Spatial local linear estimation of the L₁-conditional quantiles for functional regressors

Abstract

L₁-norm approach is used to construct the local linear estimator of the spatial regression quantile for functional regressors. Under mixing spatial condition, we establish the almost complete convergence of the constructed approach. The applicability of the constructed estimator is examined by a Monte-Carlo study. The finite sample performance of the proposed estimator is compared to the classical kernel estimator of the functional spatial quantile regression. The result indicates that our new approach is more accurate than the classical one.

KEYWORDS AND PHRASES:

• Functional data analysis (FDA)
• small ball probability
• local linear method (LLM)
• spatial data
• quantile regression
• almost complete (a.co.) convergence

Subject classifications:

• 62G08
• 62G10
• 62G35
• 62G07
• 62G32
• 62G30. Secondary: 62H12

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which improved the quality of this article substantially. They also extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under the project number R.G.P1/102/40.

Notes

1 Let ${(z_n)}_{n\in \mathbb{N}}$ be a sequence of real r.v.’s. We say that z_n converges almost-completely (a.co.) toward zero if, and only if, $\forall ϵ>0,$ ${\sum }_{n=1}^{\infty }P(|z_n|>ϵ)<\infty .$ Moreover, we say that the rate of the almost complete convergence of z_n toward zero is of order u_n (with $u_n\to 0)$ and we write $z_n=O_{a.\mathit{co}.}(u_n)$ if, and only if, $\exists ϵ>0$ such that ${\sum }_{n=1}^{\infty }P(|z_n|>ϵu_n)<\infty .$ This kind of convergence implies both almost-sure convergence and convergence in probability.