Showing papers on "Natural exponential family published in 1990"
TL;DR: In this paper, the authors describe all the natural exponential families on the real line such that the variance is a polynomial function of the mean with degree less than or equal to 3.
Abstract: Pursuing the classification initiated by Morris (1982), we describe all the natural exponential families on the real line such that the variance is a polynomial function of the mean with degree less than or equal to 3. We get twelve different types; the first six appear in the fundamental paper by Morris (1982); most of the other six appear as distributions of first passage times in the literature, the inverse Gaussian type being the most famous example. An explanation of this occurrence of stopping times is provided by the introduction of the notion of reciprocity between two measures or between two natural exponential families, and by classical fluctuation theory.
257 citations
TL;DR: In this article, a finite-parameter exponential family model based on B$-splines is constructed and the maximum likelihood estimation of the parameters of the model is obtained based on a random sample of size $n$ from $f$ from a known compact interval.
Abstract: Let $f$ be a continuous and positive unknown density on a known compact interval $\mathscr{Y}$. Let $F$ denote the distribution function of $f$ and let $Q = F^{-1}$ denote its quantile function. A finite-parameter exponential family model based on $B$-splines is constructed. Maximum-likelihood estimation of the parameters of the model based on a random sample of size $n$ from $f$ yields estimates $\hat{f, F}$ and $\hat{Q}$ of $f, F$ and $Q$, respectively. Under mild conditions, if the number of parameters tends to infinity in a suitable manner as $n \rightarrow \infty$, these estimates achieve the optimal rate of convergence. The asymptotic behavior of the corresponding confidence bounds is also investigated. In particular, it is shown that the standard errors of $\hat{F}$ and $\hat{Q}$ are asymptotically equal to those of the usual empirical distribution function and empirical quantile function.
143 citations
TL;DR: In this article, families of min-stable multivariate exponential and multivariate extreme value distributions are presented, and two families have a representation like the Marshall-Olkin multiivariate exponential distribution.
109 citations
TL;DR: In this article, the generalized beta of the second kind (GB2) family of distributions is used for modeling insurance loss processes and the results suggest that seemingly slight differences in modeling the tails can result in large differences in premiums and quantiles for the distribution of total insurance losses.
Abstract: This paper investigates the use of a four parameter family of probability distributions, the generalized beta of the second kind (GB2), for modeling insurance loss processes. The GB2 family includes many commonly used distributions such as the lognormal, gamma and Weibull. The GB2 also includes the Burr and generalized gamma distributions. Members of this family and their inverse distributions have significant potential for improving the distributional fit in many applications involving thin or heavy-tailed distributions. Members of the GB2 family can be generated as mixtures of well-known distributions and provide a model for heterogeneity in claims distributions. Examples are presented which consider models of the distribution of individual and of aggregate losses. The results suggest that seemingly slight differences in modeling the tails can result in large differences in reinsurance premiums and quantiles for the distribution of total insurance losses.
88 citations
19 citations
19 citations
TL;DR: In this paper, the authors consider the estimation of the common scale parameter of two or more independent shifted exponential distributions with unknown locations and show that the best location-scale in-variant estimator is inadmissible.
Abstract: We consider the estimation of the common scale parameter of two or more independent shifted exponential distributions with unknown locations. Under a large class of bowl-shaped loss functions, the best location-scale in-variant estimator is shown to be inadmissible. A class of improved estimators is derived. Some numerical results are presented to show the magnitude of risk reduction.
11 citations
TL;DR: In this paper, simple nonlinear models of autoregressive and moving average structure with a mixed exponential marginal distribution are described, and simple models of moving average structures with a nonlinear model of nonlinear autoregression are presented.
9 citations
01 Jan 1990
TL;DR: In this paper, the authors derived a BLO norm estimate for (Tf)2 and a pointwise estimate for Tf, where Tf denotes any one of the usual classical or generalized Littlewood-Paley functions.
Abstract: Let Tf denote any one of the usual classical or generalized Littlewood-Paley functions. This paper derives a BLO norm estimate for (Tf)2 and a pointwise estimate for Tf .
8 citations
TL;DR: In this paper, a Cacoullos-type lower bound for the variance of a random vector X belonging to the multidimensional exponential family is given, which is shown to characterize the exponential family.
7 citations
TL;DR: In this article, a natural exponential family with variance function (V, Ω) is considered, where Ω is the mean domain of F and V is its variance expressed in terms of the mean μ ϵ Ω.
TL;DR: In this paper, the asymptotic expected deficiency (AED) of the MLE relative to the uniformly minimum variance unbiased estimator (UMVUE) for a given one-parameter estimable function of an exponential family is obtained.
Abstract: Under some regularity conditions, the asymptotic expected deficiency (AED) of the maximum likelihood estimator (MLE) relative to the uniformly minimum variance unbiased estimator (UMVUE) for a given one-parameter estimable function of an exponential family is obtained. The exact expressions of the AED for normal, lognormal, inverse Gaussian, exponential (or gamma), Pareto, hyperbolic secant, Bernoulli, Poisson and geometric (or negative binomial) distributions are also derived.
TL;DR: In this article, it was shown that the maximum likelihood estimator (MLE) has asymptotic loss of information of order (n) n, as compared to constant order in regular cases.
Abstract: Fisher (1934), starting from his fundamental paper (1922), discussed estimators of the location parameter of a double exponential (two-sided exponential) distribution as a typical example of non-regular estimation. He showed that the maximum likelihood estimator (MLE), which is equal to the sample median in this case, has asymptotic loss of information of order \(\sqrt{n}\), as compared to constant order in regular cases. Let I and IT be the amounts of Fisher information in a single observation and that in a statistic T, respectively. Then the value of nI•IT as n → ∞, i.e. limn→∞ (nI — IT) is called the loss of information associated with T and its asymptotic value as n → ∞ is called the asymptotic loss of information (see, e.g. Rao (1961)).
TL;DR: In this article, bounds on the uniform distance between a χ22n distribution and the distribution of 2Σni = 1 Xi/μ, where X1, X2, Xn are n independent, identically distributed nonnegative random variables with common mean μ, are derived assuming that the Xi's are HNBUE or HNWUE or that a specific "mechanism" is "perturbing" an exponential distribution.
Dissertation•
10 May 1990
TL;DR: In this article, it was shown that the posterior covariance inequality of the predictive distribution is equivalent to ordering according to the covariance matrix of the parameters of the prior. But this is not the case for the binomial distribution with a degenerate prior.
01 Jan 1990
TL;DR: The exponential distribution is an example of a continuous distribution as mentioned in this paper, and a random variable X is said to follow the exponential distribution with parameter λ if its distribution function F is given by: F (x) = 1 − e−λ x for x > 0.
Abstract: The exponential distribution is an example of a continuous distribution. A random variable X is said to follow the exponential distribution with parameter λ if its distribution function F is given by: F (x) = 1 − e−λ x for x > 0. Recall that the distribution function F (x) = P (X ≤ x) by definition and is an increasing function of x. Since F (0) = 0, it follows that X is bigger than 0 with probability 1.
01 Jan 1990
TL;DR: In this article, the regular exponential family with polynomial variance function (PVF-this article) is discussed and the results for all seven cases of polynomially variance functions of third degree are given.
Abstract: In this paper,we discuss the regular exponential family(REF),a natural exponentialfamily with minimum dimension and with the mean function m=E_θX taken as a new parameterWepoint out the importance of the REF with polynomial variance function(PVF-REF)When m is asingle factor of PVF and the left end of its defined domain is zero,the corresponding REF is givenas the single-side lattice distribution family;other PVF-REFs can be obtained by a limit processAs an illustration,the results for all seven cases of polynomial variance functions of third degreeare given
TL;DR: In this article, a modified likelihood ratio criterion is provided to discriminate between gamma and exponential distributions for a complete sample of life data and estimates of the probability of correct selections have been obtained on the basis of a Monte Carlo study.
01 Feb 1990
TL;DR: In this article, the necessary and sufficient condition for the random variables Y1 - Y2,Yn-1-Yn and Yn to be conditionally mutually independent is that for each i+1,n,Yi has an exponential distribution.
Abstract: : Let Y1,...,Yn be n mutually independent, nonnegative random variables such that for each i=1,...,n,Yi has an absolutely continuous distribution function function F(x;0i) = F(0xi), where 0i>O, and F(.) has support O,oo). We show that given Yi - Yi+1>O for all i+1,...,n-1, the necessary and sufficient condition for the random variables Y1 - Y2,...,Yn-1-Yn and Yn to be conditionally mutually independent is that for each i+1,...,n,Yi has an exponential distribution. Characterization; Exponential distribution; Conditional independence.
TL;DR: In this paper, the proximity between exponential distributions and life distributions in various W-type classes was studied, where W type classes indicate DFR, DFRA, NWU, NWUE, IMRL and HNWUE.
Abstract: This paper is a continuation of the foregoing one [1]. In this paper, we study the proximity between exponential distributions and life distributions in variousW-type classes, whereW-type classes indicate DFR, DFRA, NWU, NWUE, IMRL and HNWUE.
TL;DR: In this paper, it was shown that the two-dimensional exponential model of the field theory is trivial for α 2 > 8π and for α ≥ 2π, and that α ≥ 4π.
Abstract: It is proved that the two-dimensional exponential model of the field theory is trivial for α2 > 8π.
TL;DR: In this paper, a class of probability distributions, which are geometrically infinitely divisible and form a generalization of the Geometric Stable Distribution (GSD), is defined.
Abstract: A class of probability distributions is indicated, which are geometrically infinitely divisible and form a generalization of the geometrically stable distributions. Two characterizations of these distributions are given.