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.

Posterior computations based on sample quantiles: one- and two-parameter exponential cases

Abstract: Statistical analyses based on incomplete ordered samples are common in many practical fields. Some data are unknown or disregarded due to restrictions on data collection, experimental difficulties or negligence. In the interest of compressing the data, only several informative order statistics are considered. Subsets of the available data, i.e. training samples, are widely used in a variety of statistical methodologies. In this paper, on the basis of some sample quantiles, a Bayesian analysis of the one- and two-parameter exponential models is developed. Using natural non-informative and conjugate priors, the posterior density and distribution functions for the location and scale parameters are obtained in closed-forms. Explicit expressions for the posterior moments of the exponential parameters and distribution function at any fixed point are derived. The existence and uniqueness of the posterior modes are established; simple and precise lower and upper bounds are also provided. Finally, an illustrative ...

Odd Exponential-Logarithmic Family of Distributions: Features and Modeling

Abstract: This paper introduces a general family of continuous distributions, based on the exponential-logarithmic distribution and the odd transformation. It is called the “odd exponential logarithmic family”. We intend to create novel distributions with desired qualities for practical applications, using the unique properties of the exponential-logarithmic distribution as an initial inspiration. Thus, we present some special members of this family that stand out for the versatile shape properties of their corresponding functions. Then, a comprehensive mathematical treatment of the family is provided, including some asymptotic properties, the determination of the quantile function, a useful sum expression of the probability density function, tractable series expressions for the moments, moment generating function, Rényi entropy and Shannon entropy, as well as results on order statistics and stochastic ordering. We estimate the model parameters quite efficiently by the method of maximum likelihood, with discussions on the observed information matrix and a complete simulation study. As a major interest, the odd exponential logarithmic models reveal how to successfully accommodate various kinds of data. This aspect is demonstrated by using three practical data sets, showing that an odd exponential logarithmic model outperforms two strong competitors in terms of data fitting.

On the Representation of Correlated Exponential Distributions by Phase Type Distributions.

Abstract: In this paper we present results for bivariate exponential distributions which are represented by phase type distributions. The paper extends results from previous publications [5, 14] on this topic by introducing new representations that require a smaller number of phases to reach some correlation coefficient and introduces different ways to describe correlation between exponentially distributed random variables. Furthermore, it is shown how Markovian Arrival Processes (MAPs) with exponential marginal distribution can be generated from the phase type representations of exponential distributions and how the results for exponential distributions can be applied to define correlated hyperexponential or Erlang distributions. As application examples we analyze two queueing models with correlated inter-arrival and service times.

A martingale characterization of Pólya-Lundberg processes

Abstract: We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

Multivariate exponential power distributionsas mixtures of normal distributions withBayesian applications

Abstract: It is shown that a multivariate exponential power distribution is a scale mixture of normal distributions, with respect to a probability distribution function, when its kurtosis parameter belongs to the interval (0,1]. The corresponding mixing probability distribution function is presented. This result is used to design a Bayesian hierarchical model and an algorithm to generate samples of the posterior distribution; these are applied to a problem of Quantitative Genetics.