Topic
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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TL;DR: In this paper, the authors present an overview of the mean value parametrization and characterization property of the variance function for NEF's and introduce the relationships existing between the NEF generating measure, Laplace transform and variance function.
Abstract: It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework The main aim of the paper is to expose the mean value parametrization for possible statistical applications The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation Some classes of polynomial variance functions are revisited for illustration The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution
15 citations
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TL;DR: In this paper, the q-exponential families of infinitely divisible laws are defined, which can be viewed as deformations of the normal, gamma, and Poisson exponential families.
Abstract: Using the technique developed in approximation theory, we construct examples of exponential families of infinitely divisible laws which can be viewed as "-deformations of the normal, gamma, and Poisson exponential families Replacing the differential equation of approximation theory by a q-differential equation, we define the q-exponential families, and we identify all q-exponential families with quadratic variance functions when |q| 1, and other related generalizations
15 citations
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15 Jan 2014
TL;DR: In this paper, the authors describe the fit of a BHS distribution to a NEF-GHS or Meixner distribution family and the BHS Distribution Family and the SHS and SASHS distribution families.
Abstract: Preface.- Hyperbolic Secant Distributions.- The GSH Distribution Family and Skew Versions.- The NEF-GHS or Meixner Distribution Family.- The BHS Distribution Family.- The SHS and SASHS Distribution Family.- Application to Finance.- R-Code: Fitting a BHS Distribution.
15 citations
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TL;DR: In this article, a generalized exponential power distribution (GEPD) is proposed to model the tail behavior of the distribution, which makes it more flexible and suitable for modeling than the usual normal distribution, while retaining sym- metry.
Abstract: In this paper, we propose to study a generalized form of the exponential power distribution which contains others in the literature as special cases. This unifying exponential power distribution is charac- terized by a parameter ω and a function h(ω) which regulates the tail behavior of the distribution, thus making it more flexible and suitable for modeling than the usual normal distribution, while retaining sym- metry. We derive several mathematical and statistical properties of this distribution and estimate the parameters using both the moments and maximum likelihood approach, obtaining the information matrix in the process. The multivariate extension of the distribution is also examined. Finally we fit the univariate generalized exponential power distribution as well as the normal distribution to data on eggs produced by chicken on each of two different poultry feeds (inorganic and organic copper-salt compositions) and show that the generalized exponential power distri- bution fit is considerably better. We then use the Kolmogorov-Smirnov two samples one-tailed test to show that there is an increase in egg weights and decrease in cholesterol level when the feed is organic.
15 citations
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TL;DR: In this paper, a generalized EME (GEME) distribution was proposed and various properties of the distribution were developed. But, the distribution is not suitable for the analysis of the hazard function of the EME random variable.
Abstract: Moment distributions have a vital role in mathematics and statistics, in particular in probability theory, in the perspective research related to ecology, reliability, biomedical field, econometrics, survey sampling and in life-testing. Hasnain (2013) developed an exponentiated moment exponential (EME) distribution and discussed some of its important properties. In the present work, we propose a generalization of EME distribution which we call it generalized EME (GEME) distribution and develop various properties of the distribution. We also present characterizations of the distribution in terms of conditional expectation as well as based on hazard function of the GEME random variable.
15 citations