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.

When does the minimum of a sample of an exponential family belong to an exponential family

Abstract: It is well known that if $({X}_{1},...,{X}_{n})$ are i.i.d. r.v.'s taken from either the exponential distribution or the geometric one, then the distribution of $\min({X}_{1},...,{X}_{n})$ is, with a change of parameter, is also exponential or geometric, respectively. In this note we prove the following result. Let $F$ be a natural exponential family (NEF) on $\mathbb{R}$ generated by an arbitrary positive Radon measure $\mu$ (not necessarily confined to the Lebesgue or counting measures on $\mathbb{R}$). Consider $n$ i.i.d. r.v.'s $({X}_{1},...,{X}_{n})$, $n \in 2$, taken from $F$ and let $Y =\min({X}_{1},...,{X}_{n})$. We prove that the family $G$ of distributions induced by $Y$ constitutes an NEF if and only if, up to an affine transformation, $F$ is the family of either the exponential distributions or the geometric distributions. The proof of such a result is rather intricate and probabilistic in nature.

On sequential distinguishability for the exponential family

Abstract: Let {X,, n ~ l } be an iid (independent and identically distributed) stochastic sequence assumed to be governed by a member of a countable family of probability measures 4 = {P,: 8 e ~} where P, are defined on an appropriate probability space and ~ is countable. Observing sequent/ally the stochastic sequence {X,, n>=l} we want to stop at some finite stage and decide in favour of a member of the family 9 with a uniformly small probability of error. The family 9 is said to be "Sequentially Distinguishable" i f for any given ~ (O l and ~,(~l~___e for a given s (0<~<1) and V].

On Approximating Distribution of the Quadratic Discriminant Function

Abstract: The quadratic discriminant function (QDF) with known parameters has been represented in terms of a weighted sum of independent noncentral chi-square variables. To approximate the density function of the QDF as m-dimensional exponential family, its moments in each order have been calculated. This is done using the recursive formula for the moments via the Stein's identity in the exponential family. We validate the performance of our method using simulation study and compare with other methods in the literature based on the real data. The finding results reveal better estimation of misclassification probabilities, and less computation time with our method.

Characterization of Uniform Distribution through Expectation of Function of Order Statistics

Abstract: For characterization of uniform distribution one needs any arbitrary non constant function only in place of approaches such as identical distributions, absolute continuity, constancy of regression of order statistics, continuity and linear regression of order statistics, non-degeneracy etc. available in the literature. Recently Bhatt characterized negative exponential distribution through expectation of non constant function of random variable. Attempt is made to extend the characterization of negative exponential distribution through expectation of any arbitrary non constant function of order statistics. Keyword: Characterization; Uniform distribution. MSC 2010 Subject Clasification : 62E10