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# Plots of the null hypothesis curve of r^=n1-1∑i=1n1e^1,i2/n2-1∑i=1n2e^2,i2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{r}}=n_{1}^{-1} \sum _{i=1}^{n_{1}}{\hat{e}}_{1,i}^{2}/n_{2}^{-1}\sum _{i=1}^{n_{2}}{\hat{e}} _{2,i}^{2}$$\end{document} (solid), SCB (thick) for σ12x/σ22x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{1}^{2}\left( x\right) /\sigma _{2}^{2}\left( x\right)$$\end{document} and the spline-kernel estimator σ^12x/σ^22x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma }_{1}^{2}\left( x\right) /\hat{\sigma } _{2}^{2}\left( x\right)$$\end{document} (dashed), with a 95% SCB for pair 1; b lowest simultaneous confidence band containing null hypothesis for pair 1; c 95% SCB for pair 2; d lowest simultaneous confidence band containing null hypothesis for pair 2

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Asymptotically correct simultaneous confidence bands (SCBs) are proposed in both multiplicative and additive form to compare variance functions of two samples in the nonparametric regression model based on deterministic designs. The multiplicative SCB is based on two-step estimation of ratio of the variance functions, which is as efficient, up to o...

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