Abstract: In the present paper the authors first make a generliza tion of Lehmann and Rojo's theorem on the kinds of ordering relationships, then show that the i-th order statistic of (X₁,...,X_n) and the i-th order statistic of (Y₁,...,Y_n) keep the ordering relation ships between X and Y for i=1,...,n. At last the authors investigate the invariance of the kinds of ordering relationships under monotone transformations and the i nvariance of dispersive ordering under convex(concave) functions.

Key words: stochastic ordering, hazard rate ordering, reversed hazard rate orderi ng, likelihood ratio ordering, dispersive ordering