Identification of the Parameters of a Multivariate Normal Vector by the Distribution of the Maximum

This paper continues the work started by Basu and Ghosh (J. Mult. Anal. (1978), 8, 413–429), by Gilliland and Hannan (J. Amer. Stat. Assoc. (1980), 75, No. 371, 651–654), and then continued on by Mukherjea and Stephens (Prob. Theory and Rel. Fields (1990), 84, 289– 296), and Elnaggar and Mukherjea (J. Stat. Planning and Inference (1990), 78, 23–37). Let (X1, X2,..., Xn) be a multivariate normal vector with zero means, a common correlation (rho)and variances (sigma)2 1, (sigma)2 2,..., (sigma)2 n such that the parameters (rho), (sigma)2 1, (sigma)2 2,..., s2 n are unknown, but the distribution of the max{Xi: 1<=i<=n} (or equivalently, the distribution of the min{Xi: 1<=i<=n}) is known. The problem is whether the parameters are identifiable and then how to determine the (unknown) parameters in terms of the distribution of the maximum (or its density). Here, we solve this problem for general n. Earlier, this problem was considered only for n<=3. Identifiability problems in related contexts were considered earlier by numerous authors including: T. W. Anderson and S. G. Ghurye, A. A. Tsiatis, H. A. David, S. M. Berman, A. Nadas, and many others. We also consider here the case where the Xiʹs have a common covariance instead of a common correlation.