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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.


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Journal ArticleDOI
01 Apr 1968
TL;DR: In this article, a characterization theorem for a subclass of the exponential family whose probability density function is given by where a(x) ≥ 0, f(ω) = ∫a (x) exp (ωx) dx and ωx is to be interpreted as a scalar product is given.
Abstract: In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

9 citations

Journal ArticleDOI
TL;DR: In this paper, the authors used a mixture of exponential distributions, with Ft being the mixing distribution, to obtain the limiting distribution of Ytt, as t → 0+, and the finiteness and form of E(Ytk) for any real k < 1, with an emphasis on distributions related to symmetric random walks and compound Poisson and extreme stable distributions.

9 citations

Journal ArticleDOI
TL;DR: In this article, the Block and Basu Bivariate generalized exponential (BGS) distribution is extended to the generalized exponential distribution, where the joint probability distribution function and the joint cumulative distribution function can be expressed in compact forms.

9 citations

Journal ArticleDOI
TL;DR: In this article, state-space models with exponential and conjugate exponential family densities are introduced, such as Poisson-Gamma, Binomial-Beta, Gamma and Normal-Normal processes.

9 citations

01 Jan 2013
TL;DR: In this article, a multivariate generalized linear model for non-normal multivariate data is proposed, which can accommodate a wide variety of different types of data, much like generalized linear models do in the univariate case.
Abstract: In order to develop a general approach for analysis of non-normal multivariate data, it would be desirable to obtain a simple-minded framework that can accommodate a wide variety of different types of data, much like generalized linear models do in the univariate case. There is no shortage of multivariate distributions available, but the main stumbling block so far has been the lack of a suitable multivariate form of exponential dispersion model. In the univariate case, an exponential dispersion model ED(μ, σ) is a twoparameter family parametrized by the mean μ and dispersion parameter σ, with variance σV (μ), where V denotes the unit variance function. The generalized linear models paradigm is based on combining a link function with a suitable linear model. Estimation uses quasi-likelihood for the regression parameters, and the Pearson statistic for estimating the dispersion parameter. We consider a new k-variate exponential dispersion model EDk(μ,Σ) aimed at providing a fully flexible covariance structure corresponding to a mean vector μ and a positive-definite dispersion matrix Σ. The covariance matrix is of the form Cov(Y ) = Σ⊙V (μ), where ⊙ denotes the Hadamard (elementwise) product between two matrices, and V (μ) denotes the (matrix) unit variance function. We consider a multivariate generalized linear model for independent response vectors Y i ∼ EDk(μi,Σ) defined by g(μ ⊤ i ) = xiB, where the link function g is applied coordinatewise to μ i , xi is an m-vector of covariates, and B is an m × k matrix of regression coefficients. We estimate the regression matrix B using a quasi-score function, and we estimate the dispersion matrix Σ using a multivariate Pearson statistic defined as a weighted sum of squares and cross-products matrix of residuals. This model specializes to the classical multivariate multiple regression model when g is the identity function and EDk(μ,Σ) is the multivariate normal distribution. The construction of the multivariate exponential dispersion model EDk(μ,Σ) is based on an extended convolution method, which makes the marginal distributions follow a given univariate exponential dispersion model. We illustrate the method by considering multivariate versions of the Poisson and gamma distributions, and discuss some of the challenges faced in the implementation of the method.

9 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202319
202262
202114
202010
20196
201823