Extensions of Some Statistical Concepts to the Complex Domain

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under Hermitian-form constraints, and similar maxima/minima problems in the complex domain are discussed. Some vector/matrix differential operators are developed to handle the above types of problems. These operators in the complex domain and the optimization problems in the complex domain are believed to be new and novel. These operators will also be useful in maximum likelihood estimation problems, which will be illustrated in the concluding remarks. Detailed steps are given in the derivations so that the methods are easily accessible to everyone.