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.

The complementary exponential power series distribution

Abstract: In this paper, we introduce the complementary exponential power series distributions, with failure rate either increasing, which is complementary to the exponential power series model proposed by Chahkandi & Ganjali (2009). The new class of distribution arises on a latent complementary risks scenarios, where the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. This new class contains several distributions as particular case. The properties of the proposed distribution class are discussed such as quantiles, moments and order statistics. Estimation is carried out via maximum likelihood. Simulation results on maximum likelihood estimation are presented. An real application illustrate the usefulness of the new distribution class.

Nonparametric regression in exponential families

Abstract: Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function, which include, for example, Poisson regression, binomial regression and gamma regression. We propose a unified approach of using a mean-matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic Gaussian regression problem. Then in principle any good nonparametric Gaussian regression procedure can be applied to the transformed data. To illustrate our general methodology, in this paper we use wavelet block thresholding to construct the final estimators of the regression function. The procedures are easily implementable. Both theoretical and numerical properties of the estimators are investigated. The estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a wide range of Besov spaces. The estimators also perform well numerically.

Prediction Intervals for the Two Parameter Exponential Distribution

Abstract: We discuss the problem of predicting, on the basis of a sample from a two parameter exponential distribution, the s'th smallest observation Ys in a future sample of n observations for the same distribution, and the mean Y of the future sample. It is shown how to obtain prediction intervals for Ys and Y, based on a Type II censored sample from the distribution.

Asymptotic Behaviour of the Variance Function

Abstract: We investigate the asymptotic behaviour of the variance function V of a natural exponential family with support S c R. If inf S = 0, we show that V(O) = 0 and that the right derivative at zero is V'(O+) = inf {S\{0}}. Using a theorem by Mora (1990) we show that if lim c -P V(cp) = uP uniformly on compact subsets in p for either c -+ oo or c -+0, then p 0 (0, 1), and the corresponding exponential dispersion model, suitably scaled, converges to a member of the Tweedie family of exponential dispersion models, corresponding to the variance function V(p) = pP. This gives a kind of central limit theory for exponential dispersion models. In the case p = 2, the limiting family is gamma, and the result essentially follows from Tauber theory. For p = 1, we obtain a version of the Poisson law of small numbers, generalizing a result for discrete models due to Jorgensen (1986). For 1 2 or p < 0 the limiting families are generated by respectively positive stable distributions or extreme stable distributions, in the latter case inf S =-oo. A number of illustrative examples are considered.