Volume 26, Issue 1 -

To test the hypothesis H 0 : f = ψ that an unknown density f is equal to a specified one, ψ , an estimate f ^ of f is compared with ψ . The total variation distance ∥ f ^- ψ ∥ 1 is used as test statistic. The density estimate f ^ considered is a peculiar one. A table of critical values is provided, this table is applicable for arbitrary ψ . Relations with other methods, Neyman´s smooth tests in particular, are discussed and power comparisons are performed. In certain situations, our test is recommendable. An example from practice is provided.

In this paper, we prove coincidence of uniform and individual convergence rates for convex density classes, which are Baire spaces with respect to the L 1 (R d )-metric, in the field of nonparametric estimation. We show that for convex density classes, which are no Baire spaces, the validity of this result is not guaranteed, in general.

Let X be a random variable taking values in a compact Riemannian manifold without boundary, and let Y be a discrete random variable valued in {0;1} which represents a classification label. We introduce a kernel rule for classification on the manifold based on n independent copies of ( X , Y ). Under mild assumptions on the bandwidth sequence, it is shown that this kernel rule is consistent in the sense that its probability of error converges to the Bayes risk with probability one.

In this paper comparison results of convex type are established for several path-dependent options in some classes of semimartingale models. The options considered are some classes of lookback options, Asian and American options and barrier options. Comparison of the path-dependent options is based on ordering properties of the local characteristics of the underlying processes as well as on suitable propagation of convexity property. These properties allow a stochastic analysis of the basic linking process which establishes a link between the value processes in the underlying models. The linking process gives a unified tool to obtain comparison results for these path-dependent options. This paper extends and unifies several results in the literature.