The exponentiated generalized inverse Gaussian distribution. (English) Zbl 1207.62028

Summary: The modeling and analysis of life time data is an important aspect of statistical work in a wide variety of scientific and technological fields. I. J. Good [Biometrika 40, 237–264 (1953; Zbl 0051.37103)] introduced a probability distribution which is commonly used in the analysis of lifetime data. For the first time, based on this distribution, we propose the so-called exponentiated generalized inverse Gaussian distribution, which extends the exponentiated standard gamma distribution [S. Nadarajah and S. Kotz, Acta Appl. Math. 92, No. 2, 97–111 (2006; Zbl 1128.62015)]. Various structural properties of the new distribution are derived, including expansions for its moments, moment generating function, moments of the order statistics, and so forth. We discuss maximum likelihood estimation of the model parameters. The usefulness of the new model is illustrated by means of a real data set.


62E10 Characterization and structure theory of statistical distributions
62G30 Order statistics; empirical distribution functions
62N05 Reliability and life testing


Mathematica; LMOMENTS; Ox
Full Text: DOI


[1] Atkinson, A.C., The simulation of generalised inverse Gaussian and hyberbolic random variables, SIAM journal on scientific and statistical computing, 3, 502-515, (1982) · Zbl 0489.65008
[2] Barakat, H.M.; Abdelkader, Y.H., Computing the moments of order statistics from nonidentical random variables, Statistical methods and applications, 13, 15-26, (2004) · Zbl 1056.62012
[3] Barndorff-Nielsen, O., Hyperbolic distributions and distributions on hyperbolae, Scandinavian journal of statistics, 5, 151-157, (1978) · Zbl 0386.60018
[4] Barndorff-Nielsen, O.; Blæsild, P.; Halgreen, C., First hitting time models for the generalized inverse Gaussian distribution, Stochastic processes and their applications, 7, 49-54, (1978) · Zbl 0373.60101
[5] Chaudhry, M.A.; Zubair, S.M., Generalized incomplete gamma functions with applications, Journal of computational and applied mathematics, 55, 99-124, (1994) · Zbl 0833.33002
[6] Chaudhry, M.A.; Zubair, S.M., On a class of incomplete gamma functions with applications, (2002), Chapman & Hall, CRC London · Zbl 1011.33002
[7] Chen, G.; Balakrishnan, N., A general purpose approximate goodness-of-fit test, Journal of quality technology, 27, 154-161, (1995)
[8] Dagpunar, J.S., An easily implemented generalised inverse Gaussian generator, Communications in statistics—simulation and computation, 18, 703-710, (1989)
[9] Doornik, J.A., An object-oriented matrix language—ox 4, (2006), Timberlake Consultants Press London
[10] Embrechts, P., A property of the generalized inverse Gaussian distribution with some applications, Journal of applied probability, 20, 537-544, (1983) · Zbl 0536.60022
[11] Garvan, F., The Maple book, (2002), Chapman & Hall, CRC London
[12] Good, I.J., The population frequencies of species and the estimation of population parameters, Biometrika, 40, 237-260, (1953) · Zbl 0051.37103
[13] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, series, and products, (2007), Academic Press New York · Zbl 1208.65001
[14] Greenwood, J.A.; Landwehr, J.M.; Matalas, N.C.; Wallis, J.R., Probability weighted moments—definition and relation to parameters of several distributions expressable in inverse form, Water resources research, 15, 1049-1054, (1979)
[15] Gupta, R.D.; Kundu, D., Generalized exponential distributions, Australian & New Zealand journal of statistics, 41, 173-188, (1999) · Zbl 1007.62503
[16] Gusmão, F.R.S.; Ortega, E.M.M.; Cordeiro, G.M., The generalized inverse Weibull distribution, Statistical papers, (2009) · Zbl 1440.62049
[17] Hosking, J.R.M., \(L\)-moments: analysis and estimation of distributions using linear combinations of order statistics, Journal of the royal statistical society. series B, 52, 105-124, (1990) · Zbl 0703.62018
[18] Iyengar, S.; Liao, Q., Modeling neural activity using the generalized inverse Gaussian distribution, Biological cybernetics, 77, 289-295, (1997) · Zbl 0887.92009
[19] Jørgensen, B., Statistical properties of the generalized inverse Gaussian distribution, (1982), Springer-Verlag New York · Zbl 0486.62022
[20] Lawrence, C.T.; Tits, A.L., A computationally efficient feasible sequential quadratic programming algorithm, SIAM journal on optimization, 11, 1092-1118, (2001) · Zbl 1035.90105
[21] Mudholkar, G.S.; Hutson, A.D., The exponentiated Weibull family: some properties and a flood data application, Communications in statistics—theory and methods, 25, 3059-3083, (1996) · Zbl 0887.62019
[22] Mudholkar, G.S.; Srivastava, D.K.; Freimer, M., The exponentiated Weibull family, Technometrics, 37, 436-445, (1995) · Zbl 0900.62531
[23] Nadarajah, S.; Kotz, S., The exponentiated type distributions, Acta applicandae mathematicae, 92, 97-111, (2006) · Zbl 1128.62015
[24] Nassar, M.M.; Eissa, F.H., On the exponentiated Weibull distribution, Communications in statistics—theory and methods, 32, 1317-1336, (2003) · Zbl 1140.62308
[25] Nichols, M.D.; Padgett, W.J., A bootstrap control chart for Weibull percentiles, Quality and reliability engineering international, 22, 141-151, (2006)
[26] Nguyen, T.T.; Chen, J.T.; Gupta, A.K.; Dinh, K.T., A proof of the conjecture on positive skewness of generalised inverse Gaussian distributions, Biometrika, 90, 245-250, (2003) · Zbl 1035.60012
[27] Sichel, H.S., On a distribution law for word frequencies, Journal of the American statistical association, 70, 542-547, (1975)
[28] Sigmon, K.; Davis, T.A., MATLAB primer, (2002), Chapman & Hall, CRC · Zbl 1017.93002
[29] Song, K.S., Rényi information, loglikelihood and an intrinsic distribution measure, Journal of statistical planning and inference, 93, 51-69, (2001) · Zbl 0997.62003
[30] Thabane, L.; Haq, M.S., Prediction from a model using a generalized inverse Gaussian prior, Statistical papers, 40, 175-184, (1999) · Zbl 0928.62026
[31] Watson, G.N., A treatise on the theory of Bessel functions, (1995), Cambridge University Press · Zbl 0849.33001
[32] Wolfram, S., The Mathematica book, (2003), Cambridge University Press
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.