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: The purpose of this correction is to state precisely the class of distributions in which the exponential distribution is characterized and to prove certain amplifications as discussed by the authors, which is the same as the one in this paper.
Abstract: The purpose of this correction is to state precisely the class of distributions in which the exponential distribution is characterized and to prove certain amplifications.
2 citations
TL;DR: In this paper, the joint probability distributions for three structure factors whose subscripts add to zero, in the general case of unequal atoms, are expressed in an exponential form for space groups P1 and P{\bar 1}.
Abstract: The joint probability distributions for three structure factors whose subscripts add to zero, in the general case of unequal atoms, are expressed in an exponential form for space groups P1 and P{\bar 1}. The latter are representative of noncentrosymmetric and centrosymmetric space groups respectively. The exponential form possesses considerably improved convergence properties over those of the standard asymptotic series, although it too remains asymptotic. With the range of values ordinarily obtained for the normalized structure-factor magnitudes, the exponential forms are quite accurate. However, the accuracy deteriorates somewhat as these magnitudes approach their largest possible values. By altering the exponential form with the use of a result from the inequality theory, joint probability distributions are obtained which are accurate over the entire range of values for the structure-factor magnitudes and are most accurate at the largest values. Several probability measures of interest are derived from the joint distribution functions such as expected values, variances and the probability that a structure factor has a positive sign. Numerical tests indicate that the derived probability measures are very reliable and that their validity extends to higher space groups than P1 and P{\bar 1}.
2 citations
21 Mar 2011
TL;DR: In this article, a new class of heavy-tailed distribution is introduced, called skewed stable, which has many inferential advantages, since it is a member of exponential family, so the Bayesian inference can be drawn similar to the exponential family of distributions.
Abstract: Stable distributions are a class of distributions which allow skewness and heavy tail. Non‐Gaussian stable random variables play the role of normal distribution in the central limit theorem, for normalized sums of random variables with infinite variance. The lack of analytic formula for density and distribution functions of stable random variables has been a major drawback to the use of stable distributions, also in the case of inference in Bayesian framework. Buckle introduced priors for the parameters of stable random variables to obtain an analytic form of posterior distribution. However, many researchers tried to solve the problem, through the Markov chain Monte Carlo methods, e.g. [8] and their references. In this paper a new class of heavy‐tailed distribution is introduced, called skewed stable. This class has two main advantages: It has many inferential advantages, since it is a member of exponential family, so the Bayesian inference can be drawn similar to the exponential family of distributions a...
2 citations
TL;DR: For the case of lattice distributions, the power of these tests is approximated by asymptotic expansions up to errors of order o(n−1) as mentioned in this paper.
2 citations
TL;DR: In this paper, a bivariate weighted exponential distribution based on the generalized exponential distribution was proposed, which generalizes the bivariate distribution with weighted exponential marginals (BWE) and provides better fit than the BWE distribution.
Abstract: In this paper, we introduce a bivariate weighted exponential distribution based on the generalized exponential distribution. This class of distributions generalizes the bivariate distribution with weighted exponential marginals (BWE). We derive different properties of this new distribution. It is a four-parameter distribution, and the maximum-likelihood estimator of unknown parameters cannot be obtained in explicit forms. One data set has been re-analyzed and it is observed that the proposed distribution provides better fit than the BWE distribution.
2 citations