Regression rank scores in nonlinear models

Abstract

Consider the nonlinear regression model $Y_i=g({\bf x}_i,\boldmath $θ$)+e_i,\quad i=1,...,n$(1) with ${\bf x}_i\in \mathbb{R}^k,$ $\boldmathθ=(θ_0,θ_1,...,θ_p)^{\prime}\in \boldmath $Θ$$ (compact in $\mathbb{R}^{p+1}$), where $g({\bf x},\boldmath $θ$)=θ_0+\tilde{g}({\bf x},θ_1,...,θ_p)$ is continuous, twice differentiable in $\boldmath $θ$$ and monotone in components of $\boldmath $θ$$. Following Gutenbrunner and Jurečková (1992) and Jurečková and Procházka (1994), we introduce regression rank scores for model (1), and prove their asymptotic properties under some regularity conditions. As an application, we propose some tests in nonlinear regression models with nuisance parameters.