Natural Exponential Families Research Articles

On Lévy measures for infinitely divisible natural exponential families

We link the infinitely divisible measure μ to its modified Lévy measure ρ = ρ ( μ ) in terms of their variance functions, where x - 2 [ ρ ( d x ) - ρ ( { 0 } ) δ 0 ( d x ) ] is the Lévy measure associated with μ . We deduce that, if the variance function of μ is a polynomial of degree p ⩾ 2 , then, the variance function of ρ is still a second degree polynomial. We illustrate these results with some Lévy processes such as positive stable and a class of Poisson stopped-sum processes.

Characterization of the simple cubic multivariate exponential families

The Letac–Mora class of real cubic natural exponential families has been characterized by a property of 2-orthogonality of an associated sequence of polynomials (see [G. Letac, M. Mora, Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990) 1–37; A. Hassairi, M. Zarai, Characterization of the cubic exponential families by orthogonality of polynomials, Ann. Probab. 32 (2004) 2463–2476]). The present paper introduces a notion of transorthogonality for a sequence of polynomial on R d to extend the characterization to the multivariate version of the Letac–Mora class of real natural exponential families.

Bhattacharyya matrices and cubic exponential families

This paper introduces a notion of semi-diagonality for a Bhattacharyya matrix to give a characterization of the Letac–Mora class of real natural exponential families having cubic variance function.

A new bivariate distribution in natural exponential family

We propose a new bivariate distribution following a GLM form i.e., natural exponential family given the constantly correlated covariance matrix. The proposed distribution can represent an independent bivariate gamma distribution as a special case. In order to derive the distribution we utilize an integrating factor method to satisfy the integrability condition of the quasi-score function. The derived distribution becomes a mixture of discrete and absolute continuous distributions. The proposal of our new bivariate distribution will make it possible to develop some bivariate generalized linear models. Further the discrete correlated bivariate distribution will also arise from an independent bivariate Poisson mass function by compounding our proposed distribution (Iwasaki and Tsubaki, 2002).

Characterizations of some polynomial variance functions byd-pseudo-orthogonality

From a notion ofd-pseudo-orthogonality for a sequence of polynomials (d ∈ {2,3, …}), this paper introduces three different characterizations of natural exponential families (NEF's) with polynomial variance functions of exact degree 2d-1. These results provide extended versions of the Meixner (1934), Shanbhag (1972), 1979) and Feinsilver (1986) characterization results of quadratic NEF's based on classical orthogonal polynomials. Some news sets of polynomials with (2d-1)-term recurrence relation are then pointed out and we completely illustrate the cases associated to the families of positive stable distributions.

On d-orthogonality of the Sheffer systems associated to a convolution semigroup

In this note we investigate which Sheffer polynomials can be associated to a convolution semigroup of probability measures, usually induced by a stochastic process with stationary and independent increments. From a recent kind of d-orthogonality ( d ∈ { 2 , 3 , … } ), we characterize the associated d-orthogonal polynomials by the class of generating probability measures, which belongs to the natural exponential family with polynomial variance functions of exact degree 2 d - 1 . This extends some results of (classical) orthogonality; in particular, some new sets of martingales are then pointed out. For each integer d ⩾ 2 we completely illustrate polynomials with ( 2 d - 1 )-term recurrence relation for the families of positive stable processes.

Classification of Riesz Exponential Families on a Symmetric Cone by Invariance Properties

Letac(5) has characterized the Wishart natural exponential families (NEFs) on the symmetric cone Ω of a Jordan algebra E by invariance under a group G of automorphisms of E preserving Ω. Hassairi and Lajmi(4) have introduced the so called Riesz NEF's as an extension of the Wishart NEF's and they have characterized them by invariance under the triangular group T. Pursuing these classifications, we construct a class of subgroups of G containing the triangular group T and we use them to classify the Riesz NEFs on Ω and therefore to characterize each class by an invariance property.

Characterization of the cubic exponential families by orthogonality of polynomials

This paper introduces a notion of 2-orthogonality for a sequence of polynomials to give extended versions of the Meixner and Feinsilver characterization results based on orthogonal polynomials. These new versions subsume the Letac-Mora characterization of the real natural exponential families having cubic variance function.

The use of the weighted likelihood in the natural exponential families with quadratic variance

AbstractThe authors propose a weighted likelihood concept for the estimation of parameters in natural exponential families with quadratic variance. They apply the results to both simulated and real data.

Conjugate Priors Represent Strong Pre‐Experimental Assumptions

Abstract. It is well known that Jeffreys’ prior is asymptotically least favorable under the entropy risk, i.e. it asymptotically maximizes the mutual information between the sample and the parameter. However, in this paper we show that the prior that minimizes (subject to certain constraints) the mutual information between the sample and the parameter is natural conjugate when the model belongs to a natural exponential family. A conjugate prior can thus be regarded as maximally informative in the sense that it minimizes the weight of the observations on inferences about the parameter; in other words, the expected relative entropy between prior and posterior is minimized when a conjugate prior is used.

Small-area estimation based on natural exponential family quadratic variance function models and survey weights

SUMMARY We propose pseudo empirical best linear unbiased estimators of small-area means based on natural exponential family quadratic variance function models when the basic data consist of survey-weighted estimators of these means, area-specific covariates and certain summary measures involving the weights. We also provide explicit approximate mean squared errors of these estimators in the spirit of Prasad & Rao (1990), and these estimators can be readily evaluated. A simulation study is undertaken to evaluate the performance of the proposed inferential procedure. We estimate also the proportion of poor children in the 5-17 years age-group for the different counties in one of the states in the United States.

Estimation of proportional covariances in the presene of certain linear restrictions

Proportionality of covariance matrices of n independent p-dimensional normal distributions with the same type of linear restrictions of the inverse covariances is considered. Conditions for existence and uniqueness of the maximum likelihood estimator are obtained through the development of general results for scale-invariant natural exponential families.

Martingales Defined by Reciprocals of Sums and Related Characterizations

We prove that linearly transformed inverses of cumulative sums form backward martingales for gamma, inverse Gaussian, and Kendall and Borel–Tanner sequences of independent, identically distributed random variables. Conversely, a characterization of the family of these four distributions by linearity of regression of inverses of sums is obtained. The results in both directions are derived via the technique of variance functions of natural exponential families.

Extension of the variance function of a steep exponential family

Let F={P(m,F); m∈M F} be a multidimensional steep natural exponential family parameterized by its domain of the means M F and let V F ( m) be its variance function. This paper studies the boundary behaviour of V F . Necessary and sufficient conditions on a point m of ∂M F are given so that V F admits a continuous extension V F( m) to the point m . It is also shown that the existence of V F( m) implies the existence of a limit distribution P( m,F) concentrated on an exposed face of M F containing m . The relation between V F( m) and P( m,F) is established and some illustrating examples are given.

Enriched conjugate and reference priors for the Wishart family on symmetric cones

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.

Reference priors for exponential families with simple quadratic variance function

Reference analysis is one of the most successful general methods to derive noninformative prior distributions. In practice, however, reference priors are often difficult to obtain. Recently developed theory for conditionally reducible natural exponential families identifies an attractive reparameterization which allows one, among other things, to construct an enriched conjugate prior. In this paper, under the assumption that the variance function is simple quadratic, the order-invariant group reference prior for the above parameter is found. Furthermore, group reference priors for the mean- and natural parameter of the families are obtained. A brief discussion of the frequentist coverage properties is also presented. The theory is illustrated for the multinomial and negative-multinomial family. Posterior computations are especially straightforward due to the fact that the resulting reference distributions belong to the corresponding enriched conjugate family. A substantive application of the theory relates to the construction of reference priors for the Bayesian analysis of two-way contingency tables with respect to two alternative parameterizations.

Unbiased estimation in the multivariate natural exponential family with simple quadratic variance function

We give expansions for the unbiased estimator of a parametric function of the mean vector in a multivariate natural exponential family with simple quadratic variance function. This expansion is giv...

Domains of attraction for exponential families

With the df F of the rv X we associate the natural exponential family of df's F λ where dF λ(x)= e λx dF(x)/E e λX for λ∈Λ≔{λ∈ R | E e λX<∞} . Assume λ ∞= sup Λ⩽∞ does not lie in Λ. Let λ↑ λ ∞, then non-degenerate limit laws for the normalised distributions F λ ( a λ x+ b λ ) are the normal and gamma distributions. Their domains of attractions are determined. Applications to saddlepoint and gamma approximations are considered.

A note on natural exponential families with cuts

Let μ be a positive measure defined on the product of two vector spaces E= E 1× E 2. Let F= F( μ) be a natural exponential family (NEF) generated by μ such that the projection of F on E 1 constitutes a NEF on E 1. This property, called a cut on E 1, has been defined and characterized by Barndorff-Nielsen (Information and Exponential Families, Wiley, Chichester) and further developed by Barndorff-Nielsen and Koudou (Theory Probab. Appl. 40 (1995) 361). Their results can be used to conclude two properties of NEFs with cuts. The first stating that a NEF F has a cut on E 1 if and only if for all random vectors ( X, Y) on E 1× E 2, having a distribution in F, the regression curve of Y on X is linear. The second property states that the linearity of the scedastic curve of Y on X is a necessary condition for F to have a cut on E 1. These two properties of linearity of the regression and scedastic curves provide, in some situations, rather easily verifiable conditions for examining whether a NEF has a cut. Moreover, they are used to provide some interesting characterizations. In particular, some characterizations of the Gaussian and Poisson NEFs are obtained as special cases.

Characterizations using weighted distributions

For one-parameter natural exponential and power series families we give some general characterizations from an affine relation connecting the mean of member distributions and their length biased versions. Examples subsume many known cases. Other characterizations are explored using random variable relations involving the length biased version of partial sums. Finally, characterizations of the law of X admitting more general weights are obtained by equating the laws of a function H( X) and the weighted version of X.