# A weighted version of Gamma distribution

Kanchan Jain; Neetu Singla; Rameshwar D. Gupta

Discussiones Mathematicae Probability and Statistics (2014)

- Volume: 34, Issue: 1-2, page 89-111
- ISSN: 1509-9423

## Access Full Article

top## Abstract

top## How to cite

topKanchan Jain, Neetu Singla, and Rameshwar D. Gupta. "A weighted version of Gamma distribution." Discussiones Mathematicae Probability and Statistics 34.1-2 (2014): 89-111. <http://eudml.org/doc/270808>.

@article{KanchanJain2014,

abstract = {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.},

author = {Kanchan Jain, Neetu Singla, Rameshwar D. Gupta},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {gamma distribution; weight function; hazard function; maximum likelihood estimator; Akaike Information criterion; Akaike information criterion},

language = {eng},

number = {1-2},

pages = {89-111},

title = {A weighted version of Gamma distribution},

url = {http://eudml.org/doc/270808},

volume = {34},

year = {2014},

}

TY - JOUR

AU - Kanchan Jain

AU - Neetu Singla

AU - Rameshwar D. Gupta

TI - A weighted version of Gamma distribution

JO - Discussiones Mathematicae Probability and Statistics

PY - 2014

VL - 34

IS - 1-2

SP - 89

EP - 111

AB - 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.

LA - eng

KW - gamma distribution; weight function; hazard function; maximum likelihood estimator; Akaike Information criterion; Akaike information criterion

UR - http://eudml.org/doc/270808

ER -

## References

top- [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 Zbl0794.62075
- [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 Zbl0969.62037
- [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 Zbl0216.22702
- [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 Zbl0497.62017
- [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 Zbl0734.62093
- [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 Zbl1007.62503
- [11] R.G. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (6) (2009) 621-634. doi: 10.1080/02331880802605346 Zbl1291.60029
- [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 Zbl0707.62197
- [13] R. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. 1, second edition (New York, Wiley Inter-Science, 1994). Zbl0811.62001
- [14] C.D. Lai and M. Xie, Stochastic Ageing and Dependence for Reliability (Germany, Springer, 2006). doi: 10.1007/0-387-34232-X Zbl1098.62130
- [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 Zbl0924.62018
- [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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.