On the properties of the Generalized Normal Distribution

• Autorzy: Thomas L. Toulias , Christos P. Kitsos
• Języki publikacji: EN

Abstrakty

EN

The target of this paper is to provide a critical review and to enlarge the theory related to the Generalized Normal Distributions (GND). This three term (position, scale shape) distribution is based in a strong theoretical background due to Logarithm Sobolev Inequalities. Moreover, the GND is the appropriate one to support the Generalized entropy type Fisher's information measure.

Słowa kluczowe

EN

entropy type Fisher's information; Shannon entropy; Normal distribution; truncated distribution

Kategorie tematyczne

• 62E15: Exact distribution theory
• 62H10: Distribution of statistics
• 60E05: Distributions: general theory

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae Probability and Statistics
• Rocznik: 2014
• Tom: 34
• Numer: 1-2
• Strony: 35-49

Daty

wydano

2014

otrzymano

2014-04-20

poprawiono

2014-06-15

Twórcy

autor

Thomas L. Toulias

• Technological Educational Institute of Athens, Informatics Department, Egaleo 12243, Athens, Greece

autor

Christos P. Kitsos

• Technological Educational Institute of Athens, Informatics Department, Egaleo 12243, Athens, Greece

Bibliografia

• [1] W.A. Benter, Generalized Poincaré inequality for the Gaussian measures, Amer. Math. Soc. 105 (2) (1989) 49-60.
• [2] T. Cacoullos and M. Koutras, Quadric forms in sherical random variables: Generalized non-central χ² distribution, Naval Research Logistics Quarterly 31 (1984) 447-461. doi: 10.1002/nav.3800310310.
• [3] T.M. Cover and J.A. Thomas, Elements of Information Theory, 2nd Ed. (Wiley, 2006).
• [4] K. Fragiadakis and S.G. Meintanis, Test of fit for asymentric Laplace distributions with applications, J. of Statistics: Advances in Theory and Applications 1 (1) (2009) 49-63.
• [5] E. Gómez, M.A. Gómez-Villegas and J.M. Marin, A multivariate generalization of the power exponential family of distributions, Comm. Statist. Theory Methods 27 (3) (1998) 589-600. doi: 10.1080/03610929808832115
• [6] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).
• [7] L. Gross, Logarithm Sobolev inequalities, Amer. J. Math. 97 (761) (1975) 1061-1083. doi: 10.2307/2373688
• [8] 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
• [9] C.P. Kitsos and N.K. Tavoularis, New entropy type information measures, in: Information Technology Interfaces (ITI 2009), Luzar, Jarec and Bekic (Ed(s)), (Dubrovnic, Croatia, 2009) 255-259.
• [10] C.P. Kitsos and T.L. Toulias, Bounds for the generalized entropy-type information measure, J. Comm. Comp. 9 (1) (2012) 56-64.
• [11] 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.
• [12] C.P. Kitsos and T.L. Toulias, On the family of the γ-ordered normal distributions, Far East J. of Theoretical Statistics 35 (2) (2011) 95-114.
• [13] C.P. Kitsos and T.L. Toulias, New information measures for the generalized normal distribution, Information 1 (2010) 13-27. doi: 10.3390/info1010013.
• [14] C.P. Kitsos and T.L. Toulias, Entropy inequalities for the generalized Gaussia, in: Information Technology Interfaces (ITI 2010), Cavtat, Croatia (Ed(s)), (2010, 551-556.
• [15] S. Kotz, Multivariate distribution at a cross-road, in: Statistical Distributions in Scientific Work Vol. 1, Patil, Kotz, Ord (Ed(s)), (Dordrecht, The Netherlands: D. Reidel Publ., 1975) 247-270. doi:10.1007/978-94-010-1842-5₂0.
• [16] S. Kullback and A. Leibler A, On information and sufficiency, Ann. Math. Statist. 22 (1951) 79-86. doi: 10.1214/aoms/1177729694.
• [17] S. Nadarajah, A generalized normal distribution, J. Appl. Stat. 32 (7) (2005) 685-694. doi: 10.1080/02664760500079464
• [18] S. Nadarajah, The Kotz type distribution with applications, Statistics 37 (4) (2003) 341-358. doi: 10.1080/0233188031000078060
• [19] T. Papaioanou and K. Ferentinos, Entropic descriptor of a complex behaviour, Comm. Stat. Theory and Methods 34 (2005) 1461-1470.
• [20] M. Del Pino, J. Dolbeault and I. Gentil, Nonlinear difussions, hypercontractivity and the optimal $L^p$-Euclidean logarithic Sobolev ineqiality, J. Math. Anal. Appl. 293 (2) (2004) 375-388. doi: 10.1016/j.jmaa.2003.10.009
• [21] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379-423, 623-656. doi: j.1538-7305.1948.tb01338.x.
• [22] S. Sobolev, On a theorem of functional analysis, English translation: AMS Transl. Ser. 2 34 (1963) 39-68.

Identyfikatory

DOI

10.7151/dmps1167