Some applications of inequalities for extreme order statistics to a genetic selection problem

Two inequalities are derived for the extreme order statistics in a special case of unequally-correlated multivariate normal variables, and their implications in a genetic selection problem which was considered recently by Rawlings (1976, Biometrics 32, 875-887) and Hill (1976, Biometrics 32, 889-902; 1977, Biometrics 33, 703-712) are discussed. The inequalities establish a monotonicity property of the extreme order statistics as a function of the intraclass correlation, and they yield a chain of bounds when the family sizes are not all equal. Thus the inequalities extend the applicability of the results given by Rawlings and Hill to the more general case.