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Simultaneous confidence bands for comparing variance functions of two samples based on deterministic designs

DOI:10.1007/s00180-020-01043-6
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Chen Zhong at Tsinghua University
Chen Zhong
Lijian Yang at Tsinghua University
Lijian Yang
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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 order n-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{-1/2}$$\end{document}, as an infeasible estimator if the two mean functions are known a priori. The additive SCB, which is the log transform of the multiplicative SCB, is location and scale invariant in the sense that the width of SCB is free of the unknown mean and variance functions of both samples. Simulation experiments provide strong evidence that corroborates the asymptotic theory. The proposed SCBs are used to analyze several strata pressure data sets from the Bullianta Coal Mine in Erdos City, Inner Mongolia, China.
Plots of 95% 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} (solid) and the 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) in Case 1, with an1=n2=300,ε∼N0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=300, \varepsilon \sim N\left( 0,1\right) $$\end{document}; bn1=n2=300,ε∼U-3,3;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=300,\varepsilon \sim U\left( -\sqrt{3},\sqrt{3}\right) ;$$\end{document}cn1=n2=600,ε∼N0,1;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=600,\varepsilon \sim N\left( 0,1\right) ;$$\end{document}dn1=n2=600,ε∼U-3,3;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n_{1}=n_{2}=600,\varepsilon \sim U\left( -\sqrt{3},\sqrt{3}\right) ;$$\end{document}en1=n2=900,ε∼N0,1;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=900,\varepsilon \sim N\left( 0,1\right) ;$$\end{document}fn1=n2=900,ε∼U-3,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=900,\varepsilon \sim U\left( -\sqrt{3},\sqrt{3}\right) $$\end{document}
Plots of 95% 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} (solid) and the 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) in Case 1, with an1=n2=300,ε∼N0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=300, \varepsilon \sim N\left( 0,1\right) $$\end{document}; bn1=n2=300,ε∼U-3,3;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=300,\varepsilon \sim U\left( -\sqrt{3},\sqrt{3}\right) ;$$\end{document}cn1=n2=600,ε∼N0,1;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=600,\varepsilon \sim N\left( 0,1\right) ;$$\end{document}dn1=n2=600,ε∼U-3,3;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n_{1}=n_{2}=600,\varepsilon \sim U\left( -\sqrt{3},\sqrt{3}\right) ;$$\end{document}en1=n2=900,ε∼N0,1;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=900,\varepsilon \sim N\left( 0,1\right) ;$$\end{document}fn1=n2=900,ε∼U-3,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{1}=n_{2}=900,\varepsilon \sim U\left( -\sqrt{3},\sqrt{3}\right) $$\end{document}
… 
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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Computational Statistics (2021) 36:1197–1218
https://doi.org/10.1007/s00180-020-01043-6
ORIGINAL PAPER
Simultaneous confidence bands for comparing variance
functions of two samples based on deterministic designs
Chen Zhong1·Lijian Yang1
Received: 8 August 2020 / Accepted: 22 October 2020 / Published online: 31 October 2020
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract
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 order n−1/2, as an infeasible estimator if the two mean functions are
known a priori. The additive SCB, which is the log transform of the multiplicative
SCB, is location and scale invariant in the sense that the width of SCB is free of
the unknown mean and variance functions of both samples. Simulation experiments
provide strong evidence that corroborates the asymptotic theory. The proposed SCBs
are used to analyze several strata pressure data sets from the Bullianta Coal Mine in
Erdos City, Inner Mongolia, China.
Keywords Brownian motion ·B-spline ·Kernel ·Oracle efficiency ·Strata pressure ·
Variance ratio
1 Introduction
Nonparametric simultaneous confidence band (SCB) is a useful tool for statistical
inference about the global properties of an entire unknown curve or function. It was
first constructed in Bickel and Rosenblatt (1973) for a kernel density function. Then
nonparametric SCB was soon extended to regression function, see Johnston (1982),
Härdle (1989), Härdle and Marron (1991), Eubank and Speckman (1993), Xia (1998),
and Claeskens and Van Keilegom (2003) for early works about SCB. SCB not only is
a theoretically beautiful construct, but also has wide applications in many areas such
as sample survey and functional data analysis, see Zhao and Wu (2008), Ma et al.
BLijian Yang
yanglijian@tsinghua.edu.cn
1Center for Statistical Science and Department of Industrial Engineering, Tsinghua University, Beijing
100084, China
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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