Let [mu](· ; [Sigma], G1) and [mu](· ; [Omega], G2) be elliptically contoured measures on k centered at 0, having scale parameters ([Sigma], [Omega]) and radial cdf's (G1, G2). Elliptical measures vm(·) and vM(·), depending on ([Sigma], [Omega], G1, G2), are constructed such that...

Let X([omega]) be a random element taking values in a linear space 4 endowed with the partial order <=; let 10 be the class of nonnegative order-preserving functions on 4 such that, for each g[set membership, variant]10, E[g(X)] is defined; and let 11n10 be the subclass of concave functions. A version of Markov's inequality for such spaces in P(X >= x) <= inf10E[g(X)]/g(x). Moreover, if E(X) = [xi] is defined and if Jensen's inequality applies, we have a further inequality P(X>=x) <= inf11E[g(X)]/g(x) <= inf11g([xi])/g(x). Applications are given using a variety or orderings of interest in statistics and applied probability.

Distribution-free results beyond Gauss-Markov theory are found under weak assumptions regarding the errors. Symmetry, unimodality, and location-scale families are studied in estimation; nonstandard versions of Gauss-Markov results are given; and distribution-free confidence sets are tightened...

Consider a stochastic sequence {Zn; n=1,2,...}, and define Pn([var epsilon])=P(Zn<[var epsilon]). Then the stochastic convergence Zn-->0 is said to be monotone whenever the sequence Pn([var epsilon])[short up arrow]1 monotonically in n for each [var epsilon]0. This mode of convergence is investigated here; it is seen to be stronger than convergence...</[var>