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

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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Statistical inference for functional time series is investigated by extending the classic concept of autocovariance function (ACF) to functional ACF (FACF). It is established that for functional moving average (FMA) data, the FMA order can be determined as the highest nonvanishing order of FACF, just as in classic time series analysis. A two-step estimator is proposed for FACF, the first step involving simultaneous B-spline estimation of each time trajectory and the second step plug-in estimation of FACF by using the estimated trajectories in place of the latent true curves. Under simple and mild assumptions, the proposed tensor product spline FACF estimator is asymptotically equivalent to the oracle estimator with all known trajectories, leading to asymptotic correct simultaneous confidence envelope (SCE) for the true FACF. Simulation experiments validate the asymptotic correctness of the SCE and data-driven FMA order selection. The proposed SCEs are computed for the FACFs of an ElectroEncephalogram (EEG) functional time series with interesting discovery of finite FMA lag and Fourier form functional principal components.