Regression rank scores in nonlinear models

TL;DR

This paper introduces regression rank scores for nonlinear models, analyzes their asymptotic properties, and develops tests for nonlinear regression with nuisance parameters.

Contribution

It extends the concept of regression rank scores to nonlinear models and establishes their asymptotic behavior, enabling new testing methods.

Findings

Regression rank scores are asymptotically normal.

New tests for nonlinear regression with nuisance parameters are proposed.

The methods are based on regularity conditions and monotonicity assumptions.

Abstract

Consider the nonlinear regression model $Y_i=g({\bf x}_i,\boldmath $\theta$)+e_i,\quad i=1,...,n$(1) with ${\bf x}_i\in \mathbb{R}^k,$ $\boldmath{\theta}=(\theta_0,\theta_1,...,\theta_p)^{\prime}\in \boldmath $\Theta$$ (compact in $\mathbb{R}^{p+1}$), where $g({\bf x},\boldmath $\theta$)=\theta_0+\tilde{g}({\bf x},\theta_1,...,\theta_p)$ is continuous, twice differentiable in $\boldmath $\theta$$ and monotone in components of $\boldmath $\theta$$. Following Gutenbrunner and Jure\v{c}kov\'{a} (1992) and Jure\v{c}kov\'{a} and Proch\'{a}zka (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.