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.

Statistics of the Chi-Square Type, with Application to the Analysis of Multiple Time-Series Power Spectra

Abstract: It is often necessary to compare the power spectra of two or more time series: one may, for instance, wish to estimate what the power spectrum of the combined data sets might have been, or one may wish to estimate the significance of a particular peak that shows up in two or more power spectra. Also, one may occasionally need to search for a complex of peaks in a single power spectrum, such as a fundamental and one or more harmonics, or a fundamental plus sidebands, etc. Visual inspection can be revealing, but it can also be misleading. This leads one to look for one or more ways of forming statistics, which readily lend themselves to significance estimation, from two or more power spectra. The familiar chi-square statistic provides a convenient mechanism for combining variables drawn from normal distributions, and one may generalize the chi-square statistic to be any function of any number of variables with arbitrary distributions. In dealing with power spectra, we are interested mainly in exponential distributions. One well-known statistic, formed from the sum of two or more variables with exponential distributions, satisfies the gamma distribution. We show that a transformation of this statistic has the convenient property that it has an exponential distribution. We introduce two additional statistics formed from two or more variables with exponential distributions. For certain investigations, we may wish to study the minimum power (as a function of frequency) drawn from two or more power spectra. In other investigations, it may be helpful to study the product of the powers. We give numerical examples and an example drawn from our solar-neutrino research.

Recurrence and Ergodicity for Exponential Family State-Space Models

Abstract: : We give two results concerning the properties of state-space models with exponential family observation distribution and conjugate state distribution. The first result gives a simple and general interpretation of the parameters of the predictive state distribution in terms of the observation forecast distribution. The second result shows how the first result can be used to check the long-term model properties of recurrence and ergodicity for a class of non-Gaussian observation distributions. In particular, these results apply to models with Poisson, binomial and multinomial observation distributions. Keywords: Bayesian forecasting; Binomial time series; Multinomial time series; Poisson time series; Recursive updating; Time series. (KR)

Some properties of 2-orthogonal polynomials associated with inverse Gaussian distribution

Abstract: In this work, we discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials. In particular, we discuss the cumulant-generating function and associated...