Concentration probabilities on ellipsoids are used to gauge efficiency in multiparameter estimation. Chebychev bounds support a distribution-free approach. The concepts are used to quantify gains in efficiency due to unbiasedness and to conditioning on another statistic.

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.

Control charts are given for monitoring an input-output model against changes in form, against changes in its coefficients, and against changes in process variance. When a process is not in control due to changes in some coefficients, monitoring shifts to a diagnostic mode to identify the...

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...

It is shown that randomization applies in repeated measurements whenever the F-test applies in the analysis. Contrary to common practice, randomization otherwise invalidates the use of Hotelling's T2 and Box's approximate test, and it presents difficulties in testing that dispersion matrices...

Let be a Studentizing transformation of the type T(Y) = [nu]A'Y/||B'Y||, and let GM(n) consist of scale mixtures of Gaussian laws on [bumpy equals]k; 1 [less-than-or-equals, slant] k [less-than-or-equals, slant] n[bumpy equals]. It is shown that GM(n) is closed under T(·). Concentration...

Suppose that experimental designs have comparable A-efficiencies for estimation, the choice among designs then depending on their comparative D-efficiencies. Using the theory of majorization, it is shown that designs having greater D-efficiencies are more ill-conditioned, using condition numbers...