Optimization of functions of matrices with an application in statistics Original Research Article

The behavior of special differentiable real-valued functions defined on a set of matrices is examined. We call these functions (M; U) functions. They are intrinsically interesting and can provide useful inequalities. We give a theorem helping to find their global extrema. The search for these extrema is performed through classical techniques using differential calculus. We concentrate on (M; U) functions that are important in statistics and econometrics. Their global extremizers are explicitly given. These functions are useful in the minimization of mean squared errors or variances of quadratic forms y′ Ay when y has a multivariate normal or, more generally, an elliptically contoured distribution. Finally, an illustrative application to the estimation of the covariance matrix in a linear regression model is given.