Mixtures of traces of Wishart and inverse Wishart matrices

AbstractTraces of Wishart matrices appear in many applications, for example in finance, discriminant analysis, Mahalanobis distances and angles, loss functions and many more. These applications typically involve mixtures of traces of Wishart and inverse Wishart matrices that are concerned in this paper. Of particular interest are the sampling moments and their limiting joint distribution. The covariance matrix of the marginal positive and negative spectral moments is derived in closed form (covariance matrix of Y=[p?1Tr{W?1},p?1Tr{W},p?1Tr{W2}]?, where W?Wp(Σ=I,n)). The results are obtained through convenient recursive formulas for E[?i=0kTr{W?mi}] and E[Tr{W?mk}?i=0k?1Tr{Wmi}]. Moreover, we derive an explicit central limit theorem for the scaled vector Y, when p/n?d<1,p,n?∞, and present a simulation study on the convergence to normality and on a skewness measure