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 $U_n\left(t\right)=\sum _{i=1}^{\left[nt\right]}\sum _{j=\left[nt\right]+1}^{n}h(X_i,X_j),0\le t\le 1,$ 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.

KEYWORDS:

• U-statistic processes
• long-range dependence
• Hermite polynomials
• empirical processes
• change-point problems
• non-central limit theorems

AMS SUBJECT CLASSIFICATION:

• 60F17
• 62G30
• 62M10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research has been partially supported by the Collaborative Research Center SFB 823 Statistical Modeling of Nonlinear Dynamic Processes of the DFG (German Research Foundation).