The asymptotic behaviors for autoregression quantile estimates

ABSTRACT

This article is concerned with the asymptotic theory of estimates of unknown parameters in autoregressive quantile processes. We assume random errors form a strictly stationary $\varphi$-mixing sequences. In view of the approach of argmins and blocking argument, we prove the parameter estimators satisfy the functional moderate deviation principle (MDP). Further, we give the law of the iterated logarithm under some standard conditions. Based on the contraction principle, the moderate deviation principles of L-estimators on the autoregression quantile (ARQ) and autoregression rank scores (ARRS’s) are also discussed. This method can be extended to a fair range of different statistical estimation problems.

KEYWORDS:

• Quantile autoregression
• reproducing kernel Hilbert space
• $\varphi$-mixing
• autoregression rank scores
• moderate deviations

MR(2000) Subject classification:

• 60F10
• 60F05
• 62M10

Acknowledgments

The authors are very grateful to two anonymous referees for their valuable comments which improve the presentation of this work.

Additional information

Funding

Project supported by the Fundamental Research Funds for the Central Universities, China University of Geosciences(Wuhan)(CUGSX01).