[Paper Review] An efficient and robust test for a change-point in correlation

Herold Dehling, Daniel Vogel+2 more|arXiv (Cornell University)|Mar 22, 2012

Statistical Methods and Inference6 citations

TL;DR

This paper proposes a nonparametric change-point test for correlation in bivariate time series using Kendall's tau, leveraging a novel U-statistic invariance principle for dependent processes. The method ensures robustness under heavy-tailed distributions and provides consistent estimation of the change-point location under a single change-point assumption.

ABSTRACT

For a bivariate time series $((X_i,Y_i))_{i=1,...,n}$ we want to detect whether the correlation between $X_i$ and $Y_i$ stays constant for all $i = 1,...,n$. We propose a nonparametric change-point test statistic based on Kendall's tau and derive its asymptotic distribution under the null hypothesis of no change by means a new U-statistic invariance principle for dependent processes. The asymptotic distribution depends on the long run variance of Kendall's tau, for which we propose an estimator and show its consistency. Furthermore, assuming a single change-point, we show that the location of the change-point is consistently estimated. Kendall's tau possesses a high efficiency at the normal distribution, as compared to the normal maximum likelihood estimator, Pearson's moment correlation coefficient. Contrary to Pearson's correlation coefficient, it has excellent robustness properties and shows no loss in efficiency at heavy-tailed distributions. We assume the data $((X_i,Y_i))_{i=1,...,n}$ to be stationary and P-near epoch dependent on an absolutely regular process. The P-near epoch dependence condition constitutes a generalization of the usually considered $L_p$-near epoch dependence, $p \ge 1$, that does not require the existence of any moments. It is therefore very well suited for our objective to efficiently detect changes in correlation for arbitrarily heavy-tailed data.

Motivation & Objective

• To develop a test that detects changes in correlation between two time series without assuming moment conditions.
• To ensure robustness against heavy-tailed distributions, where traditional correlation methods fail.
• To establish asymptotic distribution theory for the test statistic under the null hypothesis of no change.
• To propose a consistent estimator for the location of a single change-point in correlation.
• To generalize the framework to P-near epoch dependent processes, allowing for weak dependence without requiring finite moments.

Proposed method

• Uses Kendall's tau as the correlation measure due to its high efficiency under normality and robustness under heavy-tailed distributions.
• Applies a new U-statistic invariance principle tailored for dependent, stationary processes to derive the asymptotic distribution of the test statistic.
• Derives the asymptotic distribution under the null hypothesis, which depends on the long-run variance of Kendall's tau.
• Proposes a consistent estimator for the long-run variance of Kendall's tau to enable practical implementation.
• Operates under the P-near epoch dependence condition, a generalization of L_p-near epoch dependence that does not require existence of moments.
• Establishes consistency of the change-point location estimator under a single change-point model.

Experimental results

Research questions

• RQ1 Can a nonparametric test for correlation change be developed that remains valid under arbitrarily heavy-tailed distributions?
• RQ2 How can the asymptotic distribution of a Kendall's tau-based test statistic be derived under weak dependence assumptions?
• RQ3 Is the location of a single change-point in correlation consistently estimated using this test framework?
• RQ4 What is the role of the long-run variance of Kendall's tau in the asymptotic distribution and inference?
• RQ5 How does the P-near epoch dependence condition extend the applicability of the method beyond standard moment-based assumptions?

Key findings

• The proposed test statistic based on Kendall's tau has a well-defined asymptotic distribution under the null hypothesis of no change, derived via a new U-statistic invariance principle.
• The long-run variance of Kendall's tau is consistently estimated, enabling valid inference without requiring moment conditions.
• The method maintains high efficiency under normality and exhibits excellent robustness under heavy-tailed distributions.
• The location of a single change-point in correlation is consistently estimated under the assumed model.
• The P-near epoch dependence condition allows the method to be applied to a broad class of weakly dependent processes without requiring existence of finite moments.
• The test is robust and efficient, outperforming Pearson correlation in heavy-tailed settings while retaining high power under normality.