On the asymptotic normality of the R-estimators of the slope parameters of simple linear regression models with associated errors

ABSTRACT

The purpose of this paper is to prove, under mild conditions, the asymptotic normality of the rank estimator of the slope parameter of a simple linear regression model with stationary associated errors. This result follows from a uniform linearity property for linear rank statistics that we establish under general conditions on the dependence of the errors. We prove also a tightness criterion for weighted empirical process constructed from associated triangular arrays. This criterion is needed for the proofs which are based on that of Koul [Behavior of robust estimators in the regression model with dependent errors. Ann Stat. 1977;5(4):681–699] and of Louhichi [Louhichi S. Weak convergence for empirical processes of associated sequences. Ann Inst Henri Poincaré Probabilités Statist. 2000;36(5):547–567].

KEYWORDS:

• Robust estimators
• r-estimators
• linear models
• rank statistics
• dependent errors
• association
• mixing
• asymptotic normality
• weighted empirical processes

Acknowledgments

The authors are very grateful to the anonymous referee for her/his constructive comments that helped them in improving the content of this paper. The paper was completed when the corresponding author, Sana Louhichi, was visiting the Department of Mathematics of the University of Gabès to which she is very grateful for the warm hospitality and the good working conditions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Monotonicity of $S_x\left(\mathrm{\Delta }\right)$ as a function of Δ is discussed without using any probability argument: without iid assumptions and also without an underlying distribution F.