Natural exponential family

About: Natural exponential family is a research topic. Over the lifetime, 1973 publications have been published within this topic receiving 60189 citations. The topic is also known as: NEF.

A New Family of Distributions: Libby-Novick Beta

Abstract: We define a family of distributions, named the Libby-Novick beta family of distributions, which includes the classical beta generalized and exponentiated generators. The new family offers much more flexibility for modeling real data than these two generators and the Kumaraswamy family of distributions (Cordeiro \& de Castro, 2011). The extended fa\-mily provides reasonable parametric fits to real data in several areas because the additional shape parameters can control the skewness and kurtosis simultaneously, vary tail weights and provide more entropy for the generated distribution. For any given distribution, we can construct a wider distribution with three additional shape parameters which has much more flexibility than the original one. The family density function is a linear combination of exponentiated densities defined from the same baseline distribution. The proposed family also has tractable mathematical properties including moments, generating function, mean deviations and order statistics. The parameters are estimated by maximum likelihood and the observed information matrix is determined. The importance of the family is very well illustrated. By means of two real data sets we demonstrate that this family can give better fits than those ones using the McDonald, beta generalized and Kumaraswamy generalized classes of distributions.

Fast learning of gamma mixture models with k -MLE

Abstract: We introduce a novel algorithm to learn mixtures of Gamma distributions. This is an extension of the k-Maximum Likelihood Estimator algorithm for mixtures of exponential families. Although Gamma distributions are exponential families, we cannot rely directly on the exponential families tools due to the lack of closed-form formula and the cost of numerical approximation: our method uses Gamma distributions with a fixed rate parameter and a special step to choose this parameter is added in the algorithm. Since it converges locally and is computationally faster than an Expectation-Maximization method for Gamma mixture models, our method can be used beneficially as a drop-in replacement in any application using this kind of statistical models.

The Kumaraswamy Generalized Power Weibull Distribution

Abstract: A new family of distributions called Kumaraswamy-generalized power Weibull (Kgpw) distribution is proposed and studied. This family has a number of well known sub-models such as Weibull, exponentiated Weibull, Kumaraswamy Weibull, generalized power Weibull and new sub-models, namely, exponentiated generalized power Weibull, Kumaraswamy generalized power exponential distributions. Some statistical properties of the new distribution include its moments, moment generating function, quantile function and hazard function are derived. In addition, maximum likelihood estimates of the model parameters are obtained. An application as well as comparisons of the Kgpw and its sub-distributions is given. Keywords: Generalized power Weibull distribution, Kumaraswamy distribution, Maximum likelihood estimation, Moment generating function, Hazard rate function.

On entire functions of exponential type

Abstract: Le t J be a n en tire fun c ti o n a nd le t p. \"\" 1 and 1(1 , r)= {f:~ 1 f 11)(re iO) I \"dO} 1/\". for a U s uffic ie ntly la rge r, the n J is of ex pone nti a l type not exceeding. {2 log (l-t. ~) + 1 + log (2N) !} .. If thi s co ndition is re place d by re lated co nditi ons, th e n a lso is of expo ne nti a l t ype. An e ntire fun c tion f(z) is said to be of bounde d ind ex if and only if th ere exi sts a non-negative integer N (ind e pe nde nt of z) s uch thatO\"'j \",N j!-k! (1.1) for all k and all z, and the smallest s uc h integer N is calle d the index off(z) ([1], [4] , [5]).1 It is known that a function of bounde d inde x N is of ex ponenti al type not exceedin g N+ 1 [6] but that a function of expon e ntial type need not be of bounde d inde x. In fac t any e ntire fun c tion havin g ze ros of arbitrarily large multipli city is not of bound e d index and th ere exist fun c tion s with simple zeros and of exponential type whic h are not of bounded index [8]. In a recent paper [2] Fred Gross considers interesting variations of condition (1.1) and proves the following THEOREM A: Let f be entire and C a positive constant. If there exists a positive integer N such that for k=O, 1,. . , N, f satisfies one ofthefollowing,for all z with l z I sufficiently large:

Divergence measures between populations: applications in the exponential family

Abstract: Toussaint (1974) introduced a divergence measure between s populations as a generalization of the J-divergence. The asymptotic distribution of Toussaint's measure is obtained when the parameters are substituted by their maximum likelihood estimators and the s probability density functions belong to the exponential farnily. A procedure to test statistical hypotheses about s populations is given These results can also be applied to multinomial populations.