Sample covariance matrices are also of essential importance in multivariate statistical analysis because many test statistics involve their eigenvalues and/or eigenvectors. The typical example is T 2 statistic, which was proposed by Hotelling [2]. We refer to [1] and [3] for various uses of the T 2 statistic. The T 2 statistic, which is the origin of multivariate linear hypothesis tests and the...

We define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I (n)denote the set of all involutions on [n](= {1, 2, . . . , n}) and let F(2n)denote the set of all fixed point free involutions on [2n]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let [n]q = 1+q+· · ·+qn−1 and let (k)q denote the q-binomial coefficient....