Showing papers on "Natural exponential family published in 2003"
TL;DR: Asymptotic distributions of the logarithm of the RML under null hypotheses are obtained and they are used to determine the minimum sample size required in discriminating between two overlapping families of distributions for a user specified probability of correct selection and tolerance limit.
173 citations
TL;DR: In this article, it is observed that for a given gamma distribution there exists a generalized exponential distribution so that the two distribution functions are almost identical, and for all practical purposes it is possible to generate approximate gamma random numbers using generalized exponential distributions.
Abstract: Recently a new distribution, named as generalized exponential distribution or exponentiated exponential distribution was introduced and studied quite extensively by the authors. It is observed that the generalized exponential distribution can be used as an alternative to the gamma distribution in many situations. Different properties like monotonicity of the hazard functions and tail behaviors of the gamma distribution and the generalized exponential distribution are quite similar in nature, but the later one has a nice compact distribution function. It is observed that for a given gamma distribution there exists a generalized exponential distribution so that the two distribution functions are almost identical. Since the gamma distribution function does not have a compact form, efficiently generating gamma random numbers is known to be problematic. We observe that for all practical purposes it is possible to generate approximate gamma random numbers using generalized exponential distribution and ...
85 citations
01 Jan 2003
TL;DR: In this paper, interval estimation of the mean in the natural exponential family with a quadratic variance function is considered. But the results and addi- tional computation suggest that the equal tailed Jereys interval and the likelihood ratio interval are the best overall alternatives to the Wald interval.
Abstract: In this paper we consider interval estimation of the mean in the natural Exponential family with a quadratic variance function; the family comprises the binomial, Poisson, negative binomial, normal, gamma, and a sixth distribution. For the three discrete cases, the Wald condence interval and three alternative intervals are examined by means of two term Edgeworth expansions of the coverage probability and a two term expansion of the expected length. The results and addi- tional computation suggest that the equal tailed Jereys interval and the likelihood ratio interval are the best overall alternatives to the Wald interval. We also show that the poor performance of the Wald interval is not limited to the discrete cases, and a serious negative bias occurs in the nonnormal continuous cases as well. The results are complemented by various illustrative examples.
65 citations
TL;DR: It is shown that the animal group-size distribution follows a power-law decay with exponent 1, and is truncated at a cut-off size which is the expected size of the groups an arbitrary individual engages in.
59 citations
TL;DR: In this paper, an entropic measure, which is optimized by a given arbitrary distribution with the finite linear expectation value of a physical random quantity of interest, is constructed and is concave irrespective of the properties of the distribution and satisfies the H-theorem for the master equation combined with the principle of microscopic reversibility.
Abstract: An ultimate generalization of the maximum entropy principle is presented. An entropic measure, which is optimized by a given arbitrary distribution with the finite linear expectation value of a physical random quantity of interest, is constructed. It is concave irrespective of the properties of the distribution and satisfies the H-theorem for the master equation combined with the principle of microscopic reversibility. This offers a unified basis for a great variety of distributions observed in nature. As examples, the entropies associated with the stretched exponential distribution postulated by Anteneodo and Plastino (1999 J. Phys. A: Math. Gen. 32 1089) and the κ-deformed exponential distribution by Kaniadaki (2002 Phys. Rev. E 66 056125) and Naudts (2002 Physica A 316 323) are derived. To include distributions with divergent moments (e.g., the Levy stable distributions), it is necessary to modify the definition of the expectation value.
48 citations
TL;DR: In this paper, an extension to the exponential power family allows for robustness characteristics for the normal location parameter problem, previously thought to be restricted to the Student-t and a subclass of the positive stable families.
33 citations
TL;DR: In this paper, the authors consider the class of lift-time distributions and derive the corresponding forms for the reliability R = Pr (X"2 < X"1) and identify at least some 30 distributions for which the algebraic form of R is not known.
25 citations
TL;DR: In this article, a new class of bivariate distributions is constructed from a given family of distributions based on the corresponding exponential representation, which generalizes the well-known exponential representation for the univariate survival function.
22 citations
TL;DR: In this article, the structure of conditional reducibility is shown to hold for a general Wishart family on a symmetric cone, and the enriched standard conjugate family is defined and discussed.
Abstract: A general Wishart family on a symmetric cone is a natural exponential family (NEF) having a homogeneous quadratic variance function. Using results in the abstract theory of Euclidean Jordan algebras, the structure of conditional reducibility is shown to hold for such a family, and we identify the associated parameterization $\phi$ and analyze its properties. The enriched standard conjugate family for $\phi$ and the mean parameter $\mu$ are defined and discussed. This family is considerably more flexible than the standard conjugate one. The reference priors for $\phi$ and $\mu$ are obtained and shown to belong to the enriched standard conjugate family; in particular, this allows us to verify that reference posteriors are always proper. The above results extend those available for NEFs having a simple quadratic variance function. Specifications of the theory to the cone of real symmetric and positive-definite matrices are discussed in detail and allow us to perform Bayesian inference on the covariance matrix $\Sigma$ of a multivariate normal model under the enriched standard conjugate family. In particular, commonly employed Bayes estimates, such as the posterior expectation of $\Sigma$ and $\Sigma^{-1}$, are provided in closed form.
20 citations
TL;DR: Reference analysis has proved to be one of the most successful general methods to derive non-informative prior distributions as mentioned in this paper, however, reference priors are typically difficult to obtain.
14 citations
TL;DR: In this paper, it was shown that if |α− s/r| ≤ 1/r2 and 3 ≤ r ≤ (x/log x)1/2, for some coprime integers s and r, then, uniformly for all f ∈ F, we have
Abstract: where e(t) stands for e2πit. New bounds for F (x, α) have been announced by the author in [Ba1] and the purpose of this paper is to supply proofs for these estimates. The problem of obtaining bounds for F (x, α) uniform in f ∈ F has been first considered by H. Daboussi. He showed [Da1] (see also [DD1] and [DD2]) that if |α− s/r| ≤ 1/r2 and 3 ≤ r ≤ (x/log x)1/2, for some coprime integers s and r, then, uniformly for all f ∈ F , we have
01 Sep 2003
TL;DR: This work considers both tree-like networks, appropriate in biological applications, and networks in which closed loops can appear, which model communication networks and networks of human sexual interactions, and a randomly growing network, with the state of a random node observed.
Abstract: A possible explanation for the frequent occurrence of power-law distributions in biology and elsewhere comes from an analysis of the interplay between random time evolution and random observation or killing time. If the system population or its topological parameters grow exponentially with time, and observations on the system correspond to stopping the evolution at an exponentially distributed random time, power-law behaviour in one or both tails of the distribution of observed quantities may result. We pursue this theme for two speci c models. The rst model is a randomly killed birth-and-death process, with applications to the numbers of genes per gene family and proteins per protein family, the distribution of taxonomic elements in live taxa, and other areas. The second model is a randomly growing network, with the state of a random node (which thus has a random age) observed. For the growing network, we consider both tree-like networks, appropriate in biological applications, and networks in which closed loops can appear, which model communication networks and networks of human sexual interactions.
TL;DR: In this article, the exponential stability of singularly perturbed time-varying systems is investigated, and it is shown that the stability of an averaged system is the same as that of a perturbed system for small perturbation parameters.
TL;DR: In this paper, the authors associate the natural exponential family of df's Fλ where d F λ (x) = e λx d F(x)/E e ǫ λX for λ ∞ = sup Λ⩽∞ does not lie in Λ.
TL;DR: In this article, the content of f, (a1, a2,..., ar), is assumed to be relatively prime to the modulus q. The problem is to determine whether there exists an absolute constant C such that for an arbitrary positive integer q,
Abstract: (1.2) f(x) = a1x k1 + · · ·+ arx with 0 < k1 < k2 < · · · < kr. We assume always that the content of f , (a1, a2, . . . , ar), is relatively prime to the modulus q. Let d = d(f) = kr denote the degree of f and for any prime p let dp(f) denote the degree of f read modulo p. A fundamental problem is to determine whether there exists an absolute constant C such that for an arbitrary positive integer q,
01 Jan 2003
TL;DR: In this article, statistical analysis of order statistics for exponential distribution is carried out, the probability density function and moment generating function (mdf) of order statistic of exponential distribution are derived, the mean, estimation variance and ADT are also given.
Abstract: The radar background noise is usually assumed as Gaussian distribution, and square law detection is taken, thus the reference sample falling in a sliding window used to estimate the noise power level follows an exponential distribution. In this paper, statistical analysis of order statistics for exponential distribution is carried out, the probability density function and moment generating function (mdf) of order statistics for exponential distribution are derived, the mean, estimation variance and ADT of order statistics are also given. Finally, its application for constant false alarm rate (CFAR) is discussed.
TL;DR: In this paper, a relation between a family of distributions for which an unbiased estimator of a function g(θ) attains the general order Bhattacharyya lower bound and that of linear combinations of the distributions from an exponential family is discussed.
Abstract: An unbiased estimation problem of a function g(θ) of a real parameter is considered. A relation between a family of distributions for which an unbiased estimator of a function g(θ) attains the general order Bhattacharyya lower bound and that of linear combinations of the distributions from an exponential family is discussed. An example on a family of distributions involving an exponential and a double exponential distributions with a scale parameter is given. An example on a normal distribution with a location parameter is also given.
TL;DR: In this article, the problem of approximating a prior by a suitable finite mixture of distributions, with respect to Prokhorov's metric, is considered and the error of approximation when the statistical model is a suitable exponential family is given.
Journal Article•
TL;DR: Two characterizations of the exponential distribution among distributions with support the nonnegative real axis are presented, based on certain prop erties of the characteristic function of theonential random variable.
Abstract: Two characterizations of the exponential distribution among distributions with support the nonnegative real axis are presented. The characterizations are based on certain prop erties of the characteristic function of the exponential random variable. Counterexamples concerning more general possible versions of the characterizations are given.
TL;DR: In this article, the authors prove the asymptotic normality of generalized L-statistics based on a sample from the uniform distribution on [0, 1] and of L-Statistics with decomposed kernels (without any restrictions on the sample distribution type).
Abstract: We obtain exponential upper bounds for tails of distributions of generalized L-statistics based on a sample from an exponential distribution. We prove the asymptotic normality of generalized L-statistics based on a sample from the uniform distribution on [0,1] and of L-statistics with decomposed kernels (without any restrictions on the sample distribution type).
30 Jun 2003
TL;DR: In this article, the authors define three variables, Time, Status and Treat, for each case in the sample, and assume that the data have been saved in C:\Example.dat as a text file.
Abstract: For each case in the sample, we define three variables, Time, Status and Treat. Let Time denote the survival time (exact or censored), Status be a dummy variable with Status=0 if Time is censored and 1 otherwise and Treat be a variable with Treat = MP if the patient received 6-MP and P if the patient receive Placebo. Assume that the data have been saved in “C:\Example.dat” as a text file, which contains three columns (Time, in the first column, Status in the second column, and Treat in the third column), separated by space(s).
15 Sep 2003
TL;DR: In this article, the variation distance closures of exponential families and their log-convex subfamilies are characterized, and a problem left open in (I. Csiszar, et al., 2002) is settled: an exponential family is constructed whose closure in reversed information divergence, rI-closure, is neither rI closed nor log convex.
Abstract: The variation distance closures of exponential families and their log-convex subfamilies are characterized. A problem left open in (I. Csiszar, et al., 2002) is settled: an exponential family is constructed whose closure in reversed information divergence, rI-closure, is neither rI-closed nor log-convex.
Posted Content•
TL;DR: In this article, a probability plot for varname compared with a one-parameter exponential distribution, with distribution function 1 - exp(-varname / mean), is presented. The values of varname should be zero or positive.
Abstract: pexp produces a probability plot for varname compared with a one-parameter exponential distribution, with distribution function 1 - exp(-varname / mean). The values of varname should be zero or positive.
TL;DR: In this article, an empirical lack-of-memory process defined on the basis of the Lau-Rao characterization of the exponential and geometric distributions is studied and convergence to a Gaussian process is shown.
Abstract: We study an empirical lack-of-memory process defined on the basis of the Lau-Rao characterization of the exponential and geometric distributions. Using the Komlos-Major-Tusnady approximation theorem for the ‘classical’ empirical process, we prove some convergence results for the lack-of-memory process, including its convergence to a Gaussian process. We also study certain integral statistics, defined as integrals of the lack-of-memory process with respect to the empirical distribution function, and apply them to the problem of testing goodness-of-fit of the exponential and geometric models. Besides giving results on their performance, we point out the connections between these and other well-known statistics.
01 Jan 2003
Journal Article•
TL;DR: In this paper, the authors studied the ruin probability in a two-type-risk Poisson model for insurance and provided explicit expressions for Θ(0) and ǫ(u) under the condition that claims obey an exponential distribution or a combination of several exponential distributions.
Abstract: Classical risk models for insurance are single-type-risk based. However, insurance companies generally use multiple-type-risk in practice. This paper studies the ruin probabilities in a two-type-risk Poisson model. Explicit expressions for Ψ(0) and Ψ(u) under the condition that claims obey an exponential distribution or a combination of several exponential distributions.
01 Jan 2003
TL;DR: In this paper, the authors established recurrence relations of the moments, product moments, percentage points, and modes of order statistics from the transformed exponential distribution, and established some recurrence relation between moments and product moments.
Abstract: In this paper, we establish some recurrence relations of the moments, product moments, percentage points, and modes of order statistics from the transformed exponential distribution.
Posted Content•
TL;DR: In this paper, the quantiles of a given name against the quantities of a one-parameter exponential distribution were compared. But the distribution function was not defined. The values of varname should be zero or positive.
Abstract: qexp plots the quantiles of varname against the quantiles of a one-parameter exponential distribution, with distribution function 1 - exp(-varname / mean). The values of varname should be zero or positive.