Simultaneous confidence bands for comparing variance functions of two samples based on deterministic designs
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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 © 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.
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Lijian Yang [email protected] Center for Statistical Science and Department of Industrial Engineering, Tsinghua University, Beijing 100084, China
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C. Zhong, L. Yang
(2012), Cao et al. (2012, 2016), Song et al. (2014), Wang et al. (2014), Zheng et al. (2014, 2016), Gu et al. (2014), Cai and Yang (2015), Gu and Yang (2015), Wang et al. (2016), Zhang and Yang (2018), and Cai et al. (2020) for recent development on nonparametric SCBs. In the context of nonparametric regression model, adaptive SCB for the regression function was studied in Hall and Titterington (1988). A rather undesirable limitation of adaptive SCBs is their reliance on the assumption of i.i.d. Gaussian errors and heteroscedasticity (constant variance function). Alternatively, Eubank and Speckman (1993) obtained the SCB for the mean function based on kernel smoothing without Gaussianity assumption on errors; however, it was also under the restrictive assumption of homoscedasticity, and the mean function being periodic. Wang (2012) constructed a spline SCB for nonparametric mean function based on deterministic designs and strongly mixing dependent errors, but
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