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

Indranil Ghosh

doi:10.1515/eqc-2018-0029

Article

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

Indranil Ghosh

Published/Copyright: March 12, 2019

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From the journal Stochastics and Quality Control Volume 34 Issue 2

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.

References

[1] B. C. Arnold, E. Castillo and J. M. Sarabia, Conditional Specification of Statistical Models, Springer Ser. Statist., Springer, New York, 1999. Search in Google Scholar

[2] B. C. Arnold and I. Ghosh, Bivariate Kumaraswamy models involving use of Arnold–Ng copulas, J. Appl. Statist. Sci. 22 (2014), no. 3–4, 227–240. Search in Google Scholar

[3] B. C. Arnold and I. Ghosh, Some alternative bivariate Kumaraswamy models, Comm. Statist. Theory Methods 46 (2017), no. 18, 9335–9354. 10.1080/03610926.2016.1208244Search in Google Scholar

[4] A. M. Awad, M. A. Hamdan and M. M. Azzam, Some inference results on Pr⁡(X<Y) in the bivariate exponential model, Comm. Statist. Theory Methods 10 (1981), no. 24, 2515–2525. 10.1080/03610928108828206Search in Google Scholar

[5] N. Balakrishnan and C.-D. Lai, Continuous Bivariate Distributions, 2nd ed., Springer, Dordrecht, 2009. 10.1007/b101765Search in Google Scholar

[6] G. K. Bhattacharyya and R. A. Johnson, Estimation of reliability in a multicomponent stress-strength model, Technical Report No. 328, Department of Statistics, University of Wisconsin, 1988. Search in Google Scholar

[7] E. Cramer and U. Kamps, The UMVUE of P(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions, Metrika 46 (1997), no. 2, 93–121. 10.1007/BF02717169Search in Google Scholar

[8] P. Embrechts, S. I. Resnick and G. Samorodnitsky, Extreme value theory as a risk management tool, North Amer. Actuar. J. 3 (1999), no. 2, 30–41. 10.1080/10920277.1999.10595797Search in Google Scholar

[9] R. C. Gupta, Reliability of a k out of n system of components sharing a common environment, Appl. Math. Lett. 15 (2002), no. 7, 837–844. 10.1016/S0893-9659(02)00051-4Search in Google Scholar

[10] R. C. Gupta and S. Subramanian, Estimation of reliability in a bivariate normal distribution with equal coefficients of variation, Comm. Statist. Simulation Comput. 27 (1998), 675–698. 10.1080/03610919808813503Search in Google Scholar

[11] R. D. Gupta and R. C. Gupta, Estimation of Pr⁡(a′⁢x>b′⁢y) in the multivariate normal case, Statistics 21 (1990), no. 1, 91–97. 10.1080/02331889008802229Search in Google Scholar

[12] D. D. Hanagal, Some inference results in several symmetric multivariate exponential models, Comm. Statist. Theory Methods 22 (1993), no. 9, 2549–2566. 10.1080/03610928308831168Search in Google Scholar

[13] D. D. Hanagal, Testing reliability in a bivariate exponential stress-strength model, J. Indian Statist. Assoc. 33 (1995), 41–45. Search in Google Scholar

[14] D. D. Hanagal, Note on estimation of reliability under bivariate Pareto stress-strength model, Statist. Papers 38 (1997), no. 4, 453–459. 10.1007/BF02926000Search in Google Scholar

[15] D. D. Hanagal, Estimation of reliability of a component subjected to bivariate exponential stress, Statist. Papers 40 (1999), no. 2, 211–220. 10.1007/BF02925519Search in Google Scholar

[16] P. K. Jana, Estimation for P[Y<X] in the bivariate exponential case due to Marshall and Olkin, J. Indian Statist. Assoc. 32 (1994), no. 1, 35–37. Search in Google Scholar

[17] P. K. Jana and D. Roy, Estimation of reliability under stress-strength model in a bivariate exponential set-up, Calcutta Statist. Assoc. Bull. 44 (1994), no. 175–176, 175–181. 10.1177/0008068319940305Search in Google Scholar

[18] E. S. Jeevanand, Bayes estimation of P(X2<X1) for a bivariate Pareto distribution, The Statistician 46 (1997), 93–99. 10.1111/1467-9884.00062Search in Google Scholar

[19] E. S. Jeevanand, Bayes estimation of reliability under stress-strength model for the Marshall–Olkin bivariate exponential distribution, IAPQR Trans. 23 (1998), no. 2, 133–136. Search in Google Scholar

[20] Y. Jin, P. G. Hall, J. Jiang and F. J. Samaniego, Estimating component reliability based on failure time data from a system of unknown design, Statist. Sinica 27 (2017), no. 2, 479–499. Search in Google Scholar

[21] R. A. Johnson, Stress-strength models in reliability, Handb. Stat. 7 (1988), 27–54. 10.1016/S0169-7161(88)07005-1Search in Google Scholar

[22] M. C. Jones, Families of distributions arising from distributions of order statistics, TEST 13 (2004), no. 1, 1–43. 10.1007/BF02602999Search in Google Scholar

[23] S. Nadarajah, Reliability for beta models, Serdica Math. J. 28 (2002), no. 3, 267–282. Search in Google Scholar

[24] S. Nadarajah, Reliability for extreme value distributions, Math. Comput. Modelling 37 (2003), no. 9–10, 915–922. 10.1016/S0895-7177(03)00107-9Search in Google Scholar

[25] S. Nadarajah, Reliability for lifetime distributions, Math. Comput. Modelling 37 (2003), no. 7–8, 683–688. 10.1016/S0895-7177(03)00074-8Search in Google Scholar

[26] S. Nadarajah, Reliability for Laplace distributions, Math. Probl. Eng. 2004 (2004), no. 2, 169–183. 10.1155/S1024123X0431104XSearch in Google Scholar

[27] S. Nadarajah, Reliability for logistic distributions, Eng. Simul. 26 (2004), 81–98. Search in Google Scholar

[28] S. Nadarajah, Reliability for some bivariate beta distributions, Math. Probl. Eng. 2005 (2005), no. 1, 101–111. 10.1155/MPE.2005.101Search in Google Scholar

[29] S. Nadarajah and S. Kotz, Reliability for Pareto models, Metron 61 (2003), no. 2, 191–204. Search in Google Scholar

[30] S. Nadarajah and S. Kotz, Two generalized beta distributions, J. Appl. Econ. 39 (2007), no. 14, 1743–1751. 10.1080/00036840500428062Search in Google Scholar

[31] S. B. Nandi and A. B. Aich, A note on confidence bounds for P(X>Y) in bivariate normal samples, Sankhyā Ser. B 56 (1994), no. 2, 129–136. Search in Google Scholar

[32] S. B. Nandi and A. B. Aich, A note on testing hypothesis regarding P(X>Y) in bivariate normal samples, IAPQR Trans. 21 (1996), no. 2, 149–153. Search in Google Scholar

[33] M. Pandey, Reliability estimation of an s-out-of-k system with non-identical component strengths: The Weibull case, Reliab. Eng. Syst. Safety 36 (1992), 109–116. 10.1016/0951-8320(92)90091-XSearch in Google Scholar

[34] J. M. Sarabia, F. Prieto and V. Jorda, Bivariate beta-generated distributions with applications to well-being data, J. Stat. Distrib. Appl. (2014), 10.1186/2195-5832-1-15. 10.1186/2195-5832-1-15Search in Google Scholar

[35] N. Singh, MVUE of Pr(X<Y) for multivariate normal populations: An application to stress strength models, IEEE Trans. Reliab. 30 (1981), 192–193. 10.1109/TR.1981.5221030Search in Google Scholar

Received: 2018-11-03

Accepted: 2019-01-29

Published Online: 2019-03-12

Published in Print: 2019-12-01

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Articles in the same Issue

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

Keywords for this article

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