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What is the best approximation of ruin probability in infinite time?

Krzysztof Burnecki, Paweł Miśta, Aleksander Weron (2005)

Applicationes Mathematicae

We compare 12 different approximations of ruin probability in infinite time studying typical light- and heavy-tailed claim size distributions, namely exponential, mixture of exponentials, gamma, lognormal, Weibull, loggamma, Pareto and Burr. We show that approximation based on the Pollaczek-Khinchin formula gives most accurate results, in fact it can be chosen as a reference method. We also introduce a promising modification to the De Vylder approximation.

When is a Riesz distribution a complex measure?

Alan D. Sokal (2011)

Bulletin de la Société Mathématique de France

Let α be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by α . I give an elementary proof of the necessary and sufficient condition for α to be a locally finite complex measure (= complex Radon measure).

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

If the space 𝒬 of quadratic forms in n is splitted in a direct sum 𝒬 1 ... 𝒬 k and if X and Y are independent random variables of n , assume that there exist a real number a such that E ( X | X + Y ) = a ( X + Y ) and real distinct numbers b 1 , . . . , b k such that E ( q ( X ) | X + Y ) = b i q ( X + Y ) for any q in 𝒬 i . We prove that this happens only when k = 2 , when n can be structured in a Euclidean Jordan algebra and when X and Y have Wishart distributions corresponding to this structure.

Why L 1 view and what is next?

László Györfi, Adam Krzyżak (2011)

Kybernetika

N. N. Cencov wrote a commentary chapter included in the Appendix of the Russian translation of the Devroye and Györfi book [15] collecting some arguments supporting the L 1 view of density estimation. The Cencov’s work is available in Russian only and it hasn’t been translated, so late Igor Vajda decided to translate the Cencov’s paper and to add some remarks on the occasion of organizing the session “25 Years of the L 1 Density Estimation” at the Prague Stochastics 2010 Symposium. In this paper we...

Why minimax is not that pessimistic

Aurelia Fraysse (2013)

ESAIM: Probability and Statistics

In nonparametric statistics a classical optimality criterion for estimation procedures is provided by the minimax rate of convergence. However this point of view can be subject to controversy as it requires to look for the worst behavior of an estimation procedure in a given space. The purpose of this paper is to introduce a new criterion based on generic behavior of estimators. We are here interested in the rate of convergence obtained with some classical estimators on almost every, in the sense...

Widespread Immunity to Breast and Prostate Cancers is Predicted by a Novel Model that also Determines Sporadic and Hereditary Susceptible Population Sizes

I. Kramer (2010)

Mathematical Modelling of Natural Phenomena

Natural immunity to breast and prostate cancers is predicted by a novel, saturated ordered mutation model fitted to USA (SEER) incidence data, a prediction consistent with the latest ideas in immunosurveillance. For example, the prevalence of natural immunity to breast cancer in the white female risk population is predicted to be 76.5%; this immunity may be genetic and, therefore, inherited. The modeling also predicts that 6.9% of White Females are...

Wild bootstrap in RCA(1) model

Zuzana Prášková (2003)

Kybernetika

In the paper, a heteroskedastic autoregressive process of the first order is considered where the autoregressive parameter is random and errors are allowed to be non-identically distributed. Wild bootstrap procedure to approximate the distribution of the least-squares estimator of the mean of the random parameter is proposed as an alternative to the approximation based on asymptotic normality, and consistency of this procedure is established.

Wilks' factorization of the complex matrix variate Dirichlet distributions.

Xinping Cui, Arjun K. Gupta, Daya K. Nagar (2005)

Revista Matemática Complutense

In this paper, it has been shown that the complex matrix variate Dirichlet type I density factors into the complex matrix variate beta type I densities. Similar result has also been derived for the complex matrix variate Dirichlet type II density. Also, by using certain matrix transformations, the complex matrix variate Dirichlet distributions have been generated from the complex matrix beta distributions. Further, several results on the product of complex Wishart and complex beta matrices with...

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