Abstract
In this article, we employ the jackknife method of estimation and the concept of Studentized $U$-statistics to derive a new weak convergence result for nondegenerate $U$-statistics on the space $D[0,1]$. We drop the classical condition that the second moment of the kernel of the underlying $U$-statistic exists and derive a weighted weak convergence result for these Studentized statistics. This weak convergence is concluded from a weighted approximation in sup-norm $\|/q\|$, in probability, of Studentized $U$-statistics by partial sums of i.i.d. projections, where $q$ is in an appropriate class of positive weight functions.
Citation
Masoud M. Nasari. "Weighted approximations for Studentized $U$-statistics." Braz. J. Probab. Stat. 28 (1) 24 - 60, February 2014. https://doi.org/10.1214/12-BJPS192
Information