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

Sana Louhichi (1) , Ryozo Miura (2) , Dalibor Volny (3)

1 IPS - Inférence Processus Stochastiques
2 Hitotsubashi University
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

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].

Mots clés: mixing; asymptotic normality; weighted empirical processes; Robust estimators; r-estimators; linear models; rank statistics; dependent errors; association

Domaines: Statistiques [math.ST]

Sana Louhichi, Ryozo Miura, Dalibor Volny. On the asymptotic normality of the R-estimators of the slope parameters of simple linear regression models with associated errors. Statistics, 2017, Special Issue in Honour of Paul Doukhan: Selected Papers from the Conference on the Occasion of Paul Doukhan’s 60th Birthday: Dependence, Limit Theorems and Applications, 51 (1), pp.167-187. ⟨10.1080/02331888.2016.1261912⟩. ⟨hal-01892322⟩