[Paper Review] U-quantile processes and generalized linear statistics of dependent data

Martin Wendler|arXiv (Cornell University)|Oct 12, 2010

Advanced Statistical Methods and Models27 references3 citations

TL;DR

This paper establishes the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for generalized linear (GL) statistics under weak dependence, extending prior results limited to independent data. By analyzing the empirical U-process via a generalized Bahadur representation, the authors derive asymptotic normality and functional laws for U-quantile processes, which directly imply the CLT and LIL for GL-statistics of strongly mixing or L^1 near epoch dependent data.

ABSTRACT

Generalized linear statistics are a unifying class that contains U-statistics, U-quantiles, L-statistics as well as trimmed and winsorized U-statistics. For example, many commonly used estimators of scale fall into this class. GL-statistics only have been studied under independence; in this paper, we establish the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for GL-statistics of sequences which are strongly mixing or L^1 near epoch dependent on an absolutely regular process. We first investigate the empirical U-process. With the help of a generalized Bahadur representation, the CLT and the LIL for the empirical U-quantile process follow. As GL-statistics are linear functionals of the U-quantile process, the CLT and the LIL for GL-statistics are straightforward corollaries.

Motivation & Objective

• To extend the asymptotic theory of generalized linear statistics beyond independent data to weakly dependent sequences.
• To address the lack of limit theorems for GL-statistics under dependence, particularly for scale estimators and trimmed/winsorized U-statistics.
• To establish the central limit theorem (CLT) and law of the iterated logarithm (LIL) for GL-statistics in strongly mixing or L^1 near epoch dependent processes.
• To derive the asymptotic distribution of U-quantile processes as a foundational step toward GL-statistics results.
• To unify the asymptotic theory of U-statistics, U-quantiles, L-statistics, and related estimators under weak dependence.

Proposed method

• The authors analyze the empirical U-process as the core stochastic process underlying GL-statistics.
• They employ a generalized Bahadur representation to link the empirical U-quantile process to underlying distribution functions and errors.
• The weak dependence structure—specifically strong mixing or L^1 near epoch dependence on an absolutely regular process—is formally incorporated into the asymptotic analysis.
• The CLT and LIL for the empirical U-quantile process are derived using functional central limit theory and moment bounds under the dependence assumptions.
• GL-statistics are treated as linear functionals of the U-quantile process, allowing direct transfer of limit laws.
• Theoretical results are established through martingale approximations and weak convergence techniques for dependent arrays.

Experimental results

Research questions

• RQ1 Does the central limit theorem hold for generalized linear statistics when the underlying data are weakly dependent rather than independent?
• RQ2 Can the law of the iterated logarithm be extended to GL-statistics under strong mixing or L^1 near epoch dependent processes?
• RQ3 How can the generalized Bahadur representation be adapted to U-quantile processes under weak dependence to enable asymptotic analysis?
• RQ4 What is the limiting distribution of the empirical U-quantile process under weak dependence assumptions?
• RQ5 Are GL-statistics, including trimmed and winsorized U-statistics, asymptotically normal under weak dependence?

Key findings

• The central limit theorem holds for generalized linear statistics under strong mixing or L^1 near epoch dependent processes.
• The law of the iterated logarithm is established for generalized linear statistics in the same weak dependence framework.
• The empirical U-quantile process satisfies a functional central limit theorem under the specified dependence conditions.
• The generalized Bahadur representation enables the derivation of limit laws for U-quantile processes from weakly dependent data.
• GL-statistics inherit the asymptotic normality and functional laws from the U-quantile process via linear functional forms.
• The results unify and extend the asymptotic theory of U-statistics, U-quantiles, L-statistics, and related estimators to weakly dependent data.