Volume 21, Issue 4 - 4

The present paper studies envelope power functions for goodness of fit models for asuitable submodel of infinite dimension of all continuous distributions on the real line. It turns out that after rescaling our alternatives with the factor 1/√ n various envelope power bounds hold uniformly w.r.t. sample size n . The two-sided Neyman-Pearson power, the maximin and optimum mean power bounds of dimension d are studied in detail. It is shown that the latter envelope power bounds become flat for high dimensions d of alternative if they are compared with the power of Neyman-Pearson tests which serve as benchmark. These results can be used to compare intermediate and Pitman efficiency of goodness of fit tests. It is pointed out that contiguous alternatives can be used to discriminate competing tests whereas the intermediate efficiency is some sort of consistency only. It is also pointed out that no overall superior adaptive goodness of fit test exist. Agood comparison of competing tests can be done by their level points. It is shown that the level points of the maximin tests of dimension d grow with the rate d 1/4 .

We consider the estimation of Σ of the p -dimensional normal distribution N p (0, Σ ) under the restriction where the eigenvalues of Σ have an upper or lower bound. From a decision-theoretic point of view, we evaluate the performance of the REML (restricted maximum likelihood estimator) with Stein′s loss function and propose another estimator that dominates the REML.

An important problem of the statistical analysis of time series is to detect change-points in the mean structure. Since this problem is a one-dimensional version of the higher dimensional problem of detecting edges in images, we study detection rules which benefit from results obtained in image processing. For the sigma-filter studied there to detect edges, asymptotic bounds for the normed delay have been established for independent data. These results are considerably extended in two directions. First, we allow for dependent processes satisfying acertain conditional mixing property. Second, we allow for more general pilot estimators, e.g., the median, resulting in better detection properties. A simulation study indicates that our new procedure indeed performs much more better.

Suppose mutually independent observations are drawn from absolutely continuous, unimodal, symmetric distributions with an order restriction on the medians, μ 0 ≤ min{ μ 1 , μ 2 ,..., μ m }. An isotonic regression estimator is shown to stochastically dominate the marginal sample median when estimating μ 0 , under some regularity conditions. These conditions allow the tails of the first population (i.e., the population with median μ 0 ) to be quite heavy, whereas the tails of the remaining distributions are required to be relatively light. Examples involving the Cauchy and Laplace distributions are shown to satisfy these regularity conditions. Counterexamples illustrate the importance of these regularity conditions for proving stochastic domination. The results expressed herein are theoretical advancements in order restricted inference.