# zbMATH — the first resource for mathematics

The beta generalized Pareto distribution with application to lifetime data. (English) Zbl 1219.62024
Summary: The generalized Pareto (GP) distribution is useful in modeling extreme value data, because of its long tail feature. In this paper, a new generalized version of this distribution which is called the beta generalized Pareto (BGP) distribution is introduced. The new distribution is more flexible and has some interesting properties. A comprehensive mathematical treatment of the BGP distribution is provided. We give closed-form expressions for the density, cumulative distribution and hazard rate function. We derive the $$r$$ th raw moment of this distribution. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher’s information matrix. In the end, an application using three real data sets is presented.

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 62G32 Statistics of extreme values; tail inference 62N05 Reliability and life testing
##### Keywords:
estimation; hazard function; moments; unimodality
Full Text:
##### References:
 [1] Aarset, M.V., How to identify bathtub hazard rate, IEEE transactions reliability, 36, 106-108, (1987) · Zbl 0625.62092 [2] Akinsete, A.; Famoye, F.; Lee, C., The beta-Pareto distribution, Statistics, 42, 547-563, (2008) · Zbl 1274.60033 [3] Ashkar, F.; Ouarda, T.B.M.J., On some methods of Fitting the generalized Pareto distribution, Journal of hydrology, 177, 117-141, (1996) [4] Barreto-Souza, W.; Santos, A.H.S.; Cordeiro, G.M., The beta generalized exponential distribution, Journal of statistical computation and simulation, 80, 159-172, (2010) · Zbl 1184.62012 [5] Birnbaum, Z.W.; Saunders, S.C., Estimation for a family of life distributions with applications to fatigue, Journal of applied probability, 6, 328-347, (1969) · Zbl 0216.22702 [6] Chen, G.; Balakrishnan, N., A general purpose approximate goodness-of-fit test, Journal of quality technology, 27, 154-161, (1995) [7] Choulakian, V.; Stephens, M.A., Goodness-of-fit for the generalized Pareto distribution, Technometrics, 43, 478-484, (2001) [8] Cordeiro, G.M.; Nadarajah, S., Closed form expressions for moments of a class of beta generalized distributions, Brazilian journal of probability and statistics, 25, 14-33, (2011) · Zbl 1298.60024 [9] Cordeiro, G.M.; Lemonte, A.J., The β-birnbaum – saunders distribution: an improved distribution for fatigue life modeling, Computational statistics and data analysis, 55, 1445-1461, (2011) · Zbl 1328.62572 [10] Davison, A.C., Modeling excesses over high thresholds with an application, (), 461-482 [11] Eugene, N.; Lee, C.; Famoye, F., The beta-normal distribution and its applications, Communications in statistics-theory and methods, 31, 497-512, (2002) · Zbl 1009.62516 [12] Famoye, F.; Lee, C.; Olumolade, O., The beta-Weibull distribution, Journal of statistical theory application, 4, 121-136, (2005) [13] Feigl, P.; Zelen, M., Estimation of exponential survival probabilities with concomitant observation, Biometrics, 21, 826-838, (1965) [14] Hosking, J.R.M.; Wallis, J.R., Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, 29, 339-349, (1987) · Zbl 0628.62019 [15] Jones, M.C., Family of distributions arising from distribution of order statistics, Test, 13, 1-43, (2004) · Zbl 1110.62012 [16] Nadarajah, S.; Kotz, S., The beta Gumbel distribution, Mathematical problems in engineering, 10, 323-332, (2004) · Zbl 1068.62012 [17] Nadarajah, S.; Gupta, A.K., The beta frechet distribution, Far east journal of theorical statistics, 15, 15-24, (2004) · Zbl 1074.62008 [18] Nadarajah, S.; Kotz, S., The beta exponential distribution, Reliability engineering and system safety, 91, 689-697, (2006) [19] Paranaiba, P.F.; Ortega, E.M.M.; Cordeiro, G.M.; Pescim, R.R., The beta burr XII distribution with application to lifetime data, Computation statistics and data analysis, 55, 1118-1136, (2011) · Zbl 1284.62108 [20] Pescim, R.R.; Demtrio, C.G.B.; Cordeiro, G.M.; Ortega, E.M.M.; Urbano, M.R., The beta generalized half-normal distribution, Computation statistics and data analysis, 54, 945-957, (2010) · Zbl 1465.62015 [21] Pickands, J., Statistical inference using extreme order statistics, Annals of statistics, 3, 119-131, (1975) · Zbl 0312.62038 [22] Rosbjerg, D.; Madsen, H.; Rasmussen, P.F., Prediction in partial duration series with generalized Pareto-distributed exceedances, Water resources research, 28, 3001-3010, (1992) [23] Silva, G.O.; Ortega, E.M.M.; Cordeiro, G.M., The beta modified Weibull distribution, Lifetime data analysis, 16, 409-430, (2010) · Zbl 1322.62071 [24] Smith, R.L., (), 621-638 [25] E. Mahmoudi, M. Rafiei, The beta exponentiated Weibull distribution and its applications, Computational Statistics, submitted for publication. [26] Van Montfort, M.A.J.; Witter, J.V., The generalized Pareto distribution applied to rainfall depths, Hydrological sciences journal, 31, 151-162, (1986) [27] Watson, G.S., Goodness-of-fit tests on a circle, Biometrika, 48, 109-114, (1961) · Zbl 0212.21905
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.