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.

On the normality a posteriori for exponential distributions,using the Bayesian estimation

Abstract: The Bayesian estimation problem for the parameter θ of an exponential probability distribution is considered, when it is assumed that θ has a natural conjugate prior density and a loss-function depending on the squared error is used. It is shown that, with probability one, the posterior density of the Bayesian—centered and scaled parameter converges pointwise to the normal probability density. The weak convergence of the posterior distributions to the normal distribution follows directly. Both correct and incorrect models are studied and the asymptotic normality is stated respectively.

Some properties and applications of half Cauchy extended exponential distribution

Abstract: In this article, we have introduced a new probability distribution having three parameters using half Cauchy family of distribution named half Cauchy extended exponential distribution. The statistical properties and characteristics of the proposed distribution like the hazard rate function (HRF), cumulative hazard function, and the probability density function (PDF), and the cumulative distribution function (CDF), quantile function and the skewness, kurtosis are provided. The parameters of the proposed distribution are estimated using the Cramer-Von-Mises (CVM) least-square estimation (LSE), and maximum likelihood estimators (MLE) methods. A real data set is analyzed to test the goodness-of-fit of the proposed distribution. It is found that the half Cauchy extended exponential distribution performed well as compared to some competing distributions. estimation (CVME), and least-square estimation (LSE) methods are used to estimate the parameter and we found that the MLEs are relatively better than LSE and CVM methods. The curves of the PDF of the proposed distribution have shown that it can have various shapes like increasing-decreasing and right skewed and flexible for modeling real-life data. Also, the graph of the hazard function is monotonically increasing or constant or reverse j-shaped according to the value of the model parameters. The applicability and suitability of the half Cauchy extended exponential distribution has been evaluated by considering a real-life dataset and the results exposed that the proposed distribution is much flexible as compared to some other fitted distributions.

PEXP: Stata module to produce probability plot for data compared with fitted exponential distribution

Abstract: pexp produces a probability plot for varname compared with a one-parameter exponential distribution, with distribution function 1 - exp(-varname / mean). The values of varname should be zero or positive.

Graphs and the Exponential Function

Abstract: In this chapter we shall review the basic theory of graph plotting and the interpretation of certain equations from a graphical point of view. In the process we shall look again at graphs of trigonometric functions, study a new topic — the exponential function — and examine the graphs of this function and combination functions containing trigonometric and exponential components.

New Dispersion Function in the Rank Regression

Abstract: In this paper we introduce a new score generating (unction for the rank regression in the linear regression model. The score function compares the 'th and s\`th power of the tail probabilities of the underlying probability distribution. We show that the rank estimate asymptotically converges to a multivariate normal. further we derive the asymptotic Pitman relative efficiencies and the most efficient values of and s under the symmetric distribution such as uniform, normal, cauchy and double exponential distributions and the asymmetric distribution such as exponential and lognormal distributions respectively