Statistical inference in nonlinear regression under heteroscedasticity

Abstract

Nonlinear regression models are commonly used in toxicology and pharmacology. When fitting nonlinear models for such data, one needs to pay attention to error variance structure in the model and the presence of possible outliers or influential observations. In this paper, an M-estimation based procedure is considered in heteroscedastic nonlinear regression models where the standard deviation is modeled by a nonlinear function. The methodology is illustrated using toxicological data.

Appendix: Notations in Theorem 1

We state the definitions of the matrices and related quantities using in Theorem 1 as follows:

• (i)
  $$ \boldsymbol{\Gamma}_{3}({\boldsymbol \theta}, {\boldsymbol \tau})=\left( \begin{array}{cc} \boldsymbol{\Gamma}_{31}({\boldsymbol \theta},{\boldsymbol \tau}) & \mathbf{0} \\ \mathbf{0} & \boldsymbol{\Gamma}_{32}({\boldsymbol \theta},{\boldsymbol \tau}) \end{array}\right), $$

  where ${\boldsymbol \Gamma}_{31}({\boldsymbol \theta},{\boldsymbol \tau})=\lim_{n\rightarrow\infty}\frac{1}{n}{\boldsymbol \Gamma}_{31n}({\boldsymbol \theta},{\boldsymbol \tau}),$ ${\boldsymbol \Gamma}_{32}({\boldsymbol \theta},{\boldsymbol \tau})=\lim_{n\rightarrow\infty}\frac{1}{n}{\boldsymbol \Gamma}_{32n}({\boldsymbol \theta},{\boldsymbol \tau}),$

  $$ {\boldsymbol \Gamma}_{31n}({\boldsymbol \theta},{\boldsymbol \tau}) =\sigma_{\psi1}^2\sum_{i=1}^n u(\mathbf{x}_i)k^2(\mathbf{z}_i,{\boldsymbol \tau}) \mathbf{f}_{{\boldsymbol \scriptstyle \theta}} (\mathbf{x}_i,{\boldsymbol \theta}) \mathbf{f}^t_{{\boldsymbol \scriptstyle \theta}} (\mathbf{x}_i,{\boldsymbol \theta}), $$
  $$ {\boldsymbol \Gamma}_{32n}({\boldsymbol \theta},{\boldsymbol \tau}) =\sigma_{\psi2}^2\sum_{i=1}^n v(\mathbf{x}_i)k^2(\mathbf{z}_i,{\boldsymbol \tau}) {\boldsymbol \sigma}_{{\boldsymbol \scriptstyle \tau}} (\mathbf{z}_i,{\boldsymbol \tau}) {\boldsymbol \sigma}^t_{{\boldsymbol \scriptstyle \tau}} (\mathbf{z}_i,{\boldsymbol \tau}), $$

  $\sigma_{\psi1}^2u(\mathbf{x})=E\psi^2(\epsilon)(<\infty),$ $\sigma_{\psi2}^2v(\mathbf{x})=\mbox{Var}\{\psi(\epsilon)\epsilon\}(<\infty),$ and $\epsilon=\frac{y-f(\mathbf{x},{\boldsymbol \scriptstyle\theta})} {\sigma(\mathbf{z},{\boldsymbol \scriptstyle\tau })}.$

• (ii)
  $$ \boldsymbol{\Gamma}_{5}({\boldsymbol \theta},{\boldsymbol \tau})=\left( \begin{array}{cc} \boldsymbol{\Gamma}_{1}({\boldsymbol \theta},{\boldsymbol \tau}) & \mathbf{0} \\ \mathbf{0} & \boldsymbol{\Gamma}_{2}({\boldsymbol \theta},{\boldsymbol \tau}) \end{array}\right), $$

  where ${\boldsymbol \Gamma}_1({\boldsymbol \theta},{\boldsymbol \tau})=\lim_{n\rightarrow\infty}\frac{1}{n}{\boldsymbol \Gamma}_{1n}({\boldsymbol \theta},{\boldsymbol \tau}),$ ${\boldsymbol \Gamma}_2({\boldsymbol \theta},{\boldsymbol \tau})=\lim_{n\rightarrow\infty}\frac{1}{n}{\boldsymbol \Gamma}_{2n}({\boldsymbol \theta},{\boldsymbol \tau}),$

  $$ {\boldsymbol \Gamma}_{1n}({\boldsymbol \theta},{\boldsymbol \tau}) =\gamma_{2}\sum_{i=1}^n k^2(\mathbf{z}_i,{\boldsymbol \tau}) \mathbf{f}_{{\boldsymbol \scriptstyle\theta}} (\mathbf{x}_i,{\boldsymbol \theta}) \mathbf{f}^t_{{\boldsymbol \scriptstyle\theta}} (\mathbf{x}_i,{\boldsymbol \theta}), $$
  $$ {\boldsymbol \Gamma}_{2n}({\boldsymbol \theta},{\boldsymbol \tau}) =\sum_{i=1}^n \left\{\frac{2\gamma_{1}+\gamma_3-1} {\sigma^2(\mathbf{z}_i,{\boldsymbol \tau})} {\boldsymbol \sigma}_{{\boldsymbol \scriptstyle \tau}} (\mathbf{z}_i,{\boldsymbol \tau}) {\boldsymbol \sigma}^t_{{\boldsymbol \scriptstyle \tau}} (\mathbf{z}_i,{\boldsymbol \tau}) +\frac{1-\gamma_1}{\sigma(\mathbf{z}_i,{\boldsymbol \tau})} \boldsymbol{\Sigma}_{{\boldsymbol \scriptstyle \tau}} (\mathbf{z}_i,{\boldsymbol \tau})\right\}, $$

  $\boldsymbol{\Sigma}_{{\boldsymbol \scriptstyle\tau}} (\mathbf{z}_i,{\boldsymbol \tau}) =(\partial^2/\partial{\boldsymbol \tau} \partial{\boldsymbol \tau}^t)\sigma(\mathbf{z}_i,{\boldsymbol \tau}),$ γ ₁ = E{ψ(ϵ)ϵ}( ≠ 0), γ ₂ = Eψ′(ϵ)( ≠ 0), and $\gamma_3=E\{\psi'(\epsilon)\epsilon^2\}(\neq 0).$

• (iii)
  $$ {\boldsymbol \nu}_n({\boldsymbol \theta},{\boldsymbol \tau})=\left(\frac{1}{n}\boldsymbol{\Gamma}_{2n}({\boldsymbol \theta}, {\boldsymbol \tau})\right)^{-1}\frac{\gamma_1-1}{n}\sum_{i=1}^nk(\mathbf{z}_i,{\boldsymbol \tau}) {\boldsymbol \sigma}_{{\boldsymbol \scriptstyle \tau}}(\mathbf{z}_i,{\boldsymbol \tau}). $$