Tamas Varga

Elliptically contoured models in statistics

Series Editors Preface. Preface. 1. Preliminaries. 2. Basic Properties. 3. Probability Density Function and Expected Values. 4. Mixtures of Normal Distributions. 5. Quadratic Forms and other Functions of Elliptically Contoured Matrices. 6. Characterization Results. 7. Estimation. 8. Hypothesis Testing. 9. Linear Models. References. Author Index. Subject Index.

Elliptically contoured models in statistics and portfolio theory

Preliminaries.- Basic Properties.- Probability Density Function and Expected Values.- Mixtures of Normal Distributions.- Quadratic Forms and other Functions of Elliptically Contoured Matrices.- Characterization Results.- Estimation.- Hypothesis Testing.- Linear Models.- Skew Elliptically Contoured Distributions.- Application in Portfolio Theory.- Author Index.- Subject Index.

Characterization of matrix variate normal distributions

In this paper, it is shown that two random matrices have a joint matrix variate normal distribution if, conditioning each one on the other, the resulting distributions satisfy certain conditions. A general result involving more than two matrices is also proved.

A NEW CLASS OF MATRIX VARIATE ELLIPTICALLY CONTOURED DISTRIBUTIONS

A new class of matrix variate elliptically contoured distributions is defined. Properties of this class of distributions are studied. Examples of distributions which belong to this class are also presented.

Rank of a quadratic form in an elliptically contoured matrix random variable

In this paper it is shown that the quadratic form in an elliptically contoured matrix variate has a constant rank and its nonzero eigenvalues are distinct with probability one if the matrix distribution satisfies certain conditions and the matrix defining the quadratic form is symmetric.

Some Inference Problems for Matrix Variate Elliptically Contoured Distributions

In this paper, maximum likelihood estimators for a broad class of matrix variate elliptically contoured distributions have been derived. Optimality properties of estimators in the multivariate regression models are studied when the error term has a matrix variate elliptically contoured distribution. Least-square estimators are also obtained for the multivariate random effect regression model when the underlying distribution is elliptically contoured.

An introduction to actuarial mathematics

Preface. 1. Financial Mathematics. 2. Mortality. 3. Life Insurances and Annuities. 4. Premiums. 5. Reserves. Answers to Odd-Numbered Problems. Appendix 1: Compound Interest Tables. Appendix 2: Illustrative Mortality Table. References. Symbol Index. Subject Index.

Skew Elliptically Contoured Distributions

Various multivariate skew normal distributions have been proposed in the literature, with each one of them aiming to characterize a particular aspect of a given phenomenon.For example, one emphasizes invariance under quadratic forms, another one uses a general latent structure to define distributions, etc. In this chapter, we deal with matrix variate closed skew normal distributions. Their distributional properties are presented and the extension to matrix variate closed skew elliptically contoured distributions is suggested. Finally, an application to portfolio theory is provided.

Application in Portfolio Theory

The mean-variance analysis of Markowitz (1952) is important for both practitioners and researchers in finance. This theory provides an easy access to the problem of optimal portfolio selection. We consider the estimator for the optimal portfolio weights and the characteristics of the efficient frontier. Furthermore, an exact test for the weights of the global minimum variance portfolio is presented as well as the inferences for Markowitz’s efficient frontier are provided. Finally, an unbiased estimator of the efficient frontier is derived and an overall-F-test is suggested.

Mixtures of Normal Distributions

Muirhead (1982) gave a definition of scale mixture of vector variate normaldistributions. Using Corollary 2.6, the scale mixture of matrix variate normal distributionsis defined in this chapter. Furthermore, we present another way to obtain the p.d.f. of a matrix variate elliptically contoured distribution from the density functions of matrix variate normal distributions. For this purpose, Laplace transform is used.