R-estimation in Linear Model with Possible Autoregressive Errors

• Institute of Information Theory and Automation of CAS, Prague, Czech Republic [jureckova@utia.cas.cz]

• Technical University of Liberec, Czech Republic [jan.picek@tul.cz]

Keywords: Autoregression rank scores – Nuisance Autoregression – Linear Model

Abstract

In the linear regression model, the standard assumption of the serial independence and identical distribution of model errors is often violated, while this cannot be immediately verified from the observations. To prevent an eventual subsequent distortion of the conclusion of the experiment, invokes a reason to admit a possible autoregressive structure of model errors, with say $p$ unknown autoregression parameters.

a et al. [2023] constructed the nonparameric test of significance of the regression, which is invariant to the nuisance parameters. In this way the authors reduced the effect of the nuisance parameters on the conclusions. The criterion of the test is based on the autoregression rank scores based only on the observations and are invariant on the autoregression parameters. However, when we wish to construct also a point estimator of the regression parameter, it is not sufficient to verify only the independence of the null distribution of the test criterion on the nuisance autoregression, but to investigate its behavior under the alternatives. It is solved by proving the asymptotic linearity of the autoregression rank scores with respect to the regression parameter. It enables to approximate the proposed estimator, to derive its asymptotic behavior, and obtain its invariance to the possible autoregression.