| Title:
|
Stress-strength based on $m$-generalized order statistics and concomitant for dependent families (English) |
| Author:
|
Domma, Filippo |
| Author:
|
Eftekharian, Abbas |
| Author:
|
Razmkhah, Mostafa |
| Language:
|
English |
| Journal:
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Applications of Mathematics |
| ISSN:
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0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
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64 |
| Issue:
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4 |
| Year:
|
2019 |
| Pages:
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437-467 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
|
The stress-strength model is proposed based on the $m$-generalized order statistics and the corresponding concomitant. For the dependency between $m$-generalized order statistics and its concomitant, a bivariate copula expansion is considered and the stress-strength model is obtained for two special cases of order statistics and upper record values. In the particular case of copula function, the generalized Farlie-Gumbel-Morgenstern bivariate distribution function is considered with proportional reversed hazard functions as marginal functions. Based on the order statistics and record values, two estimators of stress-strength are presented using a procedure similar to the inference functions for margins. Finally, a simulation study is carried out which shows the good performance of the proposed estimators for a finite sample. (English) |
| Keyword:
|
copula function |
| Keyword:
|
Dagum distribution |
| Keyword:
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generalized order statistics |
| Keyword:
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Farlie-Gumbel-Morgenstern distribution |
| Keyword:
|
proportional reversed hazard family |
| Keyword:
|
record values |
| MSC:
|
62G30 |
| MSC:
|
62N05 |
| idZBL:
|
Zbl 07088750 |
| idMR:
|
MR3987227 |
| DOI:
|
10.21136/AM.2019.0285-18 |
| . |
| Date available:
|
2019-07-24T11:24:17Z |
| Last updated:
|
2021-09-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147799 |
| . |
| Reference:
|
[1] Adimari, G., Chiogna, M.: Partially parametric interval estimation of Pr$\{Y>X\}$.Comput. Stat. Data Anal. 51 (2006), 1875-1891. Zbl 1157.62369, MR 2307549, 10.1016/j.csda.2005.12.007 |
| Reference:
|
[2] Al-Mutairi, D. K., Ghitany, M. E., Kundu, D.: Inferences on stress-strength reliability from weighted Lindley distributions.Commun. Stat., Theory Methods 44 (2015), 4096-4113. Zbl 1333.62062, MR 3406333, 10.1080/03610926.2014.968729 |
| Reference:
|
[3] Asgharzadeh, A., Valiollahi, R., Raqab, M. Z.: Stress-strength reliability of Weibull distribution based on progressively censored samples.SORT 35 (2011), 103-124. Zbl 1284.62144, MR 2908294 |
| Reference:
|
[4] Bairamov, I., Kotz, S., Bekçi, M.: New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics.J. Appl. Stat. 28 (2001), 521-536. Zbl 0991.62032, MR 1836732, 10.1080/02664760120047861 |
| Reference:
|
[5] Baklizi, A.: Estimation of Pr$(X<Y)$ using record values in the one and two parameter exponential distributions.Commun. Stat., Theory Methods 37 (2008), 692-698 corrigendum ibid. 40 2011 4322-4323. Zbl 1309.62161, MR 2432305, 10.1080/03610920701501921 |
| Reference:
|
[6] Baklizi, A.: Inference on Pr$(X<Y)$ in the two-parameter Weibull model based on records.ISRN Probab. Stat. 2012 (2012), Article ID 263612, 11 pages. Zbl 06169692, 10.5402/2012/263612 |
| Reference:
|
[7] Basirat, M., Baratpour, S., Ahmadi, J.: Statistical inferences for stress-strength in the proportional hazard models based on progressive Type-II censored samples.J. Stat. Comput. Simulation 85 (2015), 431-449. MR 3275457, 10.1080/00949655.2013.824449 |
| Reference:
|
[8] Beg, M. I., Ahsanullah, M.: Concomitants of generalized order statistics from Farlie-Gumbel-Morgenstern distributions.Stat. Methodol. 5 (2008), 1-20. Zbl 1248.62076, MR 2416936, 10.1016/j.stamet.2007.04.001 |
| Reference:
|
[9] Bose, A., Gangopadhyay, S.: A note on concomitants of records.Stat. Methodol. 10 (2013), 103-112. Zbl 1365.62065, MR 2974814, 10.1016/j.stamet.2012.07.004 |
| Reference:
|
[10] Chacko, M., Thomas, P. Y.: Estimation of parameters of bivariate normal distribution using concomitants of record values.Stat. Pap. 49 263-275 (2008). Zbl 1168.62346, MR 2365390, 10.1007/s00362-006-0011-x |
| Reference:
|
[11] Condino, F., Domma, F., Latorre, G.: Likelihood and Bayesian estimation of $P(Y>X)$ using lower record values from a proportional reversed hazard family.Stat. Pap. 59 (2018), 467-485. Zbl 06912421, MR 3800810, 10.1007/s00362-016-0772-9 |
| Reference:
|
[12] Dagum, C.: The generation and distribution of income. The Lorenz curve and the Gini ratio.Economie Appliqu{é}e 33 (1980), 327-367. |
| Reference:
|
[13] David, H. A., Nagaraja, H. N.: Order Statistics.Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester (2003). Zbl 1053.62060, MR 1994955, 10.1002/0471722162 |
| Reference:
|
[14] Dengler, B.: On the Asymptotic Behaviour of the Estimator of Kendall's Tau.Ph.D. Thesis, TU Vienna (2010). |
| Reference:
|
[15] Domma, F., Giordano, S.: A stress-strength model with dependent variables to measure household financial fragility.Stat. Methods Appl. 21 (2012), 375-389. Zbl 1255.91328, MR 2981655, 10.1007/s10260-012-0192-5 |
| Reference:
|
[16] Domma, F., Giordano, S.: A copula-based approach to account for dependence in stress-strength models.Stat. Pap. 54 (2013), 807-826. Zbl 1307.62234, MR 3072902, 10.1007/s00362-012-0463-0 |
| Reference:
|
[17] Domma, F., Giordano, S.: Concomitants of $m$-generalized order statistics from generalized Farlie-Gumbel-Morgenstern distribution family.J. Comput. Appl. Math. 294 (2016), 413-435. Zbl 1330.62216, MR 3406991, 10.1016/j.cam.2015.08.022 |
| Reference:
|
[18] Farlie, D. J. G.: The performance of some correlation coefficients for a general bivariate distribution.Biometrika 47 (1960), 307-323. Zbl 0102.14903, MR 0119312, 10.1093/biomet/47.3-4.307 |
| Reference:
|
[19] Gradshteyn, I. S., Ryzhik, I. M.: Table of Integrals, Series, and Products.Elsevier/\hskip0ptAcademic Press, Amsterdam (2007). Zbl 1208.65001, MR 2360010, 10.1016/C2009-0-22516-5 |
| Reference:
|
[20] Gupta, R. C., Subramanian, S.: Estimation of reliability in a bivariate normal distribution with equal coefficients of variation.Commun. Stat., Simulation Comput. 27 (1998), 675-698. Zbl 0916.62065, 10.1080/03610919808813503 |
| Reference:
|
[21] Hanagal, D. D.: Estimation of reliability when stress is censored at strength.Commun. Stat., Theory Methods 26 (1997), 911-919. Zbl 0917.62081, MR 1436086, 10.1080/03610929708831958 |
| Reference:
|
[22] Jaheen, Z. F.: Empirical Bayes inference for generalized exponential distribution based on records.Commun. Stat., Theory Methods 33 (2004), 1851-1861. Zbl 1213.62014, MR 2065178, 10.1081/STA-120037445 |
| Reference:
|
[23] Joe, H.: Multivariate Models and Multivariate Dependence Concepts.Monographs on Statistics and Applied Probability 73, Chapman & Hall, London (1997). Zbl 0990.62517, MR 1462613, 10.1201/b13150 |
| Reference:
|
[24] Joe, H., Xu, J. J.: The estimation method of inference functions for margins for multivariate models.Technical Report \#166, University of British Columbia, Vancouver (1996). 10.14288/1.0225985 |
| Reference:
|
[25] Kamps, U.: A concept of generalized order statistics.J. Stat. Plann. Inference 48 (1995), 1-23. Zbl 0838.62038, MR 1366370, 10.1016/0378-3758(94)00147-n |
| Reference:
|
[26] Kotz, S., Lumelskii, Y., Pensky, M.: The Stress-Strength Model and Its Generalizations. Theory and Applications.World Scientific, River Edge (2003). Zbl 1017.62100, MR 1980497, 10.1142/5015 |
| Reference:
|
[27] Marshall, A. W., Olkin, I.: Life Distributions. Structure of Nonparametric, Semiparametric, and Parametric Families.Springer Series in Statistics, Springer, New York (2007),\99999DOI99999 10.1007/978-0-387-68477-2 \hyphenation{McGraw}. Zbl 1304.62019, MR 2344835 |
| Reference:
|
[28] Mood, A. M., Graybill, F. A., Boes, D. C.: Introduction to the Theory of Statistics.McGraw-Hill Series in Probability and Statistics, McGraw-Hill Book Company, New York (1974). Zbl 0277.62002, MR 0033470 |
| Reference:
|
[29] Nadar, M., Kızılaslan, F.: Classical and Bayesian estimation of $P(X<Y)$ using upper record values from Kumaraswamy's distribution.Stat. Pap. 55 (2014), 751-783. Zbl 1336.62077, MR 3227550, 10.1007/s00362-013-0526-x |
| Reference:
|
[30] Nadarajah, S.: Reliability for some bivariate beta distributions.Math. Probl. Eng. 2005 (2005), 101-111. Zbl 1069.62081, MR 2144110, 10.1155/mpe.2005.101 |
| Reference:
|
[31] Nadarajah, S.: Expansions for bivariate copulas.Stat. Probab. Lett. 100 (2015), 77-84. Zbl 1314.62137, MR 3324077, 10.1016/j.spl.2015.02.005 |
| Reference:
|
[32] Nelsen, R. B.: An Introduction to Copulas.Springer Series in Statistics, Springer, New York (2006). Zbl 1152.62030, MR 2197664, 10.1007/0-387-28678-0 |
| Reference:
|
[33] Nevzorov, V. B., Ahsanullah, M.: Some distributions of induced records.Biom. J. 42 (2000), 1069-1081. Zbl 0960.62052, MR 1818512, /10.1002/1521-4036(200012)42:8<1069::AID-BIMJ1069>3.0.CO;2-W |
| Reference:
|
[34] Pakdaman, Z., Ahmadi, J.: Stress-strength reliability for $P(X_{r:n_1}<Y_{k:n_2})$ in the exponential case.İstatistik 6 (2013), 92-102. MR 3241750 |
| Reference:
|
[35] Raqab, M. Z., Madi, M. T.: Inference for the generalized Rayleigh distribution based on progressively censored data.J. Stat. Plann. Inference 141 (2011), 3313-3322. Zbl 1216.62151, MR 2805647, 10.1016/j.jspi.2011.04.016 |
| Reference:
|
[36] Rezaei, S., Tahmasbi, R., Mahmoodi, M.: Estimation of $P[Y>X]$ for generalized Pareto distribution.J. Stat. Plann. Inference 140 (2010), 480-494. Zbl 1177.62024, MR 2558379, 10.1016/j.jspi.2009.07.024 |
| Reference:
|
[37] Saraçoğlu, B., Kinaci, I., Kundu, D.: On estimation of $R=P(Y>X)$ for exponential distribution under progressive type-II censoring.J. Stat. Comput. Simulation 82 (2012), 729-744. Zbl 06147497, MR 2916087, 10.1080/00949655.2010.551772 |
| Reference:
|
[38] Sengupta, S.: Unbiased estimation of $P(Y>X)$ for two-parameter exponential populations using order statistics.Statistics 45 (2011), 179-188. Zbl 1283.62017, MR 2783405, 10.1080/02331880903546431 |
| Reference:
|
[39] Tahmasebi, S., Jafari, A. A., Afshari, M.: Concomitants of dual generalized order statistics from Morgenstern type bivariate generalized exponential distribution.J. Stat. Theory Appl. 14 (2015), 1-12. MR 3341061, 10.2991/jsta.2015.14.1.1 |
| Reference:
|
[40] Tarvirdizade, B., Ahmadpour, M.: Estimation of the stress-strength reliability for the two-parameter bathtub-shaped lifetime distribution based on upper record values.Stat. Methodol. 31 (2016), 58-72. Zbl 07037202, MR 3477728, 10.1016/j.stamet.2016.01.005 |
| Reference:
|
[41] Valiollahi, R., Asgharzadeh, A., Raqab, M. Z.: Estimation of $P(Y<X)$ for Weibull distribution under progressive Type-II censoring.Commun. Stat., Theory Methods 42 (2013), 4476-4498. Zbl 1282.65017, MR 3171012, 10.1080/03610926.2011.650265 |
| Reference:
|
[42] Wong, A.: Interval estimation of $P(Y>X)$ for generalized Pareto distribution.J. Stat. Plann. Inference 142 (2012), 601-607. Zbl 1231.62039, MR 2843061, 10.1016/j.jspi.2011.04.024 |
| Reference:
|
[43] Yang, S. S.: General distribution theory of the concomitants of order statistics.Ann. Stat. (1977), 996-1002. Zbl 0367.62017, MR 0501519, 10.1214/aos/1176343954 |
| Reference:
|
[44] Yörübulut, S., Gebizlioglu, O. L.: Bivariate pseudo-Gompertz distribution and concomitants of its order statistics.J. Comput. Appl. Math. 247 (2013), 68-83. Zbl 06155442, MR 3023302, 10.1016/j.cam.2013.01.006 |
| . |
