×

zbMATH — the first resource for mathematics

Proportional reversed hazard rate model and its applications. (English) Zbl 1119.62098
Summary: The purpose of this paper is to study the structure and properties of the proportional reversed hazard rate model (PRHRM) in contrast to the celebrated proportional hazard model (PHM). The monotonicity of the hazard rate and the reversed hazard rate of the model is investigated. Some criteria of aging are presented and the inheritance of the aging notions (of the base distribution) by the PRHRM is studied. Characterizations of the model involving Fisher information are presented and the statistical inference of the parameters is discussed. Finally, it is shown that several members of the proportional reversed hazard rate class have been found to be useful and flexible in real data analysis.

MSC:
62N05 Reliability and life testing
62E10 Characterization and structure theory of statistical distributions
62N02 Estimation in survival analysis and censored data
PDF BibTeX Cite
Full Text: DOI
References:
[1] Azzalini, A., A class of distributions which includes the normal ones, Scand. J. statist., 12, 171-178, (1985) · Zbl 0581.62014
[2] Bamber, D., The area above the ordinal dominance graph and the area below the receiver operating characteristic graphs, J. math. psych., 12, 387-415, (1975) · Zbl 0327.92017
[3] Block, H.; Savits, T.; Singh, H., The reversed hazard rate function, Probab. in eng. and inform. sci., 12, 69-90, (1998) · Zbl 0972.90018
[4] Chandra, N.N.; Roy, D., Some results on reversed hazard rate, Probab. in eng. and inform. sci., 15, 95-102, (2001) · Zbl 1087.62510
[5] Crescenzo, A.D., Some results on the proportional reversed hazard model, Statist. probab. lett., 50, 313-321, (2000) · Zbl 0967.60016
[6] Davis, D.J., An analysis of some failure data, J. amer. statist. assoc., 47, 113-150, (1952)
[7] Eeckhoudt, L.; Gollier, C., Demand for risky assets and the monotone probability ratio order, J. risk and uncertainty, 11, 113-122, (1995) · Zbl 0863.90043
[8] Efron, B., Logistic regression, survival analysis, and the kaplan – meier curve, J. amer. statist. assoc., 83, 414-425, (1988) · Zbl 0644.62100
[9] Efron, B.; Johnstone, Fisher information in terms of hazard rate, Ann. statist., 18, 38-62, (1990) · Zbl 0722.62022
[10] Gupta, R.C.; Akman, O., On the reliability studies of a weighted inverse Gaussian model, J. statist. plann. inference, 48, 69-83, (1995) · Zbl 0846.62073
[11] Gupta, R.C.; Akman, O., Estimation of critical points in the mixture inverse Gaussian model, Statist. papers, 38, 445-452, (1997) · Zbl 0911.62089
[12] Gupta, R.C.; Brown, N., Reliability studies of the skew normal distribution and its application to a strength stress model, Comm. statist. theory methods, 30, 11, 2427-2445, (2001) · Zbl 1009.62513
[13] Gupta, R.C.; Wu, H., Analyzing survival data by proportional reverses hazard model, Internat. J. reliability and appl., 2, 1, 1-26, (2001)
[14] Gupta, R.C.; Kannan, N.; Raychaudhari, A., Analysis of lognormal survival data, Math. biosci., 139, 103-115, (1997) · Zbl 0900.92003
[15] Gupta, R.C.; Gupta, P.L.; Gupta, R.D., Modeling failure time data by lehman alternatives, Comm. statist.—theory methods, 27, 4, 887-904, (1998) · Zbl 0900.62534
[16] Gupta, R.C.; Ramakrishnan, S.; Zhou, X., Point and interval estimation of \(P(X < Y)\): the normal case with common coefficient of variation, Ann. inst. statist. math., 51, 571-584, (1999) · Zbl 0938.62014
[17] Gupta, R.D.; Gupta, R.C., Estimation of \(\Pr(a^\prime x > b^\prime y)\) in the multivariate normal case, Statistics, 21, 91-97, (1990) · Zbl 0699.62053
[18] Gupta, R.D., Gupta, R.C., 2007. Analyzing skewed data by power normal model. Test, to appear. · Zbl 1148.62008
[19] Gupta, R.D.; Kundu, D., Generalized exponential distributions, Austral. and New Zealand J. statist., 41, 173-188, (1999) · Zbl 1007.62503
[20] Gupta, R.D.; Kundu, D., Exponentiated exponential family; an alternative to gamma and Weibull, Biometrical J., 43, 117-130, (2001) · Zbl 0997.62076
[21] Gupta, R.D.; Kundu, D., Generalized exponential distributions: different methods of estimation, J. statist. comput. simulation, 69, 315-338, (2001)
[22] Gupta, R.D.; Kundu, D., Generalized exponential distributions: statistical inferences, J. statist. theory and appl., 1, 101-118, (2002)
[23] Gupta, R.D.; Kundu, D., Closeness of gamma and generalized exponential distribution, Comm. statist. theory methods, 32, 4, 705-722, (2003) · Zbl 1048.62013
[24] Gupta, R.D.; Nanda, A.K., \(\alpha\)- and \(\beta\)-entropies and relative entropies of distributions, J. statist. theory and appl., 1, 3, 177-190, (2002)
[25] Gupta, R.D.; Nanda, A.K., Some results on (reversed) hazard rate ordering, Comm. statist. theory methods, 30, 11, 2447-2458, (2001) · Zbl 1009.60500
[26] Gupta, R.D.; Gupta, R.C.; Sankaran, P.G., Some characterization results based on the (reversed) hazard rate function, Comm. statist.—theory methods, 33, 12, 3009-3031, (2004) · Zbl 1087.62015
[27] Halperin, M.; Gilbert, P.R.; Lachin, J.M., Distribution free confidence intervals for \(P(X_1 < X_2)\), Biometrics, 43, 71-80, (1987) · Zbl 0655.62052
[28] Kalbfleisch, J.D.; Lawless, J.F., Inference based on retrospective ascertainment: an analysis of the data on transfusion-related AIDS, J. amer. statist. assoc., 84, 360-372, (1989) · Zbl 0677.62099
[29] Kijima, M.; Ohnishi, M., Stochastic orders and their application in financial optimization, Math. methods in oper. res., 50, 351-372, (1999) · Zbl 0958.91020
[30] Kullback, S.; Leibler, R.A., Information and sufficiency, Ann. of math. statist., 22, 79-86, (1951) · Zbl 0042.38403
[31] Kundu, D.; Gupta, R.D., Characterizations of the proportional (reversed) hazard model, Comm. statist.—theory methods, 33, 3095-3102, (2004) · Zbl 1087.62018
[32] Kundu, D.; Gupta, R.D., Estimation of \(P(Y < X)\) for generalized exponential distribution, Metrika, 61, 291-308, (2005) · Zbl 1079.62032
[33] Lehman, E.L., The power of rank tests, Ann. of math. statist., 24, 28-43, (1953)
[34] Mudholkar, G.S.; Hutson, A.D., The exponentiated Weibull family: some properties and a flood data application, Comm. statist.—theory methods, 25, 12, 3059-3083, (1996) · Zbl 0887.62019
[35] Mudholkar, G.S.; Srivastava, D.K., Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE trans. reliability, 42, 2, 299-302, (1993) · Zbl 0800.62609
[36] Mudholkar, G.S.; Srivastava, D.K.; Freimer, M., The exponentiated Weibull family: a reanalysis of the bus-motor-failure data, Technometrics, 37, 4, 436-445, (1995) · Zbl 0900.62531
[37] Reiser, B.; Farragi, D., Confidence bounds for \(P(a^\prime X > b^\prime Y)\), Statistics, 25, 107-111, (1994) · Zbl 0811.62039
[38] Sengupta, D.; Nanda, A.K., Logconcave and concave distributions in reliability, Naval res. logistics quart., 46, 4, 419-433, (1999) · Zbl 0928.62098
[39] Simonoff, J.S.; Hochberg, Y.; Reiser, B., Alternative estimation procedures for \(\Pr .(X < Y)\) in categorized data, Biometrics, 42, 895-907, (1986) · Zbl 0613.62125
[40] Tsodikov, A.D.; Aselain, B.; Yakovlev, A.Y., A distribution of tumor size at detection: an application to breast cancer data, Biometrics, 53, 1495-1502, (1997) · Zbl 0911.62110
[41] Wang, Y.; Hossain, A.M.; Zimmer, W.J., Monotone log-odds rate distributions in reliability, Comm. statist. theory methods, 32, 11, 2227-2244, (2003) · Zbl 1131.62324
[42] Wang, Y.; Hossain, A.M.; Zimmer, W.J., Tables of bounds for distributions with monotone log-odds rate, Comm. statist. simulation comput., 34, 1, 1-20, (2005) · Zbl 1101.62386
[43] Zimmer, W.J.; Yang, Y.; Pathak, P.K., Log-odds rate and monotone log-odds rate distributions, J. quality tech., 34, 4, 376-385, (1998)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.