Lancaster bivariate probability distributions with Poisson, negative binomial and gamma margins

Koudou, Angelo Efoévi

doi:10.1007/bf02565104

Cited by 29 publications

(26 citation statements)

References 11 publications

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“…Similar expansions are available for other distributions such as chi square and gamma (e.g., Koudou (1998); Schwartzman (2011)). Thus it may be possible to extend our results to those distributions as well.…”

Section: Summary and Extensionsmentioning

confidence: 91%

The Empirical Distribution of a Large Number of Correlated Normal Variables

Azriel

Schwartzman

2015

Journal of the American Statistical Association

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Motivated by the advent of high dimensional highly correlated data, this work studies the limit behavior of the empirical cumulative distribution function (ecdf) of standard normal random variables under arbitrary correlation. First, we provide a necessary and sufficient condition for convergence of the ecdf to the standard normal distribution. Next, under general correlation, we show that the ecdf limit is a random, possible infinite, mixture of normal distribution functions that depends on a number of latent variables and can serve as an asymptotic approximation to the ecdf in high dimensions. We provide conditions under which the dimension of the ecdf limit, defined as the smallest number of effective latent variables, is finite. Estimates of the latent variables are provided and their consistency proved. We demonstrate these methods in a real high-dimensional data example from brain imaging where it is shown that, while the study exhibits apparently strongly significant results, they can be entirely explained by correlation, as captured by the asymptotic approximation developed here.

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Section: Summary and Extensionsmentioning

confidence: 91%

The Empirical Distribution of a Large Number of Correlated Normal Variables

Azriel

Schwartzman

2015

Journal of the American Statistical Association

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“…We define the set 17) and denote by E c its complementary. The following result gives us insight on how to choose the parameters of the reference distribution.…”

Section: Integrability Conditionmentioning

confidence: 99%

“…The Lancaster problem can be reduced to finding a proper sequence {a k } k∈N such that f is nonnegative and therefore a BPDF. The characterization of the Lancaster probabilities constructed via NEF-QVF has been widely studied by Koudou, see [15,16,17]. He gave existence conditions and explicit forms of Lancaster sequences in various cases.…”

Section: Downton Bivariate Exponential Distribution and Lancaster Promentioning

confidence: 99%

Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

Goffard¹,

Loisel²,

Pommeret³

2015

Methodol Comput Appl Probab

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A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.

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“…

f_{ν} (t_{i}, t_{i^{'}}; ρ_{i i^{'}}) = f_{ν} (t_{i}) f_{ν} (t_{i^{'}}) \sum_{k = 0}^{\infty} \frac{ρ_{i i^{'}}^{k}}{k!} \frac{normalΓ false(ν / 2 false)}{normalΓ false(ν / 2 + k false)} {scriptL}_{k}^{false(ν / 2 - 1 false)} (\frac{t_{i}}{2}) {scriptL}_{k}^{false(ν / 2 - 1 false)} (\frac{t_{i^{'}}}{2}),

where

L_{k}^{(ν / 2 - 1)} (t)

are the generalized Laguerre polynomials of degree ν /2− 1:

L_{0}^{(ν / 2 - 1)} (t) = 1, L_{1}^{(ν / 2 - 1)} (t) = - t + ν / 2, L_{2}^{(ν / 2 - 1)} (t) = t^{2} - 2 (ν / 2 + 1) t + (ν / 2) (ν / 2 + 1)

and so on (Koudou, 1998). A derivation similar to the one above for the Gaussian case gives that, for large N , the covariance function (4) takes the form…”

Section: The Distribution Of Correlated χ2 Variatesmentioning

confidence: 99%