An identity for multivariate elliptically contoured matrix distribution

In this paper, we derive the Stein–Haff identity for the multivariate elliptically contoured matrix distributions. Our results generalize the results of the papers by [Stein, C., 1977. Personal communication. Unpublished notes on estimating the covariance matrix] and [Haff, L.R., 1979a. An identity for the Wishart distribution with applications. J. Multivariate Anal. 9, 531–542] for the much more general family of the matrix variate distributions. Moreover, we simplify the proof of the Stein–Haff identity given by [Kubokawa, T., Srivastava, M.S., 1999. Robust improvement in estimation of a covariance matrix in an elliptically contoured distribution. Ann. Statist. 27, 600–609] for the vector elliptically contoured matrix distribution in some important partial cases.