Showing papers on "Natural exponential family published in 2006"
TL;DR: A comprehensive treatment of the mathematical properties of the beta exponential distribution generated from the logit of a beta random variable is provided and an expression for the Fisher information matrix is provided.
414 citations
TL;DR: In this paper, the authors introduced four more exponentiated type distributions that generalize the standard gamma, standard Weibull, standard Gumbel and the standard Frechet distributions in the same way the exponentiated exponential distribution generalizes the standard exponential distribution.
Abstract: Gupta et al. [Commun. Stat., Theory Methods 27, 887–904, 1998] introduced the exponentiated exponential distribution as a generalization of the standard exponential distribution. In this paper, we introduce four more exponentiated type distributions that generalize the standard gamma, standard Weibull, standard Gumbel and the standard Frechet distributions in the same way the exponentiated exponential distribution generalizes the standard exponential distribution. A treatment of the mathematical properties is provided for each distribution.
296 citations
TL;DR: In this paper, Gupta et al. study the family of distributions termed as exponentiated Weibull distribution and compare the behavior of the failure rate with those of the Weibbull and Gamma distributions.
Abstract: In this paper we study the family of distributions termed as exponentiated Weibull distribution. The distribution has three parameters (one scale and two shape) and the Weibull distribution and the exponentiated exponential distribution, discussed by Gupta, et al. (1998), are particular cases of it. The survival function, failure rate and moments of the distributions have been derived using certain special formulas. The behavior of the failure rate has been studied and compared with those of the Weibull and Gamma distributions. The distribution has been fitted to a real life data set and the fit has been found to be very good.
146 citations
TL;DR: In this paper, the authors bypassed the spacing lemma via a diophantine problem in four variables and obtained the expected bound in Fouvry and Iwaniec's theorem.
Abstract: Abstract Fouvry and Iwaniec's theorem concerning three-dimensional exponential sums with monomials relies on a spacing lemma whose optimal form is yet unproved. We bypass their spacing lemma via a diophantine problem in four variables and we obtain the expected bound in their theorem. In the problem of abelian groups of a given order, this yields the exponent 1/4 + ε, a result close to a conjecture of H. E. Richert (1952).
128 citations
01 Aug 2006
TL;DR: In this article, two new fading distributions, α-η-μ and α-κ-μ distributions, are presented, which include the α-μ, Nakagami-m, Weibull, Rice, Rayleigh, Exponential, and the One-Sided Gaussian distributions as special cases.
Abstract: In this paper two new fading distributions, the α-η-μ Distribution and α-κ-μ Distribution, are presented. The α-η-μ distribution includes the α-μ, Nakagami-m, Nakagami-q, Weibull, Hoyt, Rayleigh, Exponential, and the One-Sided Gaussian distributions as special cases. The α-κ-μdistribution includes the α-μ, Nakagami-m, Weibull, Rice, Rayleigh, Exponential, and the One-Sided Gaussian distributions as special cases. Furthermore, it proposes estimators for the involved parameters and uses field measurements to validate the distributions.
111 citations
TL;DR: In this article, a Bayesian analysis of a generalization of the Poisson distribution is presented, and a necessary and sufficient condition on the hyperparameters of the conjugate family for the prior to be proper is established.
Abstract: This article explores a Bayesian analysis of a generalization of the Poisson distribution. By choice of a second parameter , both under-dispersed and over-dispersed data can be modeled. The Conway-Maxwell-Poisson distribu- tion forms an exponential family of distributions, so it has sucien t statistics of xed dimension as the sample size varies, and a conjugate family of prior distribu- tions. The article displays and proves a necessary and sucien t condition on the hyperparameters of the conjugate family for the prior to be proper, and it discusses methods of sampling from the conjugate distribution. An elicitation program to nd the hyperparameters from the predictive distribution is also discussed.
104 citations
TL;DR: In this article, a new family of distributions for non-negative data, defined by means of a quantile function, is introduced, which is applied to an example from environmental engineering.
Abstract: We introduce a new, flexible family of distributions for non-negative data, defined by means of a quantile function. We describe some properties of this family, and discuss several methods for estimating the parameters. The distribution is applied to an example from environmental engineering.
91 citations
Journal Article•
TL;DR: In this article, the authors investigated low-dimensional exponential families that contain the maximizers of stochastic interdependence in their closure, based on a detailed description of the structure of probability distributions with globally maximal multi-information.
Abstract: Stochastic interdependence of a probability distribution on a product space is measured by its Kullback-Leibler distance from the exponential family of product distributions (called multi-information). Here we investigate low-dimensional exponential families that contain the maximizers of stochastic interdependence in their closure. Based on a detailed description of the structure of probability distributions with globally maximal multi-information we obtain our main result: The exponential family of pure pair-interactions contains all global maximizers of the multi-information in its closure.
51 citations
Journal Article•
TL;DR: In this paper, a multivariate gamma distribution on R n by its Laplace transform (P( -θ ) ) - λ, λ > 0) was defined, and necessary and sufficient conditions on the coefficients of P were established.
Abstract: We define a multivariate gamma distribution on R n by its Laplace transform ( P( -θ ) ) - λ , λ >0, where P ( θ )=∑ T ⊂{ 1,…,n }p T ∏ i ∈Tθ i . Under p { 1,…,n } ≠0 , we establish necessary and sufficient conditions on the coefficients of P , such that the above function is the Laplace transform of some probability distribution, for all λ >0, thus characterizing all infinitely divisible multivariate gamma distributions on R n .
38 citations
TL;DR: In this article, the TP-statistics and TE-statistic are used to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold of a lower threshold.
Abstract: We introduce a new statistical tool (the TP-statistic and TE-statistic) designed specifically to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold u One important property of these statistics is that they converge to zero for power-laws or for exponentials correspondingly, regardless of the value of the exponent or of the form parameter This is particularly useful for testing the structure of a distribution (power-law or not, exponential or not) independently of the possibility of quantifying the values of the parameters We apply these statistics to the distribution of returns of one century of daily data for the Dow Jones Industrial Average and over 1 year of 5-min data of the Nasdaq Composite index Our analysis confirms previous works showing the tendency for the tails to resemble more and more a power-law for the highest quantiles but we can detect clear deviations that suggest that the structure of the tails of the distributions of returns is more complex than usually assumed; it is clearly more complex that just a power-law Our new TP- and TE-statistic should also be useful for other applications in the natural sciences as a powerful non-parametric test for power-laws and exponentials
34 citations
TL;DR: In this article, it was pointed out that the q-exponential distribution introduced in the ground-breaking paper by Tsallis [C.Tsallis, J. Stat. Phys. 52 (1988) 479] is contained by a family of distributions known since the 1940s.
TL;DR: In this paper, the generalized variance of an infinitely divisible natural exponential family F = F(/i) in a 2D linear space is shown to be a product of k univariate Poisson and (d? &)-variate Gaussian families.
Abstract: We show that if the generalized variance of an infinitely divisible natural exponential family F = F(/i) in a ^/-dimensional linear space is of the form det K'^(0) ? exp(0Tb + c), then there exists k in {0, I, ..., d} such that F is a product of k univariate Poisson and (d ? &)-variate Gaussian families. In proving this fact, we use a suitable representation of the generalized variance as a Laplace transform and the result, due to Jorgens, Calabi and Pogorelov, that any strictly convex smooth function / defined on the whole of Ud such that det/"(0) is a positive constant must be a quadratic form.
TL;DR: The goodness-of-fit tests for two and three-parameter gamma distributions are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart as mentioned in this paper.
Abstract: This article presents goodness-of-fit tests for two and three-parameter gamma distributions that are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart. The test statistics can be computed without reliance to special functions and have asymptotic chi-squared distributions. Monte Carlo simulations are used to compare the proposed test for the two-parameter gamma distribution with goodness-of-fit tests employing empirical distribution function or spacing statistics. Two data sets are used to illustrate the various tests.
TL;DR: In this paper, the exact distributions of R = X+Y, P = X + Y and W = X/(X+Y) and corresponding moment properties when X and Y follow Downton's bivariate exponential distribution were derived.
Abstract: Motivated by environmental applications, we derive the exact distributions of R = X+Y, P = X
Y and W = X/(X+Y) and the corresponding moment properties when X and Y follow Downton’s bivariate exponential distribution. The expressions turn out to involve several special functions. For practical purposes, we also provide extensive tabulations of the percentage points associated with the distributions.
01 Jan 2006
TL;DR: In this paper, a composite Exponential-Pareto model is proposed, which combines an exponential density up to a certain threshold value, and a two parameter Pareto density for the rest of the model.
Abstract: In this paper we introduce a composite Exponential-Pareto model, which equals an exponential density up to a certain threshold value, and a two parameter Pareto density for the rest of the model. Compared with the exponential, the resulting density has a similar shape and a larger tail. This is why we expect that such a model will be a better fit than the exponential one for some heavy tailed insurance claims data (e.g. with extreme values).
Dissertation•
04 May 2006
Abstract: In many fields of study scientists are interested in estimating the number of unobserved classes. A biologist may want to find the number of rare species of an animal population in order to conserve, manage, and monitor biodiversity; a library manager may want to know how many non-circulating items are present in a library system; or a clinical investigator may want to determine the number of unseen disease occurrences. A traditional way of estimating an unknown number of classes is by using a negative binomial model (Fisher, Corbet, and Williams 1943). The negative binomial model is based on assuming that the numbers of individuals from each class are independent Poisson samples, and that the means of these Poisson random variables follow a Gamma distribution. This thesis proposes a parametric model where the law of the mean frequency of classes is a finite mixture of exponential distributions. The proposed model has the following advantages: model simplicity, efficient computation using the EM algorithm, and a straightforward interpretation of the fitted model. Also, model assessment by way of a chi-squared goodness of fit procedure can be used, a benefit this parametric model has over other commonly used non-parametric methods. A main accomplishment of this thesis is providing an efficient computation of maximum likelihood (ML) estimates for the proposed model. Without use of the EM algorithm, finding ML estimates for this model can be difficult and time consuming. The likelihood function is complicated due to high dimensionality and non-identifiability of certain parameters. Within the M step of our algorithm we embed another EM, which can effortlessly maximize the parameters in the finite mixture. We refer to the algorithm as a nested EM. Aitken’s acceleration is used to increase speed of the algorithm. Microbial samples from the coast of Massachusetts Bay near Nahant, Massachusetts are used to demonstrate data analysis using three different numbers of components in the finite mixture of the model described. It is shown that the model produces reasonable estimates and fits the data satisfactorily. This model has recently been premiered in species richness estimation (Hong et al. 2006), and its many advantages show promise for continued use in estimating the number of unobserved classes. keywords: EM algorithm, finite mixtures, species richness, Aitken’s acceleration, microorganisms
TL;DR: In this paper, an estimator of the regression parameters for generalized linear models, using the Jacobian technique, was derived for the Poisson model with log link function, and the binomial response model with the logit link function.
Abstract: In this article, we obtain an estimator of the regression parameters for generalized linear models, using the Jacobian technique. We restrict ourselves to the natural exponential family for the response variable and choose the conjugate prior for the natural parameter. Using the Jacobian of transformation, we obtain the posterior distribution for the canonical link function and thereby obtain the posterior mode for the link. Under the full rank assumption for the covariate matrix, we then find an estimator for the regression parameters for the natural exponential family. Then the proposed estimator is specially derived for the Poisson model with log link function, and the binomial response model with the logit link function. We also discuss extensions to the binomial response model when covariates are all positive. Finally, an illustrative real-life example is given for the Poisson model with log link. In order to estimate the standard error of our estimators, we use the Bernstein-von Mises theorem. Final...
TL;DR: A unified Bayesian framework addressing the two main difficulties in this context is presented, i.e., the prior distribution choice and the parameter unidentifiability problem.
Abstract: This paper deals with the Bayesian analysis of finite mixture models with a fixed number of component distributions from natural exponential families with quadratic variance function (NEF-QVF). A unified Bayesian framework addressing the two main difficulties in this context is presented, i.e., the prior distribution choice and the parameter unidentifiability problem. In order to deal with the first issue, conjugate prior distributions are used. An algorithm to calculate the parameters in the prior distribution to obtain the least informative one into the class of conjugate distributions is developed. Regarding the second issue, a general algorithm to solve the label-switching problem is presented. These techniques are easily applied in practice as it is shown with an illustrative example.
Posted Content•
TL;DR: In this article, the authors derived new recurrence relations of the single and product moments of progressively Type-II right censored order statistics from the linear exponential distribution and derived the maximum likelihood estimators (MLEs) of the location and scale parameters.
Abstract: Summary -I nthis paper, we derive new recurrence relations of the single and product moments of progressively Type-II right censored order statistics from the linear exponential distribution. These relations generalize those established by Balakrishnan and Aggarwala (2000) for the standard exponential distribution, those given by Balakrishnan and Sultan (1998) for the Rayleigh distribution and those established by Balakrishnan and Malik (1986) for the moments of order statistics from the linear exponential distribution. Also, we derive the maximum likelihood estimators (MLEs) of the location and scale parameters of the linear exponential distribution. In addition, we use the setup proposed by Balakrishnan and Aggarwala (2000) to compute the approximate best linear unbiased estimates (ABLUEs) of the location and scale parameters of the linear exponential distribution. Finally, we carry out a simulation study to compare between the techniques considered for the estimation.
TL;DR: In this paper, the authors proposed a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure.
Abstract: We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial cov...
TL;DR: In this paper, the Weibull distribution is proposed as a model for response times and the authors show that it performs better than the often-suggested power-law and logarithmic functions.
Abstract: The Weibull distribution is proposed as a model for response times. Theoretical support is offered by classical results for extreme-value distributions. Fits of the Weibull distribution to response time data in different contexts show that this distribution (and the exponential distribution on small time-scales) perform better than the often-suggested power-law and logarithmic function. This study suggests that the power-law can be viewed as an approximation, at neural level, for the aggregate strength of superposed memory traces that have different decay rates in distinct parts of the brain. As we predict, this view does not find support at the level of induced response processes. The distinction between underlying and induced processes might also be considered in other fields, such as engineering, biology and physics.
TL;DR: In this paper, a non-Markovian random walk in dimension 1 was studied and the joint probability distribution pn(x, k) in a closed form was shown to belong to the exponential family.
Abstract: We study a non-Markovian random walk in dimension 1. It depends on two parameters R and L, the probabilities to go straight on when walking to the right, and respectively to the left. The position x of the walk after n steps and the number of reversals of direction k are used to estimate R and L. We calculate the joint probability distribution pn(x, k) in a closed form and show that, approximately, it belongs to the exponential family.
TL;DR: The exact distributions of R=X+Y, P=XY and W=X/(X-Y) and the corresponding moment properties when X and Y follow Lawrence and Lewis's bivariate exponential distribution are derived.
TL;DR: In this paper, the authors discuss the log-concavity of the survival function of the minimum and maximum from Gumbel bivariate exponential models, through the logconsistency of generalized mixtures of four or fewer exponential distributions, extending the papers of Baggs and Nagaraja.
Abstract: In the classical risk theory, it is often used that different type dimensions can be aggregated into a single-dimensional statistic, as well as the assumption of properties on log-concavity of this aggregation. The extreme-order statistics, minimum and maximum, might be used as aggregate statistics. In this paper, we discuss the log-concavity of the survival function of the minimum and maximum from Gumbel bivariate exponential models, through the log-concavity of generalized mixtures of four or fewer exponential distributions, extending the papers of Baggs and Nagaraja [Baggs, G.E. and Nagaraja, H.N., 1996, Reliability properties of order statistics from bivariate exponential distributions. Communications in Statistics—Stochastic Models, 12, 611–631] and Franco and Vivo [Franco, M. and Vivo, J.M., 2002, Reliability properties of series and parallel systems from bivariate exponential models. Communications in Statistics—Theory and Methods, 31, 2349–2360] devote to the log-concavity for generalized mixtures...
TL;DR: In this paper, a notion of transorthogonality for a sequence of polynomial on R d was introduced to extend the characterization to the multivariate version of the Letac-Mora class of real cubic natural exponential families.
14 May 2006
TL;DR: This paper focuses on statistical region-based active contour models where image features are random variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special Gaussian case.
Abstract: In this paper, we focus on statistical region-based active contour models where image features (e.g. intensity) are random variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special Gaussian case. Using shape derivation tools, our effort focuses on constructing a general expression for the derivative of the energy (with respect to a domain) and derive the corresponding evolution speed. A general result is stated within the framework of multi-parameter exponential family. More particularly, when using Maximum Likelihood estimators, the evolution speed has a closed-form expression that depends simply on the probability density function, while complicating additive terms appear when using other estimators, e.g. moments method. Experimental results on both synthesized and real images demonstrate the applicability of our approach.
TL;DR: In this article, the authors link the infinitely divisible measure μ to its modified Levy measure ρ = ρ ( μ ) in terms of their variance functions, where x - 2 [ ρ(d x ) -ρ ( { 0 } ) δ 0 ( d x ) ] is the Levy measure associated with μ.
TL;DR: In this article, the probability generating function of general multivariate Bernoulli distributions and for the moment generating function for the aggregate claim amount for individual risk models with dependencies are given.
TL;DR: In this article, a method to generate normal random variable using a generalized exponential distribution is proposed, which is compared with the other existing methods and it is observed that the proposed method is quite competitive with most of the existing methods in terms of the KS − distances and corresponding p-values.
Abstract: A convenient method to generate normal random variable using a generalized exponential distribution is proposed. The new method is compared with the other existing methods and it is observed that the proposed method is quite competitive with most of the existing methods in terms of the KS − distances and the corresponding p-values.
01 Jan 2006
TL;DR: In this paper, a number of competing statistics for estimating the regression function in a family of conditionally parametric response models where, given a continuous covariate, the distribution of the response comes from a full rank exponential family with the parameter (which is in one to one correspondence with the conditional mean) being a shape constrained function of the covariate.
Abstract: We study a number of competing statistics for estimating the regression function in a family of conditionally parametric response models where, given a continuous covariate, the distribution of the response comes from a full rank exponential family with the parameter (which is in one to one correspondence with the conditional mean) being a shape constrained function of the covariate. Monotonicity, unimodality and U-shapes are considered. It is shown that the proposed competing statistics are approximately pivotal for large sample sizes and methods for constructing pointwise confidence sets for the regression function, away from a stationary point, are described. Results from a limited simulation study are presented.