Abstract: We consider a representation of the probability density function of a weighted convolution of the gamma distribution, where a confluent hypergeometric function describes how the differences between the parameters of the components of scale lead to departures from a density range. It is shown that the distributions can be characterized as the product between a gamma density and a confluent hypergeometric function. We give closed-form expressions for the cumulative, survival and hazard rate function. The corresponding moment generating function(m.g.f) and cumulant generating function(c.g.f) have been calculated and their properties have bean discussed.

Abstract: Correction on Moments of minors of Wishart matrices by M. Drton and A. Goia (Ann. Statist. 36 (2008) 2261-2283), arXiv:math/0604488

Abstract: Statistical tests not changed under an affine change of coordinate system are considered in the multivariate analysis. In the case of a multivariate linear model and a model using the canonical correlation analysis, these tests are functions of eigenvalues of matrices following a Wishart distribution. In this paper we prove the monotonicity property of test power functions being functions of elementary symmetric polynomials of eigenvalues of a matrix following a noncentral Wishart distribution.

Abstract: In this study, multivariate kernel density estimation has been investigated. Also, the applicability of multivariate kernel density function, estimation of two variable probability density function whose geometric presentation is possible has been shown by using the earthquake data in Marmara region.

Abstract: In this paper we consider some hypothesis tests within a family of Wishart distributions, where both the sample space and the parameter space are symmetric cones. For such testing problems, we first derive the joint density of the ordered eigenvalues of the generalized Wishart distribution and propose a test statistic analog to that of classical multivariate statistics for testing homoscedasticity of covariance matrix. In this generalization of Bartlett's test for equality of variances to hypotheses of real, complex, quaternion, Lorentz and octonion types of covariance structures.

Abstract: The objective of this paper is to define and study new generalized extended gamma functions. A generalized extended gamma probability density function involving generalized hypergeometric function is also defined. Closed form representations of the generalized gamma functions and the moment generating function are derived in the form of H-function using inverse Mellon transform techniques. Incomplete gamma-type functions and some special cases are discussed. Recurrence relations of generalized gamma functions are also discussed. New generalized gamma probability density function represents a unified form of several gamma and inverse Gaussian densities. Numerous well-known gamma-type functions and densities such as gamma, generalize gamma, two parameter Weibull, generalize Weibull, the Rayleigh, half-normal, Maxwell, generalize inverse Gaussian, negative binomial, chi-square, the Erlang distributions can also be obtained as special cases . Some statistical functions of a generalized gamma random variable are also derived. 2010 mathematics subject classification: 33C20 • 33E50 • 60E05

Abstract: This correspondence investigates the statistical properties of the ratio T = λ1/Σi=1mλi , where are λ1 ≥ λ2 ≥ ··· ≥ λm the m eigenvalues of an m × m complex central Wishart matrix W with n degrees of freedom. We derive new exact analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of T for complex central Wishart matrices with arbitrary dimensions. We also formulate simplified statistics of T for the special case of dual uncorrelated and dual correlated complex central Wishart matrices (m = 2) . The investigated ratio T is the most important ratio in blind spectrum sensing, since it represents a sufficient statistics for the generalized likelihood ratio test (GLRT). Thus, the derived analytical results are used to find the exact decision threshold for the desired probability of false alarm for Blind-GLRT (B-GLRT) detector. It is shown that the exact decision threshold based B-GLRT detector gives superior performance over the asymptotic decision threshold schemes proposed in the literature, which leads to efficient spectrum usage in cognitive radio.

Abstract: In this paper, we present and discuss the estimation of the Wishart Affine Stochastic Correlation (WASC) model introduced in Da Fonseca et al. (2006) under the historical measure. We review the main estimation possibilities for this continuous time process and provide elements to show that the utilization of empirical characteristic function-based estimates is advisable as this function is exponential affine in the WASC case. We thus propose to use the estimation strategy presented in Carrasco et al. (2003), using a continuum of moment conditions based on the characteristic function. We investigate the behavior of the estimates through Monte Carlo simulations. Then, we present the estimation results obtained using a dataset of equity indexes: SP500, FTSE, DAX and CAC. On the basis of these results, we show that the WASC captures many of the known stylized facts associated with financial markets, including the negative correlation between stock returns and volatility. It also helps reveal interesting patterns in the studied indexes'covariances and their correlation dynamics.