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.

Conjugate Priors for Exponential Families Having Quadratic Variance Functions

Abstract: Consider a natural exponential family parameterized by θ. It is well known that the standard conjugate prior on θ is characterized by a condition of posterior linearity for the expectation of the model mean parameter μ. Often, however, this family is not parameterized in terms of θ but rather in terms of a more usual parameter, such at the mean μ. The main question we address is: Under what conditions does a standard conjugate prior on μ induce a linear posterior expectation on μ itself? We prove that essentially this happens iff the exponential family has quadratic variance function. A consequence of this result is that the standard conjugate on μ coincides with the prior on μ induced by the standard conjugate on θ iff the variance function is quadratic. The rest of the article covers more specific issues related to conjugate priors for exponential families. In particular, we analyze the monotonicity of the expected posterior variance for μ with respect to the sample size and the hyperparameter ...

The beta generalized Rayleigh distribution with applications to lifetime data

Abstract: For the first time, we propose a new distribution so-called the beta generalized Rayleigh distribution that contains as special sub-models some well-known distributions. Expansions for the cumulative distribution and density functions are derived. We obtain explicit expressions for the moments, moment generating function, mean deviations, Bonferroni and Lorenz curves and densities of the order statistics and their moments. We estimate the parameters by maximum likelihood and provide the observed information matrix. The usefulness of the new distribution is illustrated through two real data sets that show that it is quite flexible in analyzing positive data instead of the generalized Rayleigh and Rayleigh distributions.

On the hyper-Dirichlet type 1 and hyper-Liouville distributions

Abstract: This paper concerns the characterization of a new family of multivariate beta distribution functions - the hyper-Dirichlet type 1 distribution. This family describes the joint density function of the terminal variates of an arbitrary tree constructed from finite sequences of probability vectors having independent Dirichlet type 1 distributions. Expressions for the general properties of the hyper-Dirichlet type 1 distribution are presented. In addition, the hyper-Liouville distribution is described and its properties are discussed as well as a generalization of the Liouville integral identity.

Construction of exponential tension B-splines of arbitrary order

Abstract: Some results for exponential B-splines in tension are extended to higher order exponential B-splines Based on a differentiation formula we derive a Bernstein/Bezier like representation on each interval

Exponential rings, exponential polynomials and exponential functions

Abstract: Introduction. An exponential ring, or E-ring for short, is a pair (i?, E) with R a ring—in this paper always commutative with 1—and E a morphism of the additive group of R into the multiplicative group of units of R, that is, E(x + y) = E(x)E(y) for all x9 y in i?, and E(0) = 1. Examples are (R, a), a any positive real, and (C, e). Of course, any ring R can be expanded to an £-ring (R, E) by putting E(x) — 1 for all x\\ such brings will be called trivial. Ken Manders observed that an Zί-ring whose underlying ring has no nilpotents φ 0 and has characteristic a prime/? > 0 is trivial: in such a ring each x satisfies 1 = E(0) — E(px) = E{x), so (E(x) \\y = 0, which implies E(x) = 1. Related notions of exponential ring have been considered by M. Beeson, by B. Dahn and Wolter, and by A. Wilkie, all in connection with the longstanding open problem of A. Tarski on the decidability of the field of reals with exponentiation. An effective positive solution to this problem seems unlikely without major advances in transcendental number theory: such a solution would give us a decision method to answer any question: is e = p/q>> where/?, q are positive integers. Of course there is such a decision method, but, as we don't know yet whether e is rational, we don't know how it works. Now in mathematical practice it is less the effectiveness of Tarski's decision method for the real field which matters—though this aspect is interesting—but rather the information the method provides on the algebraic-topological nature of the definable sets in R, and on the asymptotic behavior of definable functions. For example in semi-algebraic and real algebraic geometry this use is formalized in the Tarski-Seidenberg theorem (in an inconstructive version) and in a result like the finiteness of the number of connected components of a semi-algebraic set. Parts of this use of Tarski's work on the elementary theory of the reals offer more hope of being generalized to the E-ήng (R, e). The following