OPTIMIZATION OF FUNCTIONS OF MATRICES WITH AN APPLICATION IN STATISTICS

The behavior of special differentiable real-valued functions defined o n a set of matrices is examined. We call these functions (M; U) functi ons. They are intrinsically interesting and can provide useful inequal ities. We give a theorem helping to find their global extrema The sear ch for these extrema is performed through classical techniques using d ifferential calculus. We concentrate on (M; U) functions that are impo rtant in statistics and econometrics. Their global etremizers are expl icitly given. These functions are useful in the minimization of mean s quared errors or variances of quadratic forms y'Ay when y has a multiv ariate normal or, more generally, an elliptically contoured distributi on. Finally, an illustrative application to the estimation of the cova riance matrix in a linear regression model is given.