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.

Random Finite Subsets With Exponential Distributions

Abstract: Let S denote the collection of all finite subsets of **. We define an operation on S that makes S into a positive semigroup with set inclusion as the associated partial order. Positive semigroups are the natural home for probability distributions with exponential properties, such as the memoryless and constant rate properties. We show that there are no exponential distributions on S, but that S can be partitioned into subsemigroups, each of which supports a one-parameter family of exponential distributions. We then find the distribution on S that is closest to exponential, in a certain sense. This work might have applications to the problem of selecting a finite sample from a countably infinite population in the most random way.

An Extension of Banerjee and Rahim’s Model for Economic Design of $$\bar{X}$$ X ¯ Control Charts Under Generalized Exponential Shock Model

Abstract: Control chart in statistical process control is an effective way to monitor and improve quality and production cost savings in on-line activities of organizations. Economic design of these charts needs to determine sample size, sampling interval, control limit coefficient and process failure mechanism. Exponential, Gamma, and Weibull distributions are the most famous distributions used in the analysis of lifetime and failure mechanism. However, most of these distributions suffer from some drawbacks that finally affect optimal values of design parameters. Therefore, we here proposed a cost model based on generalized exponential distribution as shock model that seems to be a better alternative in many cases. Based on this new shock model, we extended Banerjee and Rahim’s model in non-uniform sampling scheme and compared that with uniform design. We also used sensitivity analysis to investigate the effect of fitting wrong Weibull, gamma, and exponential distributions instead of generalized exponential. The results showed that inappropriate fitting the wrong distribution leads to misleading values for the average cost and the design parameters.

Beta Weighted Exponential Distribution: Theory and Application

Abstract: [10] modified the idea of [2] in which they introduced a shape parameter to an exponential model to obtain the weighted exponential distribution. In this article, we introduced two shape parameters to the existing weighted exponential distribution to develop the beta weighted exponential distribution using the logit of beta function by [12]. We studied the statistical properties of the new distribution. Parameter estimation was done by the method of maximum likelihood estimation with R software code. We then used a data set on survival times of guinea pigs injected with different amount of tubercle bacilli to compare properties of well-known distributions with those of the new distribution. Our comparison showed the new distribution as the much more flexible and versatile.

Bayesian estimation of mixed weibull exponential in life testing

Abstract: This paper deals with estimating the parameters of a mixed Weibull exponential model using a Bayesian method for type I censored samples. A numerical comparison between maximum likelihood and Bayes results has been carried out, using a numerical example and computer facilities for different prior information. Bayesian results in the cases of mixed exponential (complete and censored), single exponential and single Weibull may be consider as special cases of the results of this paper. The problem can be extended to the case of more than two causes of failure.