Limit theorems for nondegenerate U-statistics of continuous semimartingales

This paper presents the asymptotic theory for nondegenerate U-statistics of high frequency observations of continuous Itô semimartingales. We prove uniform convergence in probability and show a functional stable central limit theorem for the standardized version of the U-statistic. The limiting process in the central limit theorem turns out to be conditionally Gaussian with mean zero. Finally, we indicate potential statistical applications of our probabilistic results.

Tools & Techniques - Statistics: Propensity score techniques.

Propensity score (PS) techniques are useful if the number of potential confounding pretreatment variables is large and the number of analysed outcome events is rather small so that conventional multivariable adjustment is hardly feasible. Only pretreatment characteristics should be chosen to derive PS, and only when they are probably associated with outcome. A careful visual inspection of PS will help to identify areas of no or minimal overlap, which suggests residual confounding, and trimming of the data according to the distribution of PS will help to minimise residual confounding. Standardised differences in pretreatment characteristics provide a useful check of the success of the PS technique employed. As with conventional multivariable adjustment, PS techniques cannot account for confounding variables that are not or are only imperfectly measured, and no PS technique is a substitute for an adequately designed randomised trial.

Directional statistics: introduction

In many of the natural and physical sciences, measurements are directions, either in two or three dimensions. The analysis of directional data relies on specific statistical models and procedures, which differ from the usual models and methodologies of Cartesian data. This chapter briefly introduces statistical models and inference for this type of data. The basic von Mises-Fisher distribution is introduced and nonparametric methods such as goodness-of-fit tests are presented. Further references are given for exploring related topics such as correlation and regression.