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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

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