Abstract. The Lukacs theorem is one of the most brilliant results in the area of characterizations of probability distributions. First, because it gives a deep insight into the nature of independence properties of the gamma distribution; second, because it uses beautiful and non-trivial mathematics. Originally it was proved for probability distributions concentrated on (0, ∞). In 1962 Olkin and Rubin extended it to matrix variate distributions. Since that time it has been believed that the fundamental reason such an extension is possible, is the assumed property of invariance of the distribution of the "quotient" (properly defined for matrices). The main result of this paper is that the matrix variate Lukacs theorem holds without any invariance assumption for the "quotient". The argument is based on solutions of some functional equations in matrix variate real functions, which seem to be of independent interest. The proofs use techniques of differential calculus in the cone of positive definite symmetric matrices.
IL-6 has faster kinetics and is less influenced by patient weight, therefore it seems to be more useful in clinical practice. Summary measures describe the inflammatory response well and are easier to analyze than multiple analyses of single time points.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.