The introduced three parameter (position μ, scale ∑ and shape γ) multivariate generalized Normal distribution (γ-GND) is based on a strong theoretical background and emerged from Logarithmic Sobolev Inequalities. It includes a number of well known distributions such as the multivariate Uniform, Normal, Laplace and the degenerated Dirac distributions. In this paper, the cumulative distribution, the truncated distribution and the hazard rate of the γ-GND are presented. In addition, the Maximum Likelihood Estimation (MLE) method is discussed in both the univariate and multivariate cases and asymptotic results are presented.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1168,
author = {Christos P. Kitsos and Vassilios G. Vassiliadis and Thomas L. Toulias},
title = {MLE for the $\gamma$-order Generalized Normal Distribution},
journal = {Discussiones Mathematicae Probability and Statistics},
volume = {34},
year = {2014},
pages = {143-158},
zbl = {1326.60020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1168}
}
Christos P. Kitsos; Vassilios G. Vassiliadis; Thomas L. Toulias. MLE for the γ-order Generalized Normal Distribution. Discussiones Mathematicae Probability and Statistics, Tome 34 (2014) pp. 143-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1168/
[000] [1] M. Abramovitz and I.A. Stegun, Handbook of Mathematical Functions (Washington, National Bureau of Standards, 1964).
[001] [2] G. Agrò, Maximum likelihood estimation for the exponential power distribution, Comm. in Stat. - Simulation and Computation 24 (2) (1995) 523-536. doi: 10.1080/03610919508813255
[002] [3] H. Alzer, On some inequalities for the incomplete gamma function, Mathematics of Computation 66 (228) (1997) 771-778. doi: 10.1090/S0025-5718-97-00814-4 | Zbl 0865.33002
[003] [4] M. Chiodi, Sulle distribuzioni di campionamento delle stime di massima verosimiglianza dei parametri delle curve Normali di ordine p, Technical report, Istituto di Statistica, Facoltá di Economia e Commercio di Palermo 1988.
[004] [5] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, 1994). doi: 10.1017/CBO9780511840371.
[005] [6] J.R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (New York, Wiley, 1998). | Zbl 0651.15001
[006] [7] A.M. Mineo, Un nuovo metodo di stima di p per una corretta valutazione dei parametri di intensitá e di scala di una curva Normale di ordine p, in: Atti della XXXVII Riunione Scientifica della SIS, San Remo (Ed(s)), (1994, 147-154.
[007] [8] A.M. Mineo and M. Ruggieri, A software tool for the exponential power distribution: The normalp package, J. Stat. Software (2005) 1-24.
[008] [9] T.P. Minka, Old and new matrix algebra useful for statistics, Notes 2000.
[009] [10] S.R. Searle, Matrix Algebra Useful for Statistics (New York, Wiley, 1982). | Zbl 0555.62002
[010] [11] C.P. Kitsos and N.K. Tavoularis, Logarithmic Sobolev inequalities for information measures, IEEE Trans. Inform. Theory 55 (6) (2009) 2554-2561. doi: 10.1109/TIT.2009.2018179
[011] [12] C.P. Kitsos, T.L. Toulias and C.P. Trandafir, On the multivariate γ-ordered normal distribution, Far East J. of Theoretical Statistics 38 (1) (2012) 49-73. | Zbl 1252.60020
[012] [13] G. Lunetta, Di alcune distribuzioni deducibili da una generalizzazione dello schema della curva Normale, Annali della Facoltá di Economia e Commercio di Palermo 20 (1966) 119-143.
[013] [14] D.N. Naik and K. Plungpongpun, A Kotz-Type distribution for multivariate statistical inference, Statistics for Industry and Technology (2006) 111-124. doi: 10.1007/0-8176-4487-3_7. | Zbl 05196666