DISTRIBUTION-FREE CONFIDENCE PROCEDURE FOR UMBRELLA ORDERINGS

Responses in a one-factor experiment with A; ordered treatments follow an umbrella, ordering if they consist of two piecewise monotone segments, i.e. increasing and then decreasing, or the converse. This paper proposes a non-parametric distribution-free confidence procedure for umbrella orderings, the aim being to identify the treatments that correspond to the optimal effects. It uses a method that joins the seemingly unrelated theories of U-statistics and isotonic regression. A random confidence subset of the ordered treatments is constructed, such that it contains all the unknown peaks (optimal treatments) of an umbrella ordering with any prespecified confidence level. The paper demonstrates that the proposed confidence procedure is nonparametric distribution-free and, further, that the proposed procedure naturally implies a test for umbrella alternatives. Since the proposed confidence procedure is always more informative than tests for umbrella alternatives, it should be used in their place in practice. An example illustrates the proposed procedure.