The diagonal multivariate natural exponential families and their classification

Bar‐Lev, Shaul K.; Bshouty, Daoud; Enis, Peter; Letac, Gérard; Lu, I-Li; Richards, Donald St. P.

doi:10.1007/bf02214378

Cited by 49 publications

(35 citation statements)

References 23 publications

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“…The following preliminary result shows that all multivariate gamma NEFs have a diagonal quadratic variance function (see also Bar-Lev et al [2]). …”

mentioning

confidence: 84%

“…Obviously, the crucial problem of this estimator (3) is to exhibit C n ( ) de…ned in (2). In the previous papers we only …nd the explicit expressions of C n ( ) for NEFs having homogeneous and simple quadratic variance functions of Casalis ([5] and [6]).…”

Section: Proposition 1 Let 2 M(r D )mentioning

confidence: 99%

“…and the use of a multivariate gamma NEF for mixing Poisson distribution by Ferrari et al [8], this paper is devoted to the UMVU and ML estimators of some generalized variances under the multivariate gamma hypothesis. Considered by the previous authors to be the natural multivariate extension of the real gamma NEF, the following multivariate gamma models belong to the diagonal quadratic NEFs in the sense of Bar-Lev et al [2] (see Proposition 2) and, however, does not hold the condition (4) of proportionality between UMVU and ML estimators of the generalized variance for certain quadratic NEFs. The present paper is structured as follows.…”

Section: Proposition 1 Let 2 M(r D )mentioning

confidence: 99%

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Generalized variance estimators in the multivariate gamma models

2008

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It has been shown that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. In practice, however, this estimator is often di¢ cult to obtain. This paper explicitly identi…es the results in complete bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is then pointed out that the UMVU and the maximum likelihood estimators are not proportional as conjectured for models belonging in certain quadratic exponential families.

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“…The following preliminary result shows that all multivariate gamma NEFs have a diagonal quadratic variance function (see also Bar-Lev et al [2]). …”

mentioning

confidence: 84%

Section: Proposition 1 Let 2 M(r D )mentioning

confidence: 99%

Section: Proposition 1 Let 2 M(r D )mentioning

confidence: 99%

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Generalized variance estimators in the multivariate gamma models

2008

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“…The conjugate families on µ are: Gammas for Poissons, Inverted Gammas (reciprocals of Gamma random variables) for Gammas, Betas for Binomials, F-distributions for Negative Binomials, and Skewed-t distributions (Skates 1993;Esch 2003) for the NEF-CHS. The symmetric Skewed-t is Student's t, which includes the Cauchy distribution, t 1 . We renamed what was originally known as the Skew-t (Skates 1993) as the Skewed-t, since Skew-t has since been taken to mean a Skew-Normal over a Chi, which is different than our Skewed-t, Pearson's Type IV distribution.…”

Section: Pearson Conjugates 41 Conjugate Familiesmentioning

confidence: 99%

Unifying the Named Natural Exponential Families and Their Relatives

Morris

Lock

2009

The American Statistician

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Five of the six univariate natural exponential families (NEF) with quadratic variance functions (QVF), meaning their variances are at most quadratic functions of their means, are the Normal, Poisson, Gamma, Binomial, and Negative Binomial distributions. The sixth is the NEF-CHS, the NEF generated by convolved Hyperbolic Secant distributions. These six NEF-QVFs and their relatives are unified in this paper and in the main diagram, Figure 1, which connects NEFs with many other named distributions, including Pearson's families of conjugate distributions (Inverted Gamma, Beta, F, and Skewed-t), conjugate mixtures (including two Polya urn schemes, with types I and II sampling), and conditional distributions (including the Hypergeometric and Negative Hypergeometric). Limit laws that relate these distributions are indicated by solid arrows in Figure 1.

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“…In particular, matrices Q yielding infinitely divisible PGFs were derived. Finally, Bar-Lev et al [5] introduced NMDs whose PGFs are defined as the inverse th power of any affine polynomial. Necessary and sufficient conditions on the coefficients of this affine polynomial were derived to obtain the PGF of a multivariate distribution defined on N 0 (where N 0 is the set of nonnegative integers) [6].…”

Section: Introductionmentioning

confidence: 99%

Masses of Negative Multinomial Distributions: Application to Polarimetric Image Processing

Bernardoff

Chatelain

Tourneret

2013

Journal of Probability and Statistics

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This paper derives new closed-form expressions for the masses of negative multinomial distributions. These masses can be maximized to determine the maximum likelihood estimator of its unknown parameters. An application to polarimetric image processing is investigated. We study the maximum likelihood estimators of the polarization degree of polarimetric images using different combinations of images.

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The diagonal multivariate natural exponential families and their classification

Cited by 49 publications

References 23 publications

Generalized variance estimators in the multivariate gamma models

Generalized variance estimators in the multivariate gamma models

Unifying the Named Natural Exponential Families and Their Relatives

Masses of Negative Multinomial Distributions: Application to Polarimetric Image Processing

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