Estimation of a change point in the variance function based on the χ²-distribution

ABSTRACT

Let us consider that the variance function or its νth derivative in a regression model has a change/discontinuity point at an unknown location. To use the local polynomial fits, the log-variance function which break the positivity is targeted. The location and the jump size of the change point are estimated based on a one-sided kernel-weighted local-likelihood function which is provided by the χ²-distribution. The whole structure of the log-variance function is then estimated using the data sets split by the estimated location. Asymptotic results of the proposed estimators are described. Numerical works demonstrate the performances of the methods with simulated and real examples.

Keywords:

• Discontinuity point
• Likelihood function
• Local polynomial fit
• One-sided kernel
• Rate of convergence

Mathematics Subject Classification:

• Primary 62G08
• Secondary 62G20

Acknowledgments

The author is very grateful to the editor and the referee for their many helpful comments, which helped to improve the quality of the article.

Funding

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2011-0010856).