Compartmental models with phase-type residence-time distributions

Phase-type (PH) probability distributions provide a versatile class of distributions, and are shown to fit naturally into a Markovian compartmental system, where “individuals” or “particles” move between a series of compartments, so that phase-type compartmental residence-time distributions can be incorporated simply by increasing the size of the system. Examples of PH distributions covering a variety of behaviours are given, and an application involving data analysis is included.     

Extreme behavior of multivariate phase-type distributions

This paper investigates the limiting distributions of the componentwise maxima and minima of suitably normalized iid multivariate phase-type random vectors. In the case of maxima, a large parametric class of multivariate extreme value (MEV) distributions is obtained. The flexibility of this new class is exemplified in the bivariate setup. For minima, it is shown that the dependence structure of the Marshall-Olkin class arises in the limit.     

TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts

In this paper, we consider a portfolio of n dependent risks X₁,…,X_n and we study the stochastic behavior of the aggregate claim amount S=X₁+?+X_n. Our objective is to determine the amount of economic capital needed for the whole portfolio and to compute the amount of capital to be allocated to each risk X₁,…,X_n. To do so, we use a top-down approach. For (X₁,…,X_n), we consider risk models based on multivariate compound distributions defined with a multivariate counting distribution. We use the TVaR to evaluate the total capital requirement of the portfolio based on the distribution of S, and we use the TVaR-based capital allocation method to quantify the contribution of each risk. To simplify the presentation, the claim amounts are assumed to be continuously distributed. For multivariate compound distributions with continuous claim amounts, we provide general formulas for the cumulative distribution function of S, for the TVaR of S and the contribution to each risk. We obtain closed-form expressions for those quantities for multivariate compound distributions with gamma and mixed Erlang claim amounts. Finally, we treat in detail the multivariate compound Poisson distribution case. Numerical examples are provided in order to examine the impact of the dependence relation on the TVaR of S, the contribution to each risk of the portfolio, and the benefit of the aggregation of several risks.     

Some properties and generalizations of multivariate Eyraud-Gumbel-Morgenstern distributions

The admissible values of the coefficient in a bivariate Eyraud-Gumbel-Morgenstern (EGM) distribution are found. For multivariate EGM distributions necessary and sufficient conditions are given for its coefficients, and its conditional distributions are found and shown to belong to a family of distributions further extending the multivariate EGM family.     

An algebraic approach to tempered distributions

A space of pseudoquotients is introduced that is shown to be isomorphic to the space of tempered distributions on RN. The Fourier transform is defined as a map from the space of pseudoquotients to the space of tempered distributions and as a transformation on pseudoquotients.     

Construction of multivariate distributions with given marginals

Summary We make some remarks on the problem how to construct probability measures with given marginals. Questions of this kind arise if one wants to build a stochastic model in a situation where one has some idea of the kind of dependence and knows exactly certain marginal distributions.     

An application of Kendall distributions and alternative dependence measures: SPX vs. VIX

Most of the recently-defined notions of positive or negative dependence rely upon a variety of orderings of bivariate random vectors. These orderings are generally partial orders, and thus there are many pairs of random vectors which are not comparable. By using a weakened version of stochastic domination and the concepts of Kendall distributions and metacopulas, an entirely new class of orderings, in which the comparability issue is resolved, has been recently created. Each ordering in this class can be used to construct a measure of dependence. A detailed example will be given, using data from the Standard & Poor’s 500 index and Chicago Board of Trades index for implied volatility.     

On some contributions of John Horváth to the theory of distributions

Our main task is a presentation of J. Horváth's results concerning

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singular and hypersingular integral operators,

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the analytic continuation of distribution-valued meromorphic functions, and

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a general definition of the convolution of distributions.

At some instances minor supplements to his results are given.     

Weighted Young Inequalities for Convolutions

Sufficient conditions for some weighted Young inequalities type are obtained.AMS Subject Classification (1991): 42A85, 42B20, 47G10     

Hankel convolution operators on entire functions and distributions

In this paper we study the Hankel convolution operators on the space of even and entire functions and on Schwartz distribution spaces. We characterize the Hankel convolution operators as those ones that commute with Hankel translations and with a Bessel operator. Also we prove that the Hankel convolution operators are hypercyclic and chaotic on the spaces under consideration.     

A note on characterizations of multivariate stable distributions

Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.Research supported, in part, by the Air Force Office of Scientific Research under Contract AFOSR 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Some remarks on multivariate stable distributions

This paper deals with multivariate stable distributions. Press has given an explicit algebraic representation of characteristic functions of such distributions [J. Multivariate Analysis2 (1972), 444–462]. We present counter-examples and correct proofs of some of the statements of Press. The properties of multivariate stable distributions, connected with the spectral measure Γ, present in the expression of the characteristic function, are studied.     

A class of multivariate symmetric stable distributions

Multivariate symmetric stable characteristic functions and their properties, as well as conditions for independence and an analogue of the correlation coefficient in bivariate symmetric stable distributions, are discussed.     

A characterization of spherical distributions

It is shown that when the random vector X in Rⁿ has a mean and when the conditional expectation E(u′X|v′X) = 0 for all vectors u, v Rⁿ which satisfy u′v = 0, then the distribution of X is orthogonally invariant. A version of this characterization is also established when X does not have a mean vector.     

Some recent developments on complex multivariate distributions

In this paper, the author gives a review of the literature on complex multivariate distributions. Some new results on these distributions are also given. Finally, the author discusses the applications of the complex multivariate distributions in the area of the inference on multiple time series.     

Some new approaches to multivariate probability distributions

We extend and generalize to the multivariate set-up our earlier investigations related to expected remaining life functions and general hazard measures including representations and stability theorems for arbitrary probability distributions in terms of these concepts. (The univariate case is discussed in detail in Kotz and Shanbhag, Advan. Appl. Probab. 12 (1980), 903–921.)     

Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions

In this paper, the authors derived asymptotic expressions for the null distributions of the likelihood ratio test statistics for multiple independence and multiple homogeneity of the covariance matrices when the underlying distributions are complex multivariate normal. Also, asymptotic expressions are obtained in the non-null cases for the likelihood ratio test statistics for independence of two sets of variables and the equality of two covariance matrices. The expressions obtained in this paper are in terms of beta series. In the null cases, the accuracy of the first terms alone is sufficient for many practical purposes.     

Limit distributions of upper order statistics for families of multivariate distributions

We consider a portfolio of dependent exchangeable random variables , where the dependence structure is generated by a mixture model (Archimedean copulas belong to this class of models). Define the ordered sample . We prove results of the following type: fix and choose appropriately, then converges in distribution to a random vector as , for which we can explicitly give the distribution.