A Generalized Hermite Distribution and Its Properties

Gupta, Rupesh; Jain, G. C.

doi:10.1137/0127027

Cited by 32 publications

(19 citation statements)

References 2 publications

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“…For instance, if n 1 = 2 and n 2 = 5, then the values 1 and 3 have a probability equal to 0. For this reason we only consider the situation where n 1 = 1 and n 2 = n and it corresponds to a family of distributions known as Generalized Hermite distribution (Gupta and Jain, 1974). When n = 2 this is known as Hermite distribution, which was introduced by Kemp (1965, 1966).…”

Section: On the Generalized Hermite Distributionmentioning

confidence: 99%

The Use of Score Tests for Inference on Variance Components

Verbeke¹,

Molenberghs²

2003

Biometrics

203

144

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Summary Whenever inference for variance components is required, the choice between one‐sided and two‐sided tests is crucial. This choice is usually driven by whether or not negative variance components are permitted. For two‐sided tests, classical inferential procedures can be followed, based on likelihood ratios, score statistics, or Wald statistics. For one‐sided tests, however, one‐sided test statistics need to be developed, and their null distribution derived. While this has received considerable attention in the context of the likelihood ratio test, there appears to be much confusion about the related problem for the score test. The aim of this paper is to illustrate that classical (two‐sided) score test statistics, frequently advocated in practice, cannot be used in this context, but that well‐chosen one‐sided counterparts could be used instead. The relation with likelihood ratio tests will be established, and all results are illustrated in an analysis of continuous longitudinal data using linear mixed models.

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Section: On the Generalized Hermite Distributionmentioning

confidence: 99%

The Use of Score Tests for Inference on Variance Components

Verbeke¹,

Molenberghs²

2003

Biometrics

203

144

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“…These distributions have been reported by several authors, the case r = 2 by Kemp & Kemp (1965), the two-parameter generalization by Gupta & Jain (1974) and the generalized rth-order univariate used in this paper by Milne & Westcott (1993). However, in practice, these distributions have not often been used.…”

Section: Discussionmentioning

confidence: 99%

“…For r > 2, Milne & Westcott (1993) call these distributions Generalized Hermite or rth-order univariate Hermite. The latter name is preferable because Gupta & Jain (1974) used the name 'generalized Hermite distribution' previously, related to the specific case in equation (2.2), where only l 1 and l r are non-zero. The parameters l i , i = 1, .…”

Section: The Rth-order Univariate Hermite Distributionmentioning

confidence: 99%

An application of compound Poisson modelling to biological dosimetry

Puig

Barquinero

2010

Proc. R. Soc. A.

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“…Note that for s' = 0 in the above theorem, the ^ hypergeometric polynomial is essentially the generalized Hermite polynomial which occurs in probability problems. It has been studied by Gould and Hopper [7], Gupta and Jain [8], Cohen [4], and others. …”

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confidence: 99%

Some Classes of Generating Functions for the Laguerre and Hermite Polynomials

Cohen¹

1977

Mathematics of Computation

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Abstract.In the first half of the article, we present two theorems which give, as special cases, a number of new classes of generating functions for the Laguerre polynomial. These formulae extend the recent results of Carlitz (2] and others.The latter part of our work deals with two theorems involving new generating functions for the Hermite and generalized Hermite polynomials, thus generalizing some well-known expansions.

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A Generalized Hermite Distribution and Its Properties

Cited by 32 publications

References 2 publications

The Use of Score Tests for Inference on Variance Components

The Use of Score Tests for Inference on Variance Components

An application of compound Poisson modelling to biological dosimetry

Some Classes of Generating Functions for the Laguerre and Hermite Polynomials

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