Two-Sample U-Statistic Processes for Long-Range Dependent Data

Abstract

Motivated by some common-change point tests, we investigate the asymptotic distribution of the U-statistic process $U_n(t)={\sum }_{i=1}^{[nt]}{\sum }_{j=[nt]+1}^{n}h(X_i,X_j)$, $0\le t\le 1$, when the underlying data are long-range dependent. We present two approaches, one based on an expansion of the kernel $h(x,y)$ into Hermite polynomials, the other based on an empirical process representation of the U-statistic. Together, the two approaches cover a wide range of kernels, including all kernels commonly used in applications.

References in corpus (1)

• Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes