Likelihood and Bayesian estimation of \(P(Y<X)\) using lower record values from a proportional reversed hazard family. (English) Zbl 1408.62035
A stress-strength model in reliability is meant to describe the life of a component exhibiting a random strength \(X\) and being subject to a random stress \(Y\). The usual measure of component reliability in such a model is given by \(\Phi = P(Y<X).\) This manuscript presents inferencial procedures for \(\Phi,\) when \(X\) and \(Y\) are two independent but not identically distributed random variables and when the information is based on lower record data. The distributions for \(X\) and \(Y\) are assumed to come from the proportional reversed hazard (PRH) family with parameter \(\theta.\) If \(X\) and \(Y\) are PRH distributed with parameters \(\theta_1\) and \(\theta_2,\) respectively, then \(\Phi\) is given by \(\theta_1/(\theta_1+\theta_2).\) The ML estimators and their distributions for \(\theta_1\) and \(\theta_2\) are given. From those, the ML estimator and its distribution for \(\Phi\) are derived. The uniformly minimum variance unbiased estimator of \(\Phi\) is obtained. As for the Bayesian counterpart, the Lindley approximation for the Bayes estimator for \(\Phi,\) as well as its credible set are derived. Numerical optimization techniques provide the HPD interval for such estimator. In particular, they use the Topp-Leone distribution in two real data sets as examples of applications of their theory. The estimation program is largely exploited in this problem. One more nice set of tools is available to be used in reliability.
Reviewer: Nelson I. Tanaka (São Paulo)
MSC:
| 62F10 | Point estimation |
| 62F15 | Bayesian inference |
| 62N05 | Reliability and life testing |
| 62F25 | Parametric tolerance and confidence regions |
| 90B25 | Reliability, availability, maintenance, inspection in operations research |
Keywords:
reliability; stress-strength model; point estimation; Bayesian estimation; confidence and credible intervals; Topp-Leone distributionReferences:
| [1] | Ahmad KE, Fakhry ME, Jaheen ZF (1997) Empirical Bayes estimation of \(P(Y < X)\) and characterizations of Burr-type \({X}\) model. J Stat Plan Inference 64:297-308 · Zbl 0915.62001 |
| [2] | Amin, EA, Bayesian and non-Bayesian estimation of \(P(Y < X)\) from type I generalized logistic distribution based on lower record values, Aust J Basic Appl Sci, 6, 616-621, (2012) |
| [3] | Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New York · Zbl 0914.60007 · doi:10.1002/9781118150412 |
| [4] | Asgharzadeh, A; Valiollahi, R; Raqab, MZ, Estimation of the stress-strength reliability for the generalized logistic distribution, Stat Methodol, 15, 73-94, (2013) · Zbl 1282.65017 · doi:10.1016/j.stamet.2013.05.002 |
| [5] | Babayi, S; Khorram, E; Tondro, F, Inference of \({R=P[X < Y]}\) for generalized logistic distribution, Stat J Theor Appl Stat, 48, 862-871, (2014) · Zbl 1326.62047 |
| [6] | Baklizi, A, Estimation of \({P}r{(X < Y)}\) using record values in the one and two parameter exponential distributions, Commun Stat Theory Methods, 37, 692-698, (2008) · Zbl 1309.62161 · doi:10.1080/03610920701501921 |
| [7] | Baklizi, A, Likelihood and Bayesian estimation of \({P}r{(X < Y)}\) using lower record values from the generalized exponential distribution, Comput Stat Data Anal, 52, 3468-3473, (2008) · Zbl 1452.62722 · doi:10.1016/j.csda.2007.11.002 |
| [8] | Baklizi A (2012) Inference on \({P(X < Y)}\) in the two-parameter Weibull model based on records. ISRN Probab Stat, Article ID 263612, 11 pp · Zbl 1511.62255 |
| [9] | Baklizi, A, Bayesian inference for \({P(Y < X)}\) in the exponential distribution based on records, Appl Math Model, 38, 1698-1709, (2013) · Zbl 1428.62440 |
| [10] | Baklizi, A, Interval estimation of the stress-strength reliability in the two-parameter exponential distribution based on records, J Stat Comput Simul, 84, 2670-2679, (2013) · Zbl 1453.62675 · doi:10.1080/00949655.2013.816307 |
| [11] | Balakrishnan N (1992) Handbook of the logistic distribution. Marcel Dekker, New York · Zbl 0794.62001 |
| [12] | Birnbaum, ZW; McCarty, RC, A distribution-free upper confidence bound for \({P(Y < X)},\) based on independent samples of \({X}\) and \({Y}\), Ann Math Stat, 29, 558-562, (1958) · Zbl 0087.34002 · doi:10.1214/aoms/1177706631 |
| [13] | Genç, AI, Moments of order statistics of topp-leone distribution, Stat Pap, 53, 117-131, (2012) · Zbl 1241.62077 · doi:10.1007/s00362-010-0320-y |
| [14] | Genç, AI, Estimation of \({P(X>Y)}\) with topp-leone distribution, J Stat Comput Simul, 83, 326-339, (2013) · Zbl 1348.62039 · doi:10.1080/00949655.2011.607821 |
| [15] | Ghitany, ME, Asymptotic distributions of order statistics from the topp-leone distribution, Int J Appl Math, 20, 371-376, (2007) · Zbl 1128.62021 |
| [16] | Ghitany, ME; Kotz, S; Xie, M, On some reliability measures and their stochastic orderings for the topp-leone distribution, J Appl Stat, 32, 715-722, (2005) · Zbl 1121.62374 · doi:10.1080/02664760500079613 |
| [17] | Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products. Academic, New York · Zbl 1208.65001 |
| [18] | Gupta, RD; Kundu, D, Generalized exponential distribution, Aust NZ J Stat, 41, 173-188, (1999) · Zbl 1007.62503 · doi:10.1111/1467-842X.00072 |
| [19] | Huang, K; Mi, J; Wang, Z, Inference about reliability parameter with gamma strength and stress, J Stat Plan Inference, 142, 848-854, (2012) · Zbl 1232.62131 · doi:10.1016/j.jspi.2011.10.005 |
| [20] | Jaheen, ZF, Empirical Bayes inference for generalized exponential distribution based on records, Commun Stat Theory Methods, 33, 1851-1861, (2004) · Zbl 1213.62014 · doi:10.1081/STA-120037445 |
| [21] | Kizilaslan, F; Nadar, M, Estimation with the generalized exponential distribution based on record values and inter-record times, J Stat Comput Simul, 85, 978-999, (2015) · Zbl 1457.62080 · doi:10.1080/00949655.2013.856910 |
| [22] | Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations. Theory and applications, World Scientific, Singapore · Zbl 1017.62100 · doi:10.1142/5015 |
| [23] | Kotz S, Nadarajah S (2006) J-shaped distribution, Topp and Leone’s, encyclopedia of statistical sciences, vol 6, 2nd edn. Wiley, New York, p 3786 |
| [24] | Krishnamoorthy, K; Lin, Y, Confidence limits for stress-strength reliability involving Weibull models, J Stat Plan Inference, 140, 1754-1764, (2010) · Zbl 1184.62178 · doi:10.1016/j.jspi.2009.12.028 |
| [25] | Kundu, D; Gupta, RD, Generalized exponential distribution: Bayesian estimations, Comput Stat Data Anal, 52, 1873-1883, (2008) · Zbl 1452.62182 · doi:10.1016/j.csda.2007.06.004 |
| [26] | Lindley, DV, No article title, Approximate Bayesian methods. Trab Estad, 31, 223-237, (1980) · doi:10.1007/BF02888353 |
| [27] | Nadar, M; Kizilaslan, F, Classical and Bayesian estimation of \({P(X < Y)}\) using upper record values from kumaraswamy’s distribution, Stat Pap, 55, 751-783, (2013) · Zbl 1336.62077 · doi:10.1007/s00362-013-0526-x |
| [28] | Nadarajah, S; Kotz, S, Moments of some j-shaped distributions, J Appl Stat, 30, 311-317, (2003) · Zbl 1121.62448 · doi:10.1080/0266476022000030084 |
| [29] | Pewsey, A; Gomez, HW; Bolfarine, H, No article title, Likelihood-based inference for power distributions. TEST, 21, 775-789, (2012) · Zbl 1261.62019 |
| [30] | Press SJ (2002) Subjective and objective Bayesian statistics. Principles, models, and applications, 2nd edn. Wiley-Interscience, Wiley, Hoboken · Zbl 1428.62440 |
| [31] | Rezaei, S; Tahmasbi, R; Mahmoodi, M, Estimation of \({P(Y < X)}\) for generalized Pareto distribution, J Stat Plan Inference, 140, 480-494, (2010) · Zbl 1177.62024 · doi:10.1016/j.jspi.2009.07.024 |
| [32] | Saraçoǧlu, B; Kinaci, I; Kundu, D, On estimation of \({R=P(Y < X)}\) for exponential distribution under progressive type-II censoring, J Stat Comput Simul, 82, 729-744, (2012) · Zbl 1431.62463 · doi:10.1080/00949655.2010.551772 |
| [33] | Sengupta, S, Unbiased estimation of \({P(X>Y)}\) for two-parameter exponential populations using order statistics, Statistics, 45, 179-188, (2011) · Zbl 1283.62017 · doi:10.1080/02331880903546431 |
| [34] | Topp, CW; Leone, FC, A family of j-shaped frequency functions, J Am Stat Assoc, 50, 209-219, (1955) · Zbl 0064.13601 · doi:10.1080/01621459.1955.10501259 |
| [35] | van Dorp JR, Kotz S (2006) Modeling income distributions using elevated distributions. World Scientific Press, Singapore, Distribution models theory · Zbl 1276.91068 |
| [36] | Ventura, L; Racugno, W, Recent advances on Bayesian inference for \({P(X < Y)}\), Bayesian Anal, 6, 411-428, (2011) · Zbl 1330.62157 |
| [37] | Vicari, D; Dorp, JR; Kotz, S, Two-sided generalized topp and leone (TS-GTL) distributions, J Appl Stat, 35, 1115-1129, (2008) · Zbl 1273.62041 · doi:10.1080/02664760802230583 |
| [38] | Wang, BX; Ye, ZS, Inference on the Weibull distribution based on record values, Comput Stat Data Anal, 83, 26-36, (2015) · Zbl 1507.62173 · doi:10.1016/j.csda.2014.09.005 |
| [39] | Wong, A, Interval estimation of \({P(Y < X)}\) for generalized Pareto distribution, J Stat Plan Inference, 142, 601-607, (2012) · Zbl 1231.62039 · doi:10.1016/j.jspi.2011.04.024 |
| [40] | Wong, A; Wu, YY, A note on interval estimation of \({P(X < Y)}\) using lower record data from the generalized exponential distribution, Comput Stat Data Anal, 53, 3650-3658, (2009) · Zbl 1453.62245 · doi:10.1016/j.csda.2009.03.006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
