Some Nonparametric Tests for Change-Point Detection Based on the ℙ-ℙ and ℚ-ℚ Plot Processes

Abstract

We propose nonparametric procedures for testing change-point by using the ℙ-ℙ and ℚ-ℚ plots processes. The limiting distributions of the proposed statistics are characterized under the null hypothesis of no change and also under contiguous alternatives. We give an estimator of the change-point coefficient and obtain its strong consistency. We introduce the bootstrapped version of ℙ-ℙ and ℚ-ℚ processes, requiring the estimation of quantile density, and obtain their limiting laws. Finally, we propose and investigate the exchangeable bootstrap of the empirical ℙ-ℙ plot and ℚ-ℚ plot processes which avoids the problem of the estimation of quantile density, which is of its own interest. These results are used for calculating p-values of the proposed test statistics. Emphasis is placed on the explanation of the strong approximation methodology.

Keywords:

• Bootstrap
• Brownian bridges
• Change-points
• Exchangeability
• Kernel-type-estimator
• Kiefer processes
• Nonparametric inference
• ℙ-ℙ and ℚ-ℚ plots
• Quantile and empirical processes
• Strong approximation
• Uniform consistency

Subject Classifications:

• 60F17
• 60F15
• 60G15
• 62G10
• 62G20
• 62H10
• 62H15

ACKNOWLEDGMENTS

The authors are indebted to the Editor Nitis Mukhopadhyay and an Associate Editor.

Notes

Recommended by S. Lee