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.

Generalized gamma distributions as mixed exponential laws and related limit theorems

Abstract: A theorem due to L. J. Gleser stating that a gamma distribution with shape parameter no greater than one is a mixed exponential distribution is extended to generalized gamma distributions introduced by E. W. Stacy as a special family of lifetime distributions containing both gamma distributions, exponential power and Weibull distributions. It is shown that the mixing distribution is a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the mixed Poisson distribution with the generalized gamma mixing law as a mixed geometric distribution. Limit theorems are proved establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the mixed Poisson distribution with the generalized gamma mixing law including random sums to special normal mixtures.

Bayesian precision matrix estimation for graphical Gaussian models with edge and vertex symmetries

Abstract: Graphical Gaussian models with edge and vertex symmetries were introduced by \citet{HojLaur:2008} who also gave an algorithm to compute the maximum likelihood estimate of the precision matrix for such models. In this paper, we take a Bayesian approach to the estimation of the precision matrix. We consider only those models where the symmetry constraints are imposed on the precision matrix and which thus form a natural exponential family with the precision matrix as the canonical parameter. We first identify the Diaconis-Ylvisaker conjugate prior for these models and develop a scheme to sample from the prior and posterior distributions. We thus obtain estimates of the posterior mean of the precision matrix. Second, in order to verify the precision of our estimate, we derive the explicit analytic expression of the expected value of the precision matrix when the graph underlying our model is a tree, a complete graph on three vertices and a decomposable graph on four vertices with various symmetries. In those cases, we compare our estimates with the exact value of the mean of the prior distribution. We also verify the accuracy of our estimates of the posterior mean on simulated data for graphs with up to thirty vertices and various symmetries.

Contrastive learning from pairwise measurements

Abstract: Learning from pairwise measurements naturally arises from many applications, such as rank aggregation, ordinal embedding, and crowdsourcing. However, most existing models and algorithms are susceptible to potential model misspecification. In this paper, we study a semiparametric model where the pairwise measurements follow a natural exponential family distribution with an unknown base measure. Such a semiparametric model includes various popular parametric models, such as the Bradley-Terry-Luce model and the paired cardinal model, as special cases. To estimate this semiparametric model without specifying the base measure, we propose a data augmentation technique to create virtual examples, which enables us to define a contrastive estimator. In particular, we prove that such a contrastive estimator is invariant to model misspecification within the natural exponential family, and moreover, attains the optimal statistical rate of convergence up to a logarithmic factor. We provide numerical experiments to corroborate our theory.

On Extended Generalized Exponential Distribution

Abstract: In this paper, we present a generalized exponential distribution that contains four parameters. This distribution further generalizes previously established generalized exponential distributions which now serves as special cases of the new four-parameter generalized exponential distribution. The properties of the new distribution like the cumulative distribution function, the survival function, the hazard function, the moment generating function, the median, the 100ppercentile point and the mode of distribution are established. The moment function of the distribution which cannot be obtained in close form is numerically obtained and tabulated for some selected values of the parameters. A Theorem that characterized the distribution is stated and proved.