\begin{document}$$\hat b^ \ast$$\end{document}b^* considered as a function of λ. Standard applications of SIMEX, including that of Bowden et al.23, fit a linear or quadratic model relating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b^ \ast$$\end{document}b^* to λ, extrapolating to λ = −1 to obtain the de-biased estimate. For greater accuracy, we developed a maximum-likelihood estimator of b for simple linear regression models. Our approach yields confidence intervals for b so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$var(\hat b)$$\end{document}var(b^)can be estimated. The details of our improved SIMEX approach are given in Supplementary Note 1.
