Rank-Weighted, Two-Sample U-Statistics with Application to Partial Area Under ROC Curves

• Activity Number: 149
• Type: Contributed
• Date/Time: Monday, July 30, 2007 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Chengqing (Alan) Wu*+ and Aiyi Liu and Kai Fun Yu
• Companies: National Institutes of Health and National Institutes of Health and National Institutes of Health
• Keywords: Rank Statistics ; U-Statistics ; Roc curve ; Area under ROC curves

Abstract

A rank weighted two sample U-statistic based on two independent random samples $X_1,\cdots,X_m$ and $Y_1,\cdots, Y_n$ has the form $U_{m,n}=\frac1{mn}\sum_{i=1}^m \sum_{j=1}^n K(X_i,Y_j)J(\frac{R_{xi}}m,\frac{R_{yj}}n) $ where $K$ is fixed measurable function ,$J:[0,1]^2 \rightarrow {\mathcal R} $ is a bounded measurable function and $R_{Xi}$ is the rank order of $X_i$ among $X$'s, $R_{yj}$ is the rank order of $Y_j$ among $Y$'s. A large class of statistics can be expressed as this kind U-statistics or variations thereof. This paper investigates the asymptotic properties of $U_{m,n}$.