On families of beta- and generalized gamma-generated distributions and associated inference

Abstract A general family of univariate distributions generated by beta random variables, proposed by Jones, has been discussed recently in the literature. This family of distributions possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. In a similar vein, we define here a family of univariate distributions generated by Stacy’s generalized gamma variables. For these two families of univariate distributions, we discuss maximum entropy characterizations under suitable constraints. Based on these characterizations, an expected ratio of quantile densities is proposed for the discrimination of members of these two broad families of distributions. Several special cases of these results are then highlighted. An alternative to the usual method of moments is also proposed for the estimation of the parameters, and the form of these estimators is particularly amenable to these two families of distributions.

[1]  Jafar Ahmadi,et al.  Entropy properties of record statistics , 2007 .

[2]  N. Balakrishnan.  Order statistics from the half logistic distribution , 1985 .

[3]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[4]  Rodney W. Johnson,et al.  Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy , 1980, IEEE Trans. Inf. Theory.

[5]  E. Soofi.  Principal Information Theoretic Approaches , 2000 .

[6]  M. C. Jones,et al.  Multivariate distributions with support above the diagonal , 2004 .

[7]  J. N. Kapur.  Maximum-entropy models in science and engineering , 1992 .

[8]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[9]  Lawrence B. Levy,et al.  The log F: A Distribution for All Seasons , 2002, Comput. Stat..

[10]  S. Kullback,et al.  Information Theory and Statistics , 1959 .

[11]  M. C. Jones.  Families of distributions arising from distributions of order statistics , 2004 .

[12]  Samuel Kotz,et al.  The beta exponential distribution , 2006, Reliab. Eng. Syst. Saf..

[13]  D. Kundu,et al.  EXPONENTIATED EXPONENTIAL FAMILY: AN ALTERNATIVE TO GAMMA AND WEIBULL DISTRIBUTIONS , 2001 .

[14]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[15]  R. Jain,et al.  Records , 1973, Tempo.

[16]  M. C. Jones.  On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions , 2007 .

[17]  D. Kundu,et al.  Discriminating between the log-normal and generalized exponential distributions , 2005 .

[18]  F. Famoye,et al.  BETA-NORMAL DISTRIBUTION AND ITS APPLICATIONS , 2002 .

[19]  Families of Multivariate Distributions Involving the Rosenblatt Construction , 2006 .

[20]  G. S. Mudholkar,et al.  Exponentiated Weibull family for analyzing bathtub failure-rate data , 1993 .

[21]  Mark F. J. Steel,et al.  Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers , 2004 .

[22]  M. Steel,et al.  A Constructive Representation of Univariate Skewed Distributions , 2006 .