Volume 25, Issue 3 - 3

We consider estimation for a multivariate location family. Between all confidence regions with volume less than a fixed value L , we find the equivariant confidence region with the biggest coverage probability. This region maximizes the infimum of the coverage probability over all confidence regions with volume less than L . As an application, we find an estimator of a location parameter with the property that minimizes the supremum of the probability that the error of the estimation exceeds a fixed constant. We also find a confidence region and an estimator having the previous properties, but based on the maximum likelihood estimator. In the one dimensional case, we find the Bahadur slope of the two obtained estimators. We show that except for certain families of distributions, the estimator based on the whole sample is superior to the estimator based upon the maximum likelihood estimator. Hence, we get that maximum likelihood estimators are not asymptotically sufficient.

In this paper we derive a limit theorem for recursively defined processes. For several instances of recursive processes like for depth first search processes in random trees with logarithmic height or for fractal processes it turns out that convergence can not be expected in the space of continuous functions or in the Skorohod space D . We therefore weaken the Skorohod topology and establish a convergence result in L p spaces in which D is continuously imbedded. The proof of our convergence result is based on an extension of the contraction method. An application of the limit theorem is given to the FIND process. The paper extends in particular results in [HR00] on the existence and uniqueness of random fractal measures and processes. The depth first search processes of Catalan and of logarithmically growing trees do however not fit the assumptions of our limit theorem and lead to the so far unsolved problem of degenerate limits.

We study a model with an abrupt change in the mean and dependent errors that form a linear process. Different kinds of statistics are considered, such as maximum-type statistics (particularly different CUSUM procedures) or sum-type statistics. Approximations of the critical values for change-point tests are obtained through permutation methods in the frequency domain. The theoretical results show that the original test statistics and their corresponding frequency permutation counterparts follow the same distributional asymptotics. The main step in the proof is to obtain limit theorems for the corresponding rank statistics and then deduce the permutation asymptotics conditionally on the given data. Some simulation studies illustrate that the permutation tests usually behave better than the original tests if performance is measured by the α- and β-errors respectively.