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.

Order Statistics and Benford's Law

Abstract: Fix a base and let have the standard exponential distribution; the distribution of digits of base is known to be very close to Benford's law. If there exists a such that the distribution of digits of times the elements of some set is the same as that of , we say that set exhibits shifted exponential behavior base Let be i.i.d.r.v. If the 's are Unif, then as the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If instead 's come from a compactly supported distribution with uniformly bounded first and second derivatives and a second-order Taylor series expansion at each point, then the distribution of digits of any consecutive differences and all normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford's law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.

Estimation of the Parameters of Two Parameter Exponential Distributions from Censored Samples

Abstract: In life testing a two parameter exponential distribution is often required to provide an adequate representation of the observed results. Sufficient statistics and interval estimates for the parameters are given for the case of censored data. Numerical examples are provided.

Exponential polynomial reproducing property of non-stationary symmetric subdivision schemes and normalized exponential B-splines

Abstract: An important capability for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modeling. In this regards, this study first provides necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of exponential polynomials. Then, the approximation order of such non-stationary schemes is discussed to quantify their approximation power. Based on these results, we see that the exponential B-spline generates exponential polynomials in the associated spaces, but it may not reproduce any exponential polynomials. Thus, we present normalized exponential B-splines that reproduce certain sets of exponential polynomials. One interesting feature is that the set of exponential polynomials to be reproduced is varied depending on the normalization factor. This provides us with the necessary accuracy and flexibility in designing target curves and surfaces. Some numerical results are presented to support the advantages of the normalized scheme by comparing them to the results without normalization.