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.

Moment Generating Function of the Bivariate Generalized Exponential Distribution

Abstract: Recently a new distribution, named a bivariate generalized exponential (BVGE) distribution has been introduced by Kundu and Gupta (2008). In this paper we obtain joint moments and the moment generating function for (BVGE) which is in closed form, and convenient to use in practice.

A Unified Approach for Construction of Probability Models for Bivariate Linear and Directional Data

Abstract: This article considers a unified approach based on the mixture method to construct linear bivariate models and those on the cylinder and torus involving the exponential and cardioid distributions with the truncated exponential distribution as the mixing distribution. Parameter estimation of the bivariate model on the torus is considered for the data set of phase angles of circadian-related genes in heart and liver tissues.

On Simple Exponential Sets of Polynomials

Abstract: In this paper, it is shown that certain classes of regular functions of several complex variables be represented by exponential sets of polynomials in hyperelliptical regions. Moreover, an upper bound for the order of exponential set is given.

On the Developement of Exponential Functions; Together with Several New Theorems Relating to Finite Differences

Abstract: In the year 1772, Lagrange, in a Memoir, published among those of the Berlin Academy, announced those celebrated theorems expressing the connection between simple exponential indices, and those of differentiation and integration. The demonstration of those theorems, although it escaped their illustrious discoverer, has been since accomplished by many analysts, and in a great variety of ways. Laplace set the first example in two Memoirs presented to the Academy of Sciences,* and may be supposed in the course of these researches, to have caught the first hint of the Calcul des Fonctions Generatrices with which they are so intimately connected ; as, after an interval of two years, another demonstration of them, drawn solely from the principles of that calculus appeared, together with the calculus itself, in the memoirs of the Academy. This demonstration, involving, however, the passage from finite to infinite, is therefore (although preferable perhaps in a systematic arrangement, where all is made to flow from one fundamental principle) less elegant ; not on account of any confusion of ideas, or want of evidence ; but, because the ideas of finite and infinite, as such, are extraneous to symbolic language, and, if we would avoid their use, much circumlocution, as well as very unwieldy formulæ must be introduced. Arbogast also, in his work on derivations, has given two most ingenious demonstrations of them, and added greatly to their generality ; and lastly, Dr. Brinkley has made them the subject of a paper in the Transactions of this Society,* to which I shall have occasion again to refer. Considered as insulated truths, unconnected with any other considerable branch of analysis, the method employed by the latter author seems the most simple and elegant which could have been devised. It has however the great inconvenience of not making us acquainted with the bearings and dependencies of these important theorems, which, in this instance, as in many others, are far more valuable than the mere formulæ.