On the reliability for some bivariate dependent beta and Kumaraswamy distributions: a brief survey. (English) Zbl 1432.62345
Summary: 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.
MSC:
| 62N05 | Reliability and life testing |
| 62E15 | Exact distribution theory in statistics |
| 62H12 | Estimation in multivariate analysis |
| 60E05 | Probability distributions: general theory |
| 62E10 | Characterization and structure theory of statistical distributions |
Keywords:
reliability parameter; bivariate dependent beta and Kumaraswamy distributions; reliability parameter for bivariate dependent beta and Kumaraswamy modelsReferences:
| [1] | B. C. Arnold, E. Castillo and J. M. Sarabia, Conditional Specification of Statistical Models, Springer Ser. Statist., Springer, New York, 1999.; Arnold, B. C.; Castillo, E.; Sarabia, J. M., Conditional Specification of Statistical Models (1999) · Zbl 0932.62001 |
| [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.; Arnold, B. C.; Ghosh, I., Bivariate Kumaraswamy models involving use of Arnold-Ng copulas, J. Appl. Statist. Sci., 22, 3-4, 227-240 (2014) |
| [3] | B. C. Arnold and I. Ghosh, Some alternative bivariate Kumaraswamy models, Comm. Statist. Theory Methods 46 (2017), no. 18, 9335-9354.; Arnold, B. C.; Ghosh, I., Some alternative bivariate Kumaraswamy models, Comm. Statist. Theory Methods, 46, 18, 9335-9354 (2017) · Zbl 1377.60027 |
| [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.; Awad, A. M.; Hamdan, M. A.; Azzam, M. M., Some inference results on \Pr(X<Y) in the bivariate exponential model, Comm. Statist. Theory Methods, 10, 24, 2515-2525 (1981) |
| [5] | N. Balakrishnan and C.-D. Lai, Continuous Bivariate Distributions, 2nd ed., Springer, Dordrecht, 2009.; Balakrishnan, N.; Lai, C.-D., Continuous Bivariate Distributions (2009) · Zbl 1267.62028 |
| [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.; Bhattacharyya, G. K.; Johnson, R. A., Estimation of reliability in a multicomponent stress-strength model (1988) · Zbl 0298.62026 |
| [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.; Cramer, E.; Kamps, U., The UMVUE of P(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions, Metrika, 46, 2, 93-121 (1997) · Zbl 0915.62012 |
| [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.; Embrechts, P.; Resnick, S. I.; Samorodnitsky, G., Extreme value theory as a risk management tool, North Amer. Actuar. J., 3, 2, 30-41 (1999) · Zbl 1082.91530 |
| [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.; Gupta, R. C., Reliability of a k out of n system of components sharing a common environment, Appl. Math. Lett., 15, 7, 837-844 (2002) · Zbl 1175.90121 |
| [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.; Gupta, R. C.; Subramanian, S., Estimation of reliability in a bivariate normal distribution with equal coefficients of variation, Comm. Statist. Simulation Comput., 27, 675-698 (1998) · Zbl 0916.62065 |
| [11] | R. D. Gupta and R. C. Gupta, Estimation of \Pr(a^{\prime}x>b^{\prime}y) in the multivariate normal case, Statistics 21 (1990), no. 1, 91-97.; Gupta, R. D.; Gupta, R. C., Estimation of \Pr(a^{\prime}x>b^{\prime}y) in the multivariate normal case, Statistics, 21, 1, 91-97 (1990) · Zbl 0699.62053 |
| [12] | D. D. Hanagal, Some inference results in several symmetric multivariate exponential models, Comm. Statist. Theory Methods 22 (1993), no. 9, 2549-2566.; Hanagal, D. D., Some inference results in several symmetric multivariate exponential models, Comm. Statist. Theory Methods, 22, 9, 2549-2566 (1993) · Zbl 0800.62281 |
| [13] | D. D. Hanagal, Testing reliability in a bivariate exponential stress-strength model, J. Indian Statist. Assoc. 33 (1995), 41-45.; Hanagal, D. D., Testing reliability in a bivariate exponential stress-strength model, J. Indian Statist. Assoc., 33, 41-45 (1995) |
| [14] | D. D. Hanagal, Note on estimation of reliability under bivariate Pareto stress-strength model, Statist. Papers 38 (1997), no. 4, 453-459.; Hanagal, D. D., Note on estimation of reliability under bivariate Pareto stress-strength model, Statist. Papers, 38, 4, 453-459 (1997) · Zbl 0911.62091 |
| [15] | D. D. Hanagal, Estimation of reliability of a component subjected to bivariate exponential stress, Statist. Papers 40 (1999), no. 2, 211-220.; Hanagal, D. D., Estimation of reliability of a component subjected to bivariate exponential stress, Statist. Papers, 40, 2, 211-220 (1999) · Zbl 1083.62520 |
| [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.; Jana, P. K., Estimation for P[Y<X] in the bivariate exponential case due to Marshall and Olkin, J. Indian Statist. Assoc., 32, 1, 35-37 (1994) |
| [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.; Jana, P. K.; Roy, D., Estimation of reliability under stress-strength model in a bivariate exponential set-up, Calcutta Statist. Assoc. Bull., 44, 175-176, 175-181 (1994) · Zbl 0831.62077 |
| [18] | E. S. Jeevanand, Bayes estimation of \(P(X_2<X_1)\) for a bivariate Pareto distribution, The Statistician 46 (1997), 93-99.; Jeevanand, E. S., Bayes estimation of \(P(X_2<X_1)\) for a bivariate Pareto distribution, The Statistician, 46, 93-99 (1997) |
| [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.; Jeevanand, E. S., Bayes estimation of reliability under stress-strength model for the Marshall-Olkin bivariate exponential distribution, IAPQR Trans., 23, 2, 133-136 (1998) · Zbl 0983.62074 |
| [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.; Jin, Y.; Hall, P. G.; Jiang, J.; Samaniego, F. J., Estimating component reliability based on failure time data from a system of unknown design, Statist. Sinica, 27, 2, 479-499 (2017) · Zbl 1378.62109 |
| [21] | R. A. Johnson, Stress-strength models in reliability, Handb. Stat. 7 (1988), 27-54.; Johnson, R. A., Stress-strength models in reliability, Handb. Stat., 7, 27-54 (1988) |
| [22] | M. C. Jones, Families of distributions arising from distributions of order statistics, TEST 13 (2004), no. 1, 1-43.; Jones, M. C., Families of distributions arising from distributions of order statistics, TEST, 13, 1, 1-43 (2004) · Zbl 1110.62012 |
| [23] | S. Nadarajah, Reliability for beta models, Serdica Math. J. 28 (2002), no. 3, 267-282.; Nadarajah, S., Reliability for beta models, Serdica Math. J., 28, 3, 267-282 (2002) · Zbl 1027.62070 |
| [24] | S. Nadarajah, Reliability for extreme value distributions, Math. Comput. Modelling 37 (2003), no. 9-10, 915-922.; Nadarajah, S., Reliability for extreme value distributions, Math. Comput. Modelling, 37, 9-10, 915-922 (2003) · Zbl 1045.62107 |
| [25] | S. Nadarajah, Reliability for lifetime distributions, Math. Comput. Modelling 37 (2003), no. 7-8, 683-688.; Nadarajah, S., Reliability for lifetime distributions, Math. Comput. Modelling, 37, 7-8, 683-688 (2003) · Zbl 1045.62106 |
| [26] | S. Nadarajah, Reliability for Laplace distributions, Math. Probl. Eng. 2004 (2004), no. 2, 169-183.; Nadarajah, S., Reliability for Laplace distributions, Math. Probl. Eng., 2004, 2, 169-183 (2004) · Zbl 1070.62093 |
| [27] | S. Nadarajah, Reliability for logistic distributions, Eng. Simul. 26 (2004), 81-98.; Nadarajah, S., Reliability for logistic distributions, Eng. Simul., 26, 81-98 (2004) · Zbl 1070.62093 |
| [28] | S. Nadarajah, Reliability for some bivariate beta distributions, Math. Probl. Eng. 2005 (2005), no. 1, 101-111.; Nadarajah, S., Reliability for some bivariate beta distributions, Math. Probl. Eng., 2005, 1, 101-111 (2005) · Zbl 1069.62081 |
| [29] | S. Nadarajah and S. Kotz, Reliability for Pareto models, Metron 61 (2003), no. 2, 191-204.; Nadarajah, S.; Kotz, S., Reliability for Pareto models, Metron, 61, 2, 191-204 (2003) · Zbl 1416.62571 |
| [30] | S. Nadarajah and S. Kotz, Two generalized beta distributions, J. Appl. Econ. 39 (2007), no. 14, 1743-1751.; Nadarajah, S.; Kotz, S., Two generalized beta distributions, J. Appl. Econ., 39, 14, 1743-1751 (2007) · Zbl 1117.62018 |
| [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.; Nandi, S. B.; Aich, A. B., A note on confidence bounds for P(X>Y) in bivariate normal samples, Sankhyā Ser. B, 56, 2, 129-136 (1994) · Zbl 0835.62091 |
| [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.; Nandi, S. B.; Aich, A. B., A note on testing hypothesis regarding P(X>Y) in bivariate normal samples, IAPQR Trans., 21, 2, 149-153 (1996) · Zbl 0835.62091 |
| [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.; Pandey, M., Reliability estimation of an s-out-of-k system with non-identical component strengths: The Weibull case, Reliab. Eng. Syst. Safety, 36, 109-116 (1992) |
| [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.; Sarabia, J. M.; Prieto, F.; Jorda, V., Bivariate beta-generated distributions with applications to well-being data, J. Stat. Distrib. Appl. (2014) · Zbl 1351.62057 · doi:10.1186/2195-5832-1-15 |
| [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.; Singh, N., MVUE of \(Pr(X<Y)\) for multivariate normal populations: An application to stress strength models, IEEE Trans. Reliab., 30, 192-193 (1981) · Zbl 0455.60070 |
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