The Kumaraswamy generalized gamma distribution with application in survival analysis. (English) Zbl 1219.62026
Summary: We introduce and study the so-called P. Kumaraswamy [J. Hydrol. 46, 79–88 (1980)] generalized gamma distribution that is capable of modeling bathtub-shaped hazard rate functions. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in life-time data analysis and reliability. The new distribution has a large number of well-known life-time special sub-models such as the exponentiated generalized gamma, exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma, and generalized Rayleigh, among others. Some structural properties of the new distribution are studied. We obtain two infinite sum representations for the moments and an expansion for the generating function. We calculate the density function of the order statistics and an expansion for their moments. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. The usefulness of the new distribution is illustrated in two real data sets.
MSC:
| 62E10 | Characterization and structure theory of statistical distributions |
| 62N05 | Reliability and life testing |
| 62N02 | Estimation in survival analysis and censored data |
| 62G30 | Order statistics; empirical distribution functions |
| 62F10 | Point estimation |
| 65C60 | Computational problems in statistics (MSC2010) |
Keywords:
censored data; data analysis; generalized gamma distribution; maximum likelihood estimation; moment; order statisticSoftware:
Lauricella FunctionsReferences:
| [1] | Aarset, M. V., How to identify bathtub hazard rate, IEEE Transactions on Reliability, 36, 106-108 (1987) · Zbl 0625.62092 |
| [2] | R.M. Aarts, 2000, Lauricella functions, From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. www.mathworld.wolfram.com/LauricellaFunctions.html; R.M. Aarts, 2000, Lauricella functions, From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. www.mathworld.wolfram.com/LauricellaFunctions.html |
| [3] | Ali, M. M.; Woo, J.; Nadarajah, S., Generalized gamma variables with drought application, Journal of the Korean Statistical Society, 37, 37-45 (2008) · Zbl 1196.62010 |
| [4] | Almpanidis, G.; Kotropoulos, C., Phonemic segmentation using the generalized gamma distribution and small sample Bayesian information criterion, Speech Communication, 50, 38-55 (2008) |
| [5] | Barkauskas, D. A.; Kronewitter, S. R.; Lebrilla, C. B.; Rocke, D. M., Analysis of MALDI FT-ICR mass spectrometry data: a time series approach, Analytica Chimica Acta, 648, 207-214 (2009) |
| [6] | Bekker, A.; Roux, J. J., Reliability characteristics of the Maxwell distribution: a Bayes estimation study, Communications in Statistics—Theory and Methods, 34, 2169-2178 (2005) · Zbl 1081.62087 |
| [7] | Carrasco, J. M.F.; Ortega, E. M.M.; Cordeiro, G. M., A generalized modified Weibull distribution for lifetime modeling, Computational Statistics and Data Analysis, 53, 450-462 (2008) · Zbl 1231.62015 |
| [8] | Cooray, K.; Ananda, M. M.A., A generalization of the half-normal distribution with applications to lifetime data, Communications in Statistics—Theory and Methods, 37, 1323-1337 (2008) · Zbl 1163.62006 |
| [9] | Cordeiro, G. M.; de Castro, M., A new family of generalized distributions, Journal of Statistical Computation and Simulation (2011) · Zbl 1219.62022 |
| [10] | Cordeiro, G. M.; Ortega, E. M.M.; Silva, G. O., The exponentiated generalized gamma distribution with application to lifetime data, Journal of Statistical Computation and Simulation (2010) |
| [11] | Cowles, M. K.; Carlin, B. P., Markov chain Monte Carlo convergence diagnostics: a comparative review, Journal of the American Statistical Association, 91, 133-169 (1996) · Zbl 0869.62066 |
| [12] | Cox, C., The generalized F distribution: an umbrella for parametric survival analysis, Statistics in Medicine, 27, 4301-4312 (2008) |
| [13] | Cox, C.; Chu, H.; Schneider, M. F.; Muñoz, A., Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution, Statistics in Medicine, 26, 4352-4374 (2007) |
| [14] | Eugene, N.; Lee, C.; Famoye, F., Beta-normal distribution and its applications, Communications in Statistics—Theory and Methods, 31, 497-512 (2002) · Zbl 1009.62516 |
| [15] | Exton, H., Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs (1978), Halsted Press: Halsted Press New York · Zbl 0377.33001 |
| [16] | Famoye, F.; Lee, C.; Olumolade, O., The beta-Weibull distribution, Journal of Statistical Theory and Practice, 4, 121-136 (2005) |
| [17] | Gelman, A.; Rubin, D. B., Inference from iterative simulation using multiple sequences (with discussion), Statistical Science, 7, 457-472 (1992) · Zbl 1386.65060 |
| [18] | Gomes, O.; Combes, C.; Dussauchoy, A., Parameter estimation of the generalized gamma distribution, Mathematics and Computers in Simulation, 79, 955-963 (2008) · Zbl 1162.65309 |
| [19] | Gradshteyn, I. S.; Ryzhik, I. M., (Jeffrey, A.; Zwillinger, D., Table of Integrals, Series, and Products (2000), Academic Press: Academic Press New York) · Zbl 0981.65001 |
| [20] | Gupta, R. D.; Kundu, D., Generalized exponential distributions, Australian and New Zealand Journal of Statistics, 41, 173-188 (1999) · Zbl 1007.62503 |
| [21] | Gupta, R. D.; Kundu, D., Exponentiated exponential distribution: an alternative to gamma and Weibull distributions, Biometrical Journal, 43, 117-130 (2001) · Zbl 0997.62076 |
| [22] | Gusmão, F. R.S.; Ortega, E. M.M.; Cordeiro, G. M., The generalized inverse Weibull distribution, Statistical Papers (2009) |
| [23] | Jones, M. C., Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages, Statistical Methodology, 6, 70-81 (2009) · Zbl 1215.60010 |
| [24] | Kumaraswamy, P., A generalized probability density function for double-bounded random processes, Journal of Hydrology, 46, 79-88 (1980) |
| [25] | Kundu, D.; Raqab, M. Z., Generalized Rayleigh distribution: different methods of estimation, Computational Statistics and Data Analysis, 49, 187-200 (2005) · Zbl 1429.62449 |
| [26] | Lawless, J. F., Inference in the generalized gamma and log-gamma distributions, Technometrics, 22, 409-419 (1980) · Zbl 0456.62033 |
| [27] | Lawless, J. F., Statistical Models and Methods for Lifetime Data (2003), John Wiley: John Wiley New York · Zbl 1015.62093 |
| [28] | Malhotra, J.; Sharma, A. K.; Kaler, R. S., On the performance analysis of wireless receiver using generalized-gamma fading model, Annals of Telecommunications, 64, 147-153 (2009) |
| [29] | Mudholkar, G. S.; Srivastava, D. K., Exponentiated Weibull family for analyzing bathtub failure-real data, IEEE Transaction on Reliability, 42, 299-302 (1993) · Zbl 0800.62609 |
| [30] | Mudholkar, G. S.; Srivastava, D. K.; Freimer, M., The exponentiated Weibull family: a reanalysis of the bus-motor-failure data, Technometrics, 37, 436-445 (1995) · Zbl 0900.62531 |
| [31] | Mudholkar, G. S.; Srivastava, D. K.; Kollia, G. D., A generalization of the Weibull distribution with application to the analysis of survival data, Journal of American Statistical Association, 91, 1575-1583 (1996) · Zbl 0881.62017 |
| [32] | Nadarajah, S., On the use of the generalised gamma distribution, International Journal of Electronics, 95, 1029-1032 (2008) |
| [33] | Nadarajah, S., Explicit expressions for moments of order statistics, Statistics and Probability Letters, 78, 196-205 (2008) · Zbl 1290.62023 |
| [34] | Nadarajah, S.; Gupta, A. K., A generalized gamma distribution with application to drought data, Mathematics and Computers in Simulation, 74, 1-7 (2007) · Zbl 1108.60011 |
| [35] | Ortega, E. M.M.; Bolfarine, H.; Paula, G. A., Influence diagnostics in generalized log-gamma regression models, Computational Statistics and Data Analysis, 42, 165-186 (2003) · Zbl 1429.62336 |
| [36] | Ortega, E. M.M.; Cancho, V. G.; Paula, G. A., Generalized log-gamma regression models with cure fraction, Lifetime Data Analysis, 15, 79-106 (2009) · Zbl 1302.62236 |
| [37] | Ortega, E. M.M.; Paula, G. A.; Bolfarine, H., Deviance residuals in generalized log-gamma regression models with censored observations, Journal of Statistical Computation and Simulation, 78, 747-764 (2008) · Zbl 1145.62081 |
| [38] | G.S.C. Perdoná, Modelos de Riscos Aplicados á Análise de Sobrevivência. Doctoral Thesis, Institute of Computer Science and Mathematics, University of São Paulo, Brasil (in Portuguese), 2006.; G.S.C. Perdoná, Modelos de Riscos Aplicados á Análise de Sobrevivência. Doctoral Thesis, Institute of Computer Science and Mathematics, University of São Paulo, Brasil (in Portuguese), 2006. |
| [39] | Pescim, R. R.; Demétrio, C. G.B.; Cordeiro, G. M.; Ortega, E. M.M.; Urbano, M. R., The beta generalized half-normal distribution, Computational Statistics and Data Analysis, 54, 945-957 (2009) · Zbl 1465.62015 |
| [40] | Prudnikov, A. P.; Brychkov, Y. A.; Marichev, O. I., Integrals and Series, vol. 1 (1986), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Amsterdam · Zbl 0606.33001 |
| [41] | Shankar, P. M.; Dumane, V. A.; Reid, J. M.; Genis, V.; Forsberg, F.; Piccoli, C. W.; Goldberg, B. B., Classification of ultrasonic b-mode images of breast masses using Nakagami distribution, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 48, 569-580 (2001) |
| [42] | A.N.F. Silva, Estudo evolutivo das crianças expostas ao HIV e notificadas pelo núcleo de vigilância epidemiológica do HCFMRP-USP. M.Sc. Thesis. University of São Paulo, Brazil, 2004.; A.N.F. Silva, Estudo evolutivo das crianças expostas ao HIV e notificadas pelo núcleo de vigilância epidemiológica do HCFMRP-USP. M.Sc. Thesis. University of São Paulo, Brazil, 2004. |
| [43] | Stacy, E. W., A generalization of the gamma distribution, Annals of Mathematical Statistics, 33, 1187-1192 (1962) · Zbl 0121.36802 |
| [44] | Stacy, E. W.; Mihram, G. A., Parameter estimation for a generalized gamma distribution, Technometris, 7, 349-358 (1965) · Zbl 0129.11107 |
| [45] | Trott, M., The Mathematica Guidebook for Symbolics. With 1 DVD-ROM (Windows, Macintosh and UNIX) (2006), Springer: Springer New York · Zbl 1095.68001 |
| [46] | Xie, M.; Lai, C. D., Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function, Reliability Engineering & System Safety, 52, 87-93 (1995) |
| [47] | Xie, X.; Liu, X., Analytical three-moment autoconversion parameterization based on generalized gamma distribution, Journal of Geophysical Research, 114, D17201 (2009) |
| [48] | Xie, M.; Tang, Y.; Goh, T. N., A modified Weibull extension with bathtub failure rate function, Reliability Engineering & System Safety, 76, 279-285 (2002) |
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