We derive a first order approximation for the probability that a general two-dimensional random walk crosses a circular boundary. Simulations are performed to assess the accuracy of the approximation, and statistical applications are discussed.

In a recent paper of He and Wang (1997, Ann. Statist. 25, 495-504), they considered depth contours based on data depth and they proved a uniform contour convergence theorem under some conditions on the depth measure. In this paper we prove n-1/2 rate of convergence of depth contours using...

We study asymptotic properties of location estimators based on the projection depth. A rigorous proof of the limiting distribution of projection median is provided using the result of Bai and He (Ann. Statist. 27 (1999) 1616) under a general setting. The rates of convergence of the trimmed mean...

This paper proposes a method to estimate the conditional quantile function using an epsilon-insensitive loss in a reproducing kernel Hilbert space. When choosing a smoothing parameter in nonparametric frameworks, it is necessary to evaluate the complexity of the model. In this regard, we provide...