Geodesic
A weighted version of Gamma distribution
Discussiones Mathematicae. Probability and Statistics, Tome 34 (2014) no. 1-2, pp. 89-111.

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Weighted Gamma (WG), a weighted version of Gamma distribution, is introduced. The hazard function is increasing or upside-down bathtub depending upon the values of the parameters. This distribution can be obtained as a hidden upper truncation model. The expressions for the moment generating function and the moments are given. The non-linear equations for finding maximum likelihood estimators (MLEs) of parameters are provided and MLEs have been computed through simulations and also for a real data set. It is observed that WG fits better than its submodels (WE), Generalized Exponential (GE), Weibull and Exponential distributions.
Keywords: gamma distribution, weight function, hazard function, maximum likelihood estimator, Akaike Information criterion 
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Jain, Kanchan; Singla, Neetu; Gupta, Rameshwar. A weighted version of Gamma distribution. Discussiones Mathematicae. Probability and Statistics, Tome 34 (2014) no. 1-2, pp. 89-111. http://geodesic.mathdoc.fr/item/DMPS_2014_34_1-2_a6/

[1] B.C. Arnold, R.J. Beaver, R.A. Groeneveld and W.Q. Meeker, The nontruncated marginal of a bivariate normal distribution, Psychometrika 58 (3) (1993) 471-488. doi: 10.1007/BF02294652

[2] B.C. Arnold and R.J. Beaver, The Skew Cauchy distribution, Statistics and Probability Letters 49 (2000) 285-290. doi: 10.1016/S0167-7152(00)00059-6

[3] B.C. Arnold, Flexible univariate and multivariate models based on hidden truncation, Proceedings of the 8th Tartu Conference on Multivariate Statistics, June 26-29 (Tartu, Estonia, 2007).

[4] A. Azzalini, A class of distributions which include the normal ones, Scandinavian J. Stat. 12 (1985) 171-178.

[5] R.E. Barlow and F. Proschan F, Statistical Theory of Reliability and Life Testing: Probability Models. To Begin with (Silver Springs, MD, 1981).

[6] Z.W. Birnbaum and S.C. Saunders, Estimation for a family of life distributions with applications to fatigue, J. Appl. Prob. 6 (1969) 328-347. doi: 10.2307/3212004

[7] R.A. Fisher, The effects of methods of ascertainment upon the estimation of frequencies, Annals of Eugenics 6 (1934) 13-25. doi: 10.1111/j.1469-1809.1934.tb02105.x

[8] R.E. Glaser, Bathtub and related failure rate characterizations, J. Amer. Assoc. 75 (371) (1980) 667-672. doi: 10.2307/2287666

[9] R.C. Gupta and S.N.U.A. Kirmani, The role of weighted distributions in stochastic modelling, Communications in Statisics - Theory and Methods 19 (9) (1990) 3147-3162. doi: 10.1080/03610929008830371

[10] R.D. Gupta and D. Kundu, Generalized exponential distributions, Australian New Zealand J. Stat. 41 (1999) 173-188. doi: 10.1111/1467-842X.00072

[11] R.G. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (6) (2009) 621-634. doi: 10.1080/02331880802605346

[12] K. Jain, H. Singh and I. Bagai, Relations for reliability measures of weighted distribution, Communications in Statistics - Theory and Methods 18 (12) (1989) 4393-4412. doi: 10.1080/03610928908830162

[13] R. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. 1, second edition (New York, Wiley Inter-Science, 1994).

[14] C.D. Lai and M. Xie, Stochastic Ageing and Dependence for Reliability (Germany, Springer, 2006). doi: 10.1007/0-387-34232-X

[15] M. Mahfoud and G.P. Patil, On Weighted Distributions, in: G. Kallianpur, P.R. Krishnaiah and J.K. Ghosh, eds., Statistics and Probability: Essays in Honor of C.R. Rao (North-Holland, Amsterdam, 1982) 479-492.

[16] A.K. Nanda and K. Jain, Some weighted distribution results on univariate and bivariate cases, J. Stat. Planning and Inference 77 (2) (1999) 169-180. doi: 10.1016/S0378-3758(98)00190-6

[17] G.P. Patil, Weighted Distributions, Encyclopaedia of Environmetrics, eds., A.H. El-Shaarawi and W.W. Piegorsch, 2369-2377 (John Wiley Sons, Ltd, Chichester, 2002).

[18] C.R. Rao, On discrete distributions arising out of methods of ascertainment, in: G.P. Patil (ed.), Classical and Contagious Discrete Distributions, Permagon Press, Oxford and Statistical Publishing Society (Calcutta, 1965) 320-332.

[19] M. Shaked and J.G. Shanthikumar, Stochastic Orders (Germany, Springer, 2007). doi: 10.1007/978-0-387-34675-5