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The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and j th largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes. [ABSTRACT FROM AUTHOR]

This paper proposes several methodologies whose objective consists of securing copula density estimates. More specifically, this aim will be achieved by differentiating bivariate least-squares polynomials fitted to Deheuvels' empirical copulas, by making use of Bernstein's approximating polynomials of appropriately selected orders; by differentiating linearized distribution functions evaluated at optimally spaced grid points; and by implementing the kernel density estimation technique in conjunction with a repositioning of the pseudo-observations and a certain criterion for determining suitable bandwidths. Smoother representations of such density estimates can further be secured by approximating them by means of moment-based bivariate polynomials. The various copula density estimation techniques being advocated herein are successfully applied to an actual dataset as well as a random sample generated from a known distribution. [ABSTRACT FROM AUTHOR]

The implications of a system's integrity, safety and failure modes were studied by using generalized model referred as q-analogs of the distribution of origin. It was shown graphically that the proposed univariate distributions could generate a wide array of unimodal or symmetric bimodal curves.

This paper discusses the fitting of suitable models to rainfall observations. Daily rainfall amounts were aggregated to monthly data using the Thiessen polygons method and multivariate seasonal vector integrated autoregressive moving average models (sVARIMA) were fitted to the monthly cumulative rainfall volume. The data were obtained from the 12 Palestinian meteorological gauge stations located across the 5 governorates of the Gaza Strip and incorporated 42 years (from 1973 to 2014) of irregular daily precipitation. It can be concluded that the use of sVARIMA models in the environmental sciences provide a useful approach for forecasting rainfall data as a preliminary guideline toward short and long-term sustainable water resources management. [ABSTRACT FROM AUTHOR]

This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are readily computable. Some numerical examples corroborate the validity of the results derived herein. [ABSTRACT FROM AUTHOR]