Showing papers on "Natural exponential family published in 1998"

A matrix variate generalization of the power exponential family of distributions

Abstract: This paper proposes a matrix variate generalization of the power exponential distribution family, which can be useful in generalizing statistical procedures in multivariate analysis and in designing robust alternatives to them. An example is added to show an application of the generalization.

Exponential Family Nonlinear Models

Abstract: Exponential Family.- Exponential Family Nonlinear Models.- Geometric Framework.- Some Second Order Asymptotics.- Confidence Regions.- Diagnostics and Influence Analysis.- Extension.- Appendices.- Bibliography.- Author Index.- Subject Index.

Tail probabilities for the null distribution of scanning statistics

Abstract: This paper is concerned with statistics that scan a multidimensional spatial region to detect a signal against a noisy background. The background is modelled as independent observations from an exponential family of distributions with a known 'null' value of the natural parameter, while the signal is given by independent observations from the same exponential family, but with a different value of the parameter on a particular subregion of the spatial domain. The main result is an extension to multidimensional time of the method of Pollak and Yakir, which relies on a change of measure motivated by change-point analysis, to evaluate approximately the null distribution of the likelihood ratio statistic. Both large-deviation and Poisson approximations are obtained.

Exponential and scale mixtures and equilibrium distributions

Abstract: In this article we discuss mixed exponential distributions and, more generally, scale mixtures with specific consideration the purpose of insurance modeling. Results are derived for equilibrium distributions (defined via stop-loss transforms) of mixed distributions. Some recursive relations are identified for the stop-loss transforms and moments of mixed exponential distributions. Explicit expressions are obtained for equilibrium gamma distributions with arbitrary shape parameter.

Exponential dispersion models and credibility

Abstract: The Exponential Dispersion Family is a rich family of distributions, comprised of several distributions, some of which are heavy-tailed and as such could be of significant relevance to actuarial science. The family draws its richness from a dispersion parameter σ 2 = 1/λ which is equal to 1 in the case of the Natural Exponential Family. We consider three cases. In the first λ is assumed known, in the second a prior distribution for λ is given, and in the third the prior distribution of λ is not known and is derived by means of the maximum entropy principle, assuming the prior mean of λ can be specified. For these cases, a conjugate prior distribution for the risk parameter is assumed and credibility formulae are derived for the estimation of the fair premium.

Distributional Properties of Exceedance Statistics

Abstract: Behaviour of a sequence of independent identically distributed random variables with respect to a random threshold is investigated. Three statistics connected with exceeding the threshold are introduced, their exact and asymptotic distributions are derived. Also distribution-free properties, leading to some common and some new discrete distributions, are considered. Identification of equidistribution of observations and the threshold are discussed. In this context relations between the exponential and gamma distributions are studied and a new derivation of the celebrated Laplace expansion for the standard normal distribution function is given.

Laplace Approximations for Natural Exponential Families with Cuts

Abstract: Standard and fully exponential form Laplace approximations to marginal densities are described and conditions under which these give exact answers are investigated. A general result is obtained and is subsequently applied in the case of natural exponential families with cuts, in order to derive the marginal posterior density of the mean parameter corresponding to the cut, the canonical parameter corresponding to the complement of the cut and transformations of these. Important cases of families for which a cut exists and the approximations are exact are presented as examples.

2 Higher order moments of order statistics from exponential and right-truncated exponential distributions and applications to life-testing problems

Abstract: Publisher Summary This chapter discusses the concept of higher order moments of order statistics from exponential and right-truncated exponential distributions and its applications to life-testing problems. It presents several recurrence relations satisfied by the single, double, triple, and quadruple moments of order statistics from the standard exponential distribution, respectively. It considers the scale-parameter exponential distribution and use the results derived to determine the mean, variance, and coefficients of skewness and kurtosis of the best linear unbiased estimator of the scale parameter based on doubly Type-II censored samples. While it is well known that for the case when the available sample is Type-II right censored the best linear unbiased estimator of the scale parameter has exactly a chi-square distribution, the chapter shows that a chi-square distribution provides a very close approximation to the distribution of the best linear unbiased estimator even when the available sample is doubly Type-II censored. The chapter also discusses three examples involving life-testing data and illustrates the usefulness of the chi-square approximation for the distribution of the best linear unbiased estimator by constructing approximate confidence intervals for the mean life-time.

Asymptotic Properties of a Class of Mixture Models for Failure Data: The Interior and Boundary Cases

Abstract: We analyse an exponential family of distributions which generalises the exponential distribution for censored failure time data, analogous to the way in which the class of generalised linear models generalises the normal distribution. The parameter of the distribution depends on a linear combination of covariates via a possibly nonlinear link function, and we allow another level of heterogeneity: the data may contain "immune" individuals who are not subject to failure. Thus the data is modelled by a mixture of a distribution from the exponential family and a "mass at infinity" representing individuals who never fail. Our results include large sample distributions for parameter estimators and for hypothesis test statistics obtained by maximising the likelihood of a sample. The asymptotic distribution of the likelihood ratio test statistic for the hypothesis that there are no immunes present in the population is shown to be "non-standard"; it is a 50-50 mixture of a chi-squared distribution on 1 degree of freedom and a point mass at 0. Our analysis clearly shows how "negligibility" of individual covariate values and "sufficient followup" conditions are required for the asymptotic properties.

Natural Exponential Families and Umbral Calculus

Abstract: We use the Umbral Calculus to investigate the relation between natural exponential families and Sheffer polynomials. As a corollary, we obtain a new transparent proof of Feinsilver’s theorem which says that natural exponential families have a quadratic variance function if and only if their associated Sheffer polynomials are orthogonal.

Generating Matrix Exponential Random Variates

Abstract: In this paper we present a technique for generating random variates from an empirical distribution using the matrix exponential representation of the distribution. In our experience, a matrix exponential representation of an empirical distribution produces random variates with an excellent fit with the empirical distribution. This technique is particularly important when the empirical data is very bursty, i.e., has a high variance. In this paper we discuss how to find the matrix exponential representation of an empirical distribution and we present our technique for generating random variates from the empirical distribution using its matrix exponential representation. We show how the matrix exponential representation of an empirical distribution is found through an example and then we show that matrix exponential random variates are an excellent fit with the empirical data through an χ2 goodness-of-fit test.

Global statistical information in exponential experiments and selection of exponential models

Abstract: The concept of global statistical information in the classical statistical experiment with independent exponentially distributed samples is investigated. Explicit formulas are evaluated for common exponential families. It is shown that the generalized likelihood ratio test procedure of model selection can be replaced by a generalized information procedure. Simulations in a classical regression model are used to compare this procedure with that based on the Akaike criterion.

Characterization of discrete mixtures of exponential distribution based on conditional expectation of order statistics

Abstract: In a random sample of size two, we use the relation between the conditional expectation of order statistics and the hazard rate function of the distribution to characterize the discrete mixtures of exponential distribution. In addition, we also n ention some related theorems to characterize the discrete mixtures of exponential distribution. Moreover, when the sample size is n, the above results are also valid. Finally, we give a relative result of the different formula for Nassar [14] and an application to parallel system in engineering practice.

Characterization of some types of distributions

Abstract: A class of truncated and non-truncated distributions have been characterized using the recurrence relations between expectations of a function of one and two order statistics. The specific distributions considered as a particular case of the general class of distributions are Power function, Weibull, Pareto, Beta of first kind, Burr type XII, Rectangular, Rayleigh, Exponential, Lomax, Logistic, Cauchy and Inverse Weibull.

Adaptive Estimation of Distributions Using Exponential Sub-Families

Abstract: An algorithm is presented that, for a large-dimensional exponential family G, finds a lower dimension exponential sub-family of G which contains distributions best fitting groups of identically distributed observations within a set of data. The data are therefore fitted to a family of distributions that has been adaptively chosen as representative of them. The algorithm is implemented in the special case in which G is a logspline family of distributions. An example data set is analyzed using the method.

Ordering of the Mandelbrot-like set of the exponential map

Abstract: We graphically study the Mandelbrot-like set of the complex exponential family of maps Eλ(z)=λez, which we call the Baker–Rippon–Devaney (BRD) set. We observe that the period of every hyperbolic component can be deduced with the naked eye by using two simple rules.

Solution of statistical problems for a class of exponential distributions of random variables

Abstract: For a class of exponential-type distributions with the power index ranging from 0.25 to 8, a simple formula for an approximate evaluation of the distribution function is presented. The formula is then used for solving certain statistical problems.

1 Order statistics in exponential distribution

Abstract: Publisher Summary This chapter discusses the properties of order statistics and uses the results for estimating the parameters of the one and two parameter exponential distributions. It summarizes the important properties of order statistics from the exponential distribution and describes various types of censoring. The estimates of scale parameter of the exponential distribution for Type I, Type II, and randomly censored data are derived in the chapter. The inferences concerning the two-parameter exponential distribution are also considered in the chapter. These results are extended to two or more independent Type II censored samples. Order restricted inference for the scale parameters of exponential distributions are also discussed in the chapter. Bayesian inference and Bayesian estimates of scale parameter, for Type I and Type H censored samples, are presented in the chapter.

Necessary and sufficient conditions for consistency of approximate M-estimators in nonlinear models

Abstract: The concept of M-estimator η n = arg min Σ ρ(Y i -ρ(η)) i=1 is widely used in mathematical statistics to construct robust estimators. We apply this concept to a special class of nonlinear models with non random regressors x i . The model consists of a one parametric parent family of distributions {F η } and a function of f(θ, α) which links the parameter θ and the regressors to the distribution. Hence Y i ∼ F f(θ0 , α i ) with some θ 0 ∈ Θ which is to be estimated. Special attention is paid to the pseudolinear models where the function f has the special structure f(θ,α) = g(θ'α). The M-estimator of θ is constructed by the minimization n of M n (θ) = Σ ρ(Y i - φ(f(θ, α i ))) where φ is a suitably i=1 chosen monotone function. In this paper we study the larger class of approximate M-estimators θ n which attain the minimum of M n (θ) only approximately for large n. Sufficient conditions for the consistency of approximate M-estimators are derived, motivated by sufficient conditions for the simpler parent estimator η n . In the class of approximate M-estimators which is larger than the class of ordinary M-estimators we are able to show that the sufficient conditions are necessary for the consistency of all approximate M-estimators. If {F η } is a natural exponential family and f(θ,α) = g(θ'α) has a pseudolinear structure our model reduces to the well known generalized linear model. Another special case is the class of nonlinear regression models.

On a Test of Independence in a Multivariate Exponential Distribution

Abstract: In this article the problem of testing independence in a multivariate exponential distribution with identical marginals is considered. Following Bhattacharyya and Johnson (1973) the null hypothesis of independence is transformed into a hypothesis concerning the equality of scale parameters of several exponential distributions. The conditional test for the transformed hypothesis proposed here is the likelihood ratio test. The powers of this test are estimated for selected values of the parameters, using Monte Carlo simulation and non-central chi-square approximation. The powers of the overall test are estimated using a simple formula involving the power function of the conditional test. Application to reliability problems is also discussed in some detail.

Algebraic Models in Statistics

Abstract: Exponential families are an important class of statistical models, i.e., parameterized families of probability distributions. It has been noted that in the case where the probability distributions live on a nite set most exponential families which occur in applications are actually solution sets of binomial polynomials. In fact they can be identied with the nonnegative real part of projective toric varieties. These toric varieties are not necessarily normal. This talk will explain this connection and give some examples. If time permits I will comment on how the generators of the binomial ideal dening the toric variety can be used in statistical testing.

Order statistics of independent identically distributed variables when the sum is known

Abstract: In this paper we consider the situation where we know the sum of n independent observations from the same probability distribution. We investigate how to empirically determine the marginal probability distributions of the different order statistics conditional upon knowing the sum. This research was motivated by explorations in process improvement where we know the total expected value or variance of a key measure of an n-step process and would like to estimate the proportion of the expected value or variance that is contributed by the most important step (i.e. the single step having the largest expected value or variance), the two most important steps, etc. Both graphical and tabular results are presented for exponential, gamma and normal distributions.