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. (19 78). 8. 413 439), 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 (X-1, X-2,...,X-n) b e a multivariate normal vector with zero means, a common correlation p and variances sigma (2)(1), sigma (2)(2),..., sigma (2)(n) such that the parame ters p, sigma (2)(1), sigma (2,)(2)..., delta (2)(n) are unknown, but the d istribution of the max \ X-1 : 1 less than or equal to l less than or equal to n\ (or equivalently, the distribution of the min \X-1 : 1 less than or equal to i less than or equal to n\)is known. The problem is whether the pa rameters are identifiable and then how to determine the (unknown) parameter s in terms of the distribution of the maximum (or its density). Here, we so lve this problem for general n. Earlier, this problem was considered only f or n less than or equal to3. Identifiability problems in related contexts w ere 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 ot hers. We also consider here the case where the X-1's have a common covarian ce instead of a common correlation.