribution is a weighted average of probability distribution of positive weights that sum to one. The weights themselves constitutes a probability distribution. The mixing distribution gx;θ is a weighted average of a distribution in either in ∑i=1kfx;θigθi form or in ∫fx;θgθdθ form.

3. MAIN RESULTS

Here in this paper, we define the beta second kind mixtures of some well-known distributions such as normal, lognormal, gamma, exponential, beta second kind, rectangular, erlang, chi-square, t, and F distributions. Then some characteristics of these distributions such as characteristic functions, moments, and shape characteristics are also obtained. The main results of the paper are presented in form of definitions and theorems.

Definition 3.1

A random variable X is said to have a beta second kind mixtured distribution if its probability density function is defined as

fx ;p,q,α=∫0∞1Bp,qr p−11+rp+qgx ;αdr(1)
where gx;α is a probability density function. The name of second kind mixture distribution comes from the fact that the distribution (1) is the weighted average of gx;α with weights equal to the ordinates of second kind distribution.

Definition 3.2

If X follows a beta second kind mixture of normal distribution with parameters p and q, then the density function is given by

fx;p,q=∫0∞1Bp,qrp−11+rp+qe−12x2x2r2r+12Γr+12dr; −∞<x<∞(2)
with parameters p and q since
∫−∞∞fx ;p,qdx=1(3)

The characteristic function and moments of the same distribution are presented in the theorem below.

Theorem 3.1.

If X has a beta second kind mixture of normal distributions with parameters p and q then its characteristic function is represented as

∫0∞1Bp,qrp−11+rp+qe−12t22r+12Γr+12∑m=0r2r2mit2m2r+12−mΓr+12−mdr(4)
and the 2sth moment about origin is ∫0∞1Bp,qrp−11+rp+q2sΓr+12+sΓr+12dr and 2s+1th moment about origin is zero. Mean=0, Variance=1+2pq−1,
β1=0,   β2=3+8pq−1+4pp+1q−1q−21+2pq−12.

Remark

If p=q=0 then all the values of ϕxt,μ2s+1∕,μ2s∕,μ1,μ2,μ3,μ4,β1 and β2 are true for normal distribution with mean zero and variance unity.

Definition 3.3

If a random variable X has the density function

fx;p,q=∫0∞1Bp,qrp−11+rp+qe−12logx2logx2rx2r+12Γr+12dr;  x>0,(5)
then it is said to have a beta second kind mixture of lognormal distribution with parameter p and q since
∫0∞fx ;p,qdx=1(6)

Various moments of the distribution are given in the next theorem.

Theorem 3.2.

If X is a beta second kind mixture of lognormal variable with parameters p and q then its characteristic function is given by

∫0∞1Bp,qrp−11+rp+q12r+12Γr+12∑k=0∞itkk!e12k2∑m=0r2r2mk2r−2m2m+12Γm+12dr(7)
and the sth moment about origin is
μs/=∫0∞1Bp,qrp−11+rp+qe12s22r−mΓr+12∑m=0r2r2ms2r−2mΓm+12dr

Definition 3.4

A random variable X having the density function

fx;p,q,α,β=∫0∞1Bp,qrp−11+rp+qβα+re−βxxα+r−1Γα+rdr;  x>0,(8)
is defined a beta second kind mixture of Gamma distribution with parameters p,q,α and β since
∫0∞fx ;p,q,α,βdx=1.(9)

The characteristic function and moments are provided in the theorem below.

Theorem 3.3.

If X denotes a beta second kind mixture of gamma variate with parameters p,q,α and β then its characteristic function is obtained as

1Bp,q1−itβ−α∫0∞rp−11+rp+qe−rln1−itβdr(10)
and Mean=1βα+pq−1, Variance=1β2α+pq−1+pp+1q−1q−2−p2q−12,
β1=2α+2pq−1+3pp+1q−1q−2+pp+1p+2q−1q−2q−3−3p2q−12−3p2p+1q−12q−2+2p3q−132α+pq−1+pp+1q−1q−2−p2q−123,
β2=3α2+6α+6α+6pq−1+6α+11pp+1q−1q−2+6pp+1p+2q−1q−2q−3+pp+1p+2p+3q−1q−2q−3q−4−6α+8p2q−12−12p2p+1q−12q−2−4p2p+1p+2q−12q−2q−3+6p3q−13+6p3p+1q−13q−2−3p4q−14α+pq−1+pp+1q−1q−2−p2q−122.

Remark

If p=q=0 then all the values of ϕxt,μs∕,μ1,μ2,μ3,μ4,β1 and β2 are true for Gamma distribution with parameters α and β.

Estimates of parameters by the method of moments: Let X1,X2,…..Xm be a random sample from the distribution (8). We assume that parameters p,q and β are known. Then the distribution contains only one unknown parameter α. We have μ′1=1βα+pq−1 and m′1=∑xim=X¯. Hence by the method of moments, we get 1βα+pq−1=X¯. Therefore,

α^=X¯β−pq−1(11)

Definition 3.5

A random variable X having the density function

fx;p,q,α=∫0∞1Bp,qrp−11+rp+qαr+1e−αxxrΓr+1dr;  x>0,(12)
is said to have a beta second kind mixture of Exponential distribution with parameters p,q and α since
∫0∞fx;p,q,αdx=1(13)

Various characteristics of the above distribution are described in the following theorem.

Theorem 3.4.

If X follows beta second kind mixture of exponential distributions with parameters p,q and α then its characteristic function is given by

1Bp,q1−itα−1∫0∞rp−11+rp+qe−rln1−itαdr(14)
and Mean=1α1+pq−1, Variance=1α21+pq−1+pp+1q−1q−2−p2q−12,
β1=2+2pq−1+3pp+1q−1q−2+pp+1p+2q−1q−2q−3−3p2q−12−3p2p+1q−12q−2+2p3q−1321+pq−1+pp+1q−1q−2−p2q−123,
β2=9+12pq−1+17pp+1q−1q−2+6pp+1p+2q−1q−2q−3+pp+1p+2p+3q−1q−2q−3q−4−14p2q−12−12p2p+1q−12q−2−4p2p+1p+2q−12q−2q−3+6p3q−13+6p3p+1q−13q−2−3p4q−141+pq−1+pp+1q−1q−2−p2q−122.

Remark

If p=q=0 then all the values of ϕxt,μs/,μ1,μ2,μ3,μ4,β1 and β2 are true for Exponential distribution with parameter α.

Method of moments: If X1,X2,…..Xm be a random sample drawn from the distribution (12) and parameter p,q is assumed known, then the distribution contains only one unknown parameter α. According to the method of moments, we get 1α1+pq−1=X¯. Therefore,

α^=1+pq−1X¯(15)

Definition 3.6

If a random variable X has the density function

fx;p,q,α,β=∫0∞1Bp,qrp−11+rp+qαβα+re−αβxxα+r−1Γα+rdr;  x>0,(16)
then it is said to have a beta second kind mixture of Erlang distribution with parameters p,q,α and β since
∫0∞fx;p,q,α,βdx=1(17)

The characteristic function as well as the moments is stated in the following theorem.

Theorem 3.5.

If X has beta second kind mixture of erlang distributions with parameters p,q,α and β then its characteristic function is given by

1Bp,q1−itαβ−α∫0∞rp−11+rp+qe−r ln1−itαβdr(18)
and Mean=1αβα+pq−1, Variance=1αβ2α+pq−1+pp+1q−1q−2−p2q−12,
β1=2α+2pq−1+3pp+1q−1q−2+pp+1p+2q−1q−2q−3−3p2q−12−3p2p+1q−12q−2+2p3q−132α+pq−1+pp+1q−1q−2−p2q−123
β2=3α2+6α+6α+6pq−1+6α+11pp+1q−1q−2+6pp+1p+2q−1q−2q−3+pp+1p+2p+3q−1q−2q−3q−4−6α+8p2q−12−12p2p+1q−12q−2−4p2p+1p+2q−12q−2q−3+6p3q−13+6p3p+1q−13q−2−3p4q−14α+pq−1+pp+1q−1q−2−p2q−122.

Remark

If p=q=0 then all the values of ϕxt,μs∕,μ1,μ2,μ3,μ4,β1 and β2 are true for Erlang distribution with parameters α and β.

Estimating parameters: For a random sample X1,X2,…..Xm from the distribution (16), we assume that parameters p,q and β is known and α unknown parameter. Now, μ′1=1αβα+pq−1 and m′1=∑xim=X¯. We obtain 1αβα+pq−1=X¯. Therefore,

α^=pq−1X¯β−1(19)

Definition 3.7

A random variable X having the density function

fx;p,q,m=∫0∞1Bp,qrp−11+rp+qr+1xrmr+1dr;  0<x<m,(20)
is said as beta second kind mixture of Rectangular distribution with parameters p,q and m since
∫0mfx;p,q,mdx=1.(21)

Different moments of the abovementioned distribution are expressed in the theorem below.

Theorem 3.6.

If X follows a beta second kind mixture of rectangular distribution with parameters p,q and m then its characteristic function is obtained as

∫0∞1Bp,qrp−11+rp+q∑k=0∞itkk!r+1mr+1mr+k+1r+k+1dr(22)
and the sth moment about origin is
ms∫0∞1Bp,qrp−11+rp+qr+1r+s+1dr.

Remark

If p=q=0 then all the values of ϕxt,μs∕,μ1∕,μ2∕,μ2, are true for Rectangular distribution with parameter m.

Definition 3.8

A random variable X having the density function

fx,p,q,α,β=∫0∞1Bp,qrp−11+rp+qxα+r−11−xβ−1Bα+r,βdr;  0<x<1,(23)
is called a beta second kind mixture of Beta distribution of first kind with parameters p,q,α and β since
∫0∞fx;p,q,α,βdx=1.(24)

Different moments of the same distribution are provided in the following theorem.

Theorem 3.7.

If X follows beta second kind mixture of beta distribution of first kind with parameters p,q,α and β then its sth moment about origin is given by

∫0∞1Bp,qrp−11+rp+qBα+s+r,βBα+r,βdr.(25)

Remark

If we put p=q=0 then all the values of μs∕,μ1∕,μ2∕,and μ2 are true for beta distribution of first kind with parameters α and β.

Definition 3.8

A random variable χ2 with the density function

fχ2;p,q,n=∫0∞1Bp,qrp−11+rp+qe−12χ2χ2n2+r−12n2+rΓn2+rdr;  χ2>0,(26)
is said to have a beta second kind mixture of chi-square distribution having the parameters p,q and n since
∫0∞fχ2;p,q,ndχ2=1(27)

Some characteristics of the distribution are represented in the theorem below.

Theorem 3.7.

If χ2 has beta second kind mixture chi-square distribution with parameters p,q and n then its characteristic function is expressed as

1Bp,q1−2it−n2∫0∞rp−11+rp+qe−rln1−2itdr(28)
and Mean=n+2pq−1, Variance=2n+4pq−1+4pp+1q−1q−2−4p2q−12
β1=8n+16pq−1+24pp+1q−1q−2+8pp+1p+2q−1q−2q−3−24p2q−12−24p2p+1q−12q−2+16p3q−1322n+4pq−1+4pp+1q−1q−2−4p2q−123,
β2=12n2+48n+48n+96pq−1+48n+176pp+1q−1q−2+96pp+1p+2q−1q−2q−3+16pp+1p+2p+3q−1q−2q−3q−4−48n+128p2q−12−192p2p+1q−12q−2−64p2p+1p+2q−12q−2q−3+96p3q−13+96p3p+1q−13q−2−48p4q−142n+4pq−1+4pp+1q−1q−2−4p2q−122.

Remark

Putting p=q=0 we find that all the values of ϕxt,μ1,μ2,μ3,μ4,β1 and β2 are true for chi-square distribution with parameters n.

Estimates of parameters: Let X1,X2,…..Xm be a random sample from the distribution (26). We assume that parameters p and q is known and n is unknown. Now, μ′1=n+2pq−1 and m′1=∑xim=X¯. Hence, we get n+2pq−1=X¯. Therefore,

n^=X¯−2pq−1(29)

Definition 3.9

If t as a random variable has the density function

ft;p,q,n=∫0∞1Bp,qrp−11+rp+qt2rn12+rB12+r,n21+t2nn+12+rdr;  −∞<t<∞,(30)
then it is said to have a beta second kind mixture of t distribution with parameters p,q and n provided
∫−∞∞ft;p,q,ndt=1.(31)

The following theorem expresses here some of the properties of the distribution.

Theorem 3.8.

If t is beta second kind mixture of t distribution with parameters p,q and n then the 2sth moment about origin is given by

ns∫0∞1Bp,qrp−11+rp+qΓr+s+12Γn2−sΓ12+rΓn2dr(32)
and the 2s+1th moment about origin is zero. β1=0, β2=n−2n−43+8pq−1+4pp+1q−1q−21+2pq−12.

Remark

If p=q=0 then all the values of μ2s+1,μ2s,μ1,μ2,μ3,μ4,β1 and β2 are true for t distribution with parameter n.

Definition 3.10

A random variable F having the density function

fF;p,q,n1,n2=∫0∞1Bp,qrp−11+rp+qn1n2n12+rFn12+r−1Bn12+r,n221+n1n2Fn1+n22+rdr;  F>0,(33)
is said to have a beta second kind mixture of F distribution with parametersp,q,n1 and n2 since
∫0∞fF;p,q,n1,n2dF=1(34)

The following theorem presents the characteristic function and moments of this distribution.

Theorem 3.9.

If F follows beta second kind mixture of F distribution with parameters p,q,n1 and n2 then the its characteristic function is given by

∫0∞1Bp,qrp−11+rp+q∑x=0∞itn2n1xx!Γn12+r+xΓn22−xΓn12+rΓn22(35)
and the sth moment about origin is
n2n1s∫0∞1Bp,qrp−11+rp+qΓn12+r+sΓn22−sΓn12+rΓn22dr.(36)

Then, Mean=n22n12(n2−2)(n2−4)n1n1+2+4n1+1pq−1+4pp+1q−1q−2 Variance=n22n12(n2−2)(n2−4)n1n1+2+4n1+1pq−1+4pp+1q−1q−2−n2n1(n2−2)n1+2pq−12.

Remark

If p=q=0 then all the values of ϕxt,μs∕,μ1∕,μ2∕ and μ2, are true for F distribution with parameters n1 and n2.

4. CONCLUSION

Various integral representations of beta functions can be found using the mixture beta distributions. These integrals are useful in finding mathematical and statistical properties of the various beta behavioral populations. A comparison among various features of the different beta second kind mixture distributions is shown in the tables of Appendix.

CONFLICTS OF INTEREST

Author has no conflicts of interest.

APPENDIX

A comparison among various features of the different beta second kind mixtured distributions is shown in the following Tables A1 and A2.

Sl. Name of the Distribution Probability Density Function fx Support Parameters
1 Beta 2nd kind mixture normal ∫0∞1Bp,qrp−11+rp+qe−12x2x2r2r+12Γr+12dr −∞<x<∞ p,q
2 Beta 2nd kind mixture lognormal ∫0∞1Bp,qrp−11+rp+qe−12logx2logx2rx 2r+12Γr+12dr x>0 p,q
3 Beta 2nd kind mixture gamma ∫0∞1Bp,qrp−11+rp+qβα+re−βxxα+r−1Γα+rdr x>0 p,q,α,β
4 Beta 2nd kind mixture exponential ∫0∞1Bp,qrp−11+rp+qαr+1e−αxxrΓr+1dr x>0 p,q,α
5 Beta 2nd kind mixture Erlang ∫0∞1Bp,qrp−11+rp+qαβα+re−αβxxα+r−1Γα+rdr x>0 p,q,α,β
6 Beta 2nd kind mixture rectangular ∫0∞1Bp,qrp−11+rp+qr+1xrmr+1dr 0<x<m p,q,m
7 Beta 2nd kind mixture beta 1st kind ∫0∞1Bp,qrp−11+rp+qxα+r−11−xβ−1Bα+r,βdr 0<x<1 p,q,α,β
8 Beta 2nd kind mixture chi-square ∫0∞1Bp,qrp−11+rp+qe−12χ2(χ2)n2+r−12n2+rΓn2+rdr χ2>0 p,q,n
9 Beta 2nd kind mixture t ∫0∞1Bp,qrp−11+rp+qt2rn12+rB12+r,n21+t2nn+12+rdr −∞<t<∞ p,q,n
10 Beta 2nd kind mixture F ∫0∞1Bp,qrp−11+rp+qn1n2n12+rFn12+r−1Bn12+r,n221+n1n2Fn1+n22+rdr F>0 p,q,n1,n2
Table A1

Comparison of density functions of different beta second kind mixture distributions.

Sl. Name of the Distribution Mean Variance
1 Beta 2nd kind mixture normal 0 1+2pq−1
2 Beta 2nd kind mixture lognormal Can be obtained from Equation (7) Can be obtained from Equation (7)
3 Beta 2nd kind mixture gamma 1βα+pq−1 1β2α+pq−1+pp+1q−1q−2−p2q−12
4 Beta 2nd kind mixture exponential 1α1+pq−1 1α21+pq−1+pp+1q−1q−2−p2q−12
5 Beta 2nd kind mixture Erlang 1αβα+pq−1 1αβ2α+pq−1+pp+1q−1q−2−p2q−12
6 Beta 2nd kind mixture rectangular Can be achieved from Equation (22) Can be achieved from Equation (22)
7 Beta 2nd kind mixture beta 1st kind Equation (25) provides Equation (25) provides
8 Beta 2nd kind mixture chi-square n+2pq−1 2n+4pq−1+4pp+1q−1q−2−4p2q−12
9 Beta 2nd kind mixture t 0 nn−21+2pq−1
10 Beta 2nd kind mixture F n22n12n2−2n2−4n1n1+2+4n1+1pq−1+4pp+1q−1q−2 n22n12n2−2n2−4n1n1+2+4n1+1pq−1+ 4pp+1q−1q−2−n2n1n2−2n1+2pq−12
Table A2

Comparison among first two moments of different beta second kind mixture distributions.

Comments If α=1, then the beta 2nd kind mixture of gamma distribution and Beta 2nd kind mixture of Erlang distribution becomes Beta 2nd kind mixture of exponential distribution.

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518 - 525
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2021/01/04
ISSN (Online)
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DOI
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TY  - JOUR
AU  - Mian Arif Shams Adnan
AU  - Humayun Kiser
PY  - 2021
DA  - 2021/01/04
TI  - A Class of Beta Second Kind Mixture Distributions
JO  - Journal of Statistical Theory and Applications
SP  - 518
EP  - 525
VL  - 19
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.201229.001
DO  - 10.2991/jsta.d.201229.001
ID  - Adnan2021
ER  -