On the Reliability for Some Bivariate Dependent Beta and Kumaraswamy Distributions: A Brief Survey

Indranil Ghosh

doi:10.1515/eqc-2018-0029

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On the Reliability for Some Bivariate Dependent Beta and Kumaraswamy Distributions: A Brief Survey

Indranil Ghosh

Veröffentlicht/Copyright: 12. März 2019

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Abstract

In the area of stress-strength models, there has been a large amount of work regarding the estimation of the reliability R=Pr⁡(X<Y). The algebraic form for R=Pr⁡(X<Y) has been worked out for the vast majority of the well-known distributions when X and Y are independent random variables belonging to the same univariate family. In this paper, forms of R are considered when (X,Y) follow bivariate distributions with dependence between X and Y. In particular, explicit expressions for R are derived when the joint distribution are dependent bivariate beta and bivariate Kumaraswamy. The calculations involve the use of special functions.

Keywords: Reliability Parameter; Bivariate Dependent Beta and Kumaraswamy Distributions; Reliability Parameter For Bivariate Dependent Beta and Kumaraswamy Models

MSC 2010: 60Exx; 62Fxx

A Appendix

For the bivariate beta distribution [22], let us consider the integral

I1=∫x1f⁢(x,y)⁢d⁢y=∫x1C⁢yb-1⁢(1-x)b+c-1⁢(1-y)a+c-1⁢(1-x⁢y)-a-b-c⁢d⁢y=C⁢xa-1⁢∑k=0∞∫x1(a+b+c-1+kk)⁢yb-1⁢(1-y)b+c-1⁢(x⁢y)k⁢d⁢y=C⁢xa+k-1⁢(a+b+c-1+kk)⁢∑k=0∞∫x1yb+k-1⁢(1-y)a+c-1⁢d⁢y=C⁢xa+k-1⁢(a+b+c-1+kk)⁢[∑k=0∞B⁢(k+b,a+c)-∫0xyb+k-1⁢(1-y)a+c-1⁢d⁢y]=C⁢xa+k-1⁢(a+b+c-1+kk)⁢[∑k=0∞B⁢(k+b,a+c)-∑j=k+ba+c+k+b-1(a+b+c-1+kj)⁢xj⁢(1-x)a+c+k+b-1-j],

on expanding the incomplete beta function. Therefore, the expression for R will be

R=C⁢∫01xa-1⁢I1⁢d⁢x=C(a+b+c-1+kk)[∑k=0∞[B(k+b,a+c)∫01xa-1+kdx]-∑j=k+ba+c+k+b-1(a+b+c-1+kj)∫01xa-1+j(1-x)a+c+k+b-j-1dx]=C(a+b+c-1+kk)[∑k=0∞B⁢(k+b,a+c)a+k-∑j=k+ba+c+k+b-1(a+b+c-1+kj)B(a+j,a+c+k+b-j)].

For the bivariate Libby–Novick–Jones–Olkin–Liu Kumaraswamy model [3], let us consider the integral

(A.1)I1=∫x1yδ2-1⁢(1-yδ2)α⁢(1-xδ1⁢yδ2)α+2⁢d⁢y=∫x1yδ2-1⁢(1-yδ2)α⁢[∑k=0∞(α+2+k-1k)⁢(xδ1⁢yδ2)k]⁢d⁢y=∑k=0∞(α+2+k-1k)⁢xk⁢δ1⁢∫x1yδ2⁢(k+1)-1⁢(1-yδ2)α=∑k=0∞(α+2+k-1k)⁢xk⁢δ1⁢I2,say,

where

(A.2)I2=∫x1yδ2-1⁢(1-yδ2)α⁢d⁢y=∫xδ21wk⁢(1-w)α⁢d⁢w,* on substituting⁢w=yδ2⁢d⁢y,=B⁢(k+1,α+1)-∑j=k+1α+k-1xj⁢δ2⁢(1-xδ2)α+k-1-j.

Therefore, from (A.1) and (A.2), the expression for R will be

R=α⁢(α+1)⁢∫01xδ1⁢(1-xδ1)α×[∑k=0∞(α+2+k-1k)⁢xk⁢δ1⁢{B⁢(k+1,α+1)-∑j=k+1α+k-1xj⁢δ2⁢(1-xδ2)α+k-1-j}]⁢d⁢x=α⁢(α+1)⁢[∑k=0∞(α+2+k-1k)⁢{(B⁢(k+1,α+1))2-∑j=k+1α+k-1∑ℓ=0∞(-1)ℓ⁢(α+k-1-jℓ)⁢(2⁢j⁢δ2δ1,α+1)}],

after some algebraic simplification.

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Received: 2018-11-03

Accepted: 2019-01-29

Published Online: 2019-03-12

Published in Print: 2019-12-01

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https://doi.org/10.1515/eqc-2018-0029

Schlagwörter für diesen Artikel

Reliability Parameter; Bivariate Dependent Beta and Kumaraswamy Distributions; Reliability Parameter For Bivariate Dependent Beta and Kumaraswamy Models