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Simultaneous confidence band for the differenceof regression functions of two samples

Jiakun Jiang , Li Cai & Lijian Yang

To cite this article: Jiakun Jiang , Li Cai & Lijian Yang (2020): Simultaneous confidence band forthe difference of regression functions of two samples, Communications in Statistics - Theory andMethods, DOI: 10.1080/03610926.2020.1800039

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Simultaneous confidence band for the differenceof regression functions of two samples

Jiakun Jianga, Li Caib#, and Lijian Yanga

aCenter for Statistical Science & Department of Industrial Engineering, Tsinghua University, Beijing,China; bSchool of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, China

ABSTRACTThis paper concerns the comparison of two sample non parametricregression. An asymptotically correct simultaneous confidence band(SCB) is proposed for the difference of two-sample non parametricregression functions to achieve the goal of comparison. Simulationexperiments provide strong evidence that corroborates the asymp-totic theory. The proposed SCB is used to analyze different samplesof strata pressure data from the Bullianta Coal Mine in Erdos City,Inner Mongolia, China.

ARTICLE HISTORYReceived 14 November 2019Accepted 19 July 2020

KEYWORDSB-spline; kernel; Brownianmotion; simultaneousconfidence band;strata pressure

1. Introduction

Simultaneous confidence band (SCB) is a powerful and vital inference tool for an entireunknown curve or function. It is a direct analogy to a confidence interval, regarded as acollection of confidence intervals over the whole range of functions. Non parametricSCB methodology has become increasingly important in the statistical literature. Xia(1998) proposed bias-corrected SCBs based on local polynomial fitting under theassumption of homoscedasticity. Wang and Yang (2009) proposed SCBs for non para-metric regression function based on polynomial splines. Cai and Yang (2015) proposeda spline-kernel oracally efficient two-step estimator to construct SCB for heteroscedasticvariance function. Gu and Yang (2015) established oracle efficiency of an SCB for thesingle-index link function. Wang et al. (2014) proposed kernel estimator for the distri-bution function of unobserved errors in autoregressive time series. Cao, Yang, andTodem (2012), Ma, Yang, and Carroll (2012), Song et al. (2014), Zheng, Yang, andH€ardle (2014), and Cao et al. (2016) constructed various SCBs for functional data. Thusfar, existing non parametric SCBs are mostly focused on the one-sample problem.However, an interesting problem in practical application is whether the non parametricregression functions of different samples are equal, or whether they are subject to a cer-tain relationship (Neumeyer and Sperlich, 2006). An intuitive and commonly usedmethod is to investigate the difference of their non parametric regression functions. Fordense function data, Cao, Yang, and Todem (2012), Cao et al. (2016), Song et al. (2014)had accomplished this goal by oracle efficiency. Huang et al. (2008) applied this idea to

CONTACT Lijian Yang [email protected] Center for Statistical Science & Department of IndustrialEngineering, Tsinghua University, Beijing, China.#Co-first author.� 2020 Taylor & Francis Group, LLC

COMMUNICATIONS IN STATISTICS—THEORY AND METHODShttps://doi.org/10.1080/03610926.2020.1800039

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investigate the difference in crop yield wetness relationship for different soil types, butthere were no rigorous theoretical justifications. Much effort has also been devoted tothe problem of testing the equality of multivariate regression curves, for example, Dette& Neumeyer (2001), Lavergne (2001), Gørgens (2002), Neumeyer & Dette (2003). Inparticular, Neumeyer and Sperlich (2006) considered the problem of comparing parts ofthe regression function in multivariate non parametric regression model. To our bestknowledge, the SCB has not been established for the difference of two one-dimensionalnon parametric regression functions. The aim of this paper is therefore to close thedescribed gap between the needs of practitioners and what the present litera-ture provides.To be more precise, denote by ðXs, i,Ys, iÞ

� �nsi¼1, s ¼ 1, 2 the two samples, where each

has sample size ns: Existing literature on SCBs for non parametric regression is mostlyconcerned with the random design model. Often encountered in applications (e.g., thestrata pressure data discussed in Subsection 5.2) is the deterministic design non para-metric regression model:

Ys, i ¼ ms ins

� �þ rs ins

� �es, i, i ¼ 1, :::, ns, s ¼ 1, 2 (1)

in which the Ys, i’s are responses at equally spaced design points i=ns, 1 � i � ns, andes, if gnsi¼1 are unobserved i.i.d. random errors with Eðes, 1Þ ¼ 0, varðes, 1Þ ¼ 1: Assume thatthere are smooth but unknown mean and variance functions msð�Þ and r2s ð�Þ that satisfymodel (1). In this paper, we aim to construct asymptotically correct SCB for the differ-ence of regression functions m1ð�Þ �m2ð�Þ: As an illustration, 95% SCB for m1ð�Þ �m2ð�Þ are constructed for several strata pressure data sets collected from the BuliantaCoal Mine located in Erdos City, Inner Mongolia, China, see Figure 3.For single population problem, based on model Y ¼ mðxÞ þ rðxÞ�, much research

has been done. Donoho and Johnstone (1996) and Angelini, De Canditiis, andFr�ed�erique (2003) studied non parametric estimation for the regression function mð�Þ:The SCB for the regression function mð�Þ was studied in Hall and Titterington (1988).A limitation of this adaptive SCB is its reliance on assumption that the eif gni¼1 are i.i.d.Nð0, 1Þ and the variance function r2ð�Þ are constant. Alternatively, Eubank andSpeckman (1993) obtained the SCB for the mean function mð�Þ based on kernelsmoothing; however, this was under the restrictive assumption of homoscedasticity(r2ð�Þ � r2) and the mean function mð�Þ being periodic. Wang (2012) constructed aspline SCB for the mean function mð�Þ based on deterministic designs and eif gni¼1 beingstrongly (or a) mixing, but its asymptotically conservative coverage limits its usefulnessfor testing hypotheses. All these works on SCB are focused on single-population prob-lems. In this work, we propose an asymptotically correct simultaneous confidence band(SCB) for the difference of two-sample non parametric regression functions.The remainder of the paper is organized as follows. Section 2 establishes the main

asymptotic theoretical results. Section 3 provides insight into proofs and Section 4presents concrete steps to implement the SCB. Section 5 reports some simulation resultsand analysis of the strata pressure data. A brief conclusion is made in Section 6.Technical proofs are in Appendix.

2 J. JIANG ET AL.

2. Main results

We formulate in this section the SCB for the difference of non parametric regressionfunctions m1ð�Þ �m2ð�Þ in model (1). One begins with smoothing each data setði=ns ,Ys, iÞ� �ns

i¼1 to obtain m̂sð�Þ, s¼ 1, 2; then, a plug-in estimator for m1ð�Þ �m2ð�Þ ism̂1ð�Þ � m̂2ð�Þ: The current paper constructs the SCB of m1ð�Þ �m2ð�Þ by investigatingthe asymptotic extreme value distribution of m̂1ð�Þ � m̂2ð�Þ: In the first step, the basicidea is to find a locally weighted least squares estimate m̂sðxÞ, s ¼ 1, 2, which solves theminimization problem:

minh

n�1sXnsi¼1

Ys, i � hð Þ2Kh i=ns � xð Þ ¼ n�1sXnsi¼1

Ys, i � m̂sðxÞ� �2

Kh i=ns � xð Þ,

in which KðuÞ is a kernel function, h ¼ minðhn1 , hn2Þ, where hn1 , hn2 are sequences ofsmoothing parameters called bandwidths, and KhðuÞ ¼ h�1Kðu=hÞ is the kernel functionrescaled by h. Clearly,

m̂sðxÞ ¼n�1s

Pnsi¼1Kh i=ns � xð ÞYs, i

f̂ sðxÞ, (2)

where f̂ sðxÞ ¼ n�1sPns

i¼1 Khði=ns � xÞ: The same bandwidth h ¼ hn is used for estimatingbothms, s ¼ 1, 2 to ensure that corresponding error processes in (3) and (4) are footed on acommon away-from-the-boundary interval In ¼ h, 1� h½ �:We denote by wðsÞðxÞ the s-th order derivative of a function wðxÞ: For h 2 ð0, 1� and

integer p � 0, let Cp, h 0, 1½ � be the space of functions with h�H€older continuous p-th-order derivatives on [0,1] with seminorm jj � jjp, h

Cp, h 0, 1½ � ¼ /ðxÞ : jj/jjp, h ¼ supx 6¼x0, x, x02 0, 1½ �

j/ðpÞðxÞ � /ðpÞ x0ð Þjjx � x0jh < þ1

( ),

and denote by CðpÞ 0, 1½ � the space of p-times continuously differentiable functions. Forsequences of positive real numbers cn and dn, cn � dn means cn=dn ! 0 as n ! 1:We need the following assumptions to construct the SCB for m1ð�Þ �m2ð�Þ:

(M1) The functions msð�Þ 2 Cp�1, h 0, 1½ �, s ¼ 1, 2 for h 2 ð0, 1� and integer p � 1:(M2) The errors es, s ¼ 1, 2 satisfy EðesÞ ¼ 0, E ðe2s Þ ¼ 1 and r2s ð�Þ 2 Cð1Þ½0, 1� with 0 <cr � r2s ðxÞ � Cr < þ1 for any x 2 ½0, 1�.(M3) There exist bs 2 ð0, 1=2� 1=ð4hþ 4p� 2ÞÞ,Cs 2 ð0, þ1Þ, cs 2 ð1, þ1Þ andi.i.d. Nð0, 1Þ variables Zs, insf gnsi¼1, s ¼ 1, 2 such that

P max1�l�ns

��Xli¼1

es, i �Xli¼1

Zs, ins

�� > nbss( )

< Csn�css :

(M4) The kernel function K 2 Cð1ÞðRÞ, is of order p, and is supported on �1, 1½ �:(M5) The bandwidth hns , s ¼ 1, 2, satisfies log hns=ð� log nsÞ ! t > 0 as ns ! 1 and

max n�1=2s log1=2ns, n

2bs�1s log ns

n o� hns � ns log nsð Þ�1= 2hþ2p�1ð Þ:

Hence 1=ð2hþ 2p� 1Þ � t � min 1=2, 1� 2max b1, b2f g� �

:

COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 3

(M6) There exist constants a, c,C 2 ð0,1Þ such that r2ð�Þ � ar1ð�Þ, and 0 < c �n1=n2 � C < 1 as n1, n2 ! þ1:

Assumptions (M1), (M2), (M4) are typical for kernel smoothing, adapted from H€ardle(1989), Eubank and Speckman (1993), and Cai et al. (2019). (M5) is the general conditionon the choice of bandwidth hn1 , hn2 : It is more convenient to make the inequalities on tstrict in (M5). Assumption (M6) is weaker than Hall and Titterington (1988) and Cai et al.(2014) on variance functions, both of which required the variance function to be constant.Assumption (M6) is also weaker than Cao, Yang, and Todem (2012), Cao et al. (2016) onthe ratio of sample sizes, requiring only that they be comparable rather than proportional.Thus, a common bandwidth for two estimators is reasonable. Assumption (M3) providesthe Gaussian approximation of the error process, which allows for error distribution muchmore general than Gaussian. According to Lemma S.2 in the supplement of Cai et al.(2019), Assumption (M3) is ensured by an elementary Assumption (M30):

(M30) There exists gs > 2=bs � 2, bs 2 ð0, 1=2� 1=ð4hþ 4p� 2ÞÞ, s ¼ 1, 2 such thatEje1, 1j2þgs < þ1, Eje2, 1j2þgs < þ1:Our SCB for m1ð�Þ �m2ð�Þ follows directly from Lemma 1 in the Appendix.

Theorem 1. If (M1)–(M6) hold, for any z 2 R, as n1, n2 ! 1,

P ah supx2In

jm̂1ðxÞ � m̂2ðxÞ �m1ðxÞ þm2ðxÞjvnðxÞ � bh

" #� z

( )! exp �2 exp �zð Þ� �,

where ah ¼ 2 log ðh�1Þ� �1=2

, bh ¼ ah þ a�1h 2�1 log ðCK=ð2p2ÞÞ� �

,

CK ¼ð1�1

Kð1ÞðvÞ2dv=ð1�1

KðvÞ2dv,

vnðxÞ ¼ h�1=2r1ðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin�11 þ a2n�12

q ð1�1

K2ðuÞdu( )1=2

:

Then, for any a 2 ð0, 1Þ,P m1ðxÞ �m2ðxÞ 2 m̂1ðxÞ � m̂2ðxÞ6vnðxÞ a�1h qa þ bh

� , 8x 2 In

� �! 1� a,

where qa ¼ � log �1=2 log ð1� aÞ� �

:

Theorem 1 implies that the SCB contracts to zero at the rate ðn�11 þ a2n�12 Þ1=2h�1=2 log 1=2h�1: In the special case p ¼ 2, h ¼ 1, as in Subsection 4.1, the implementedorder of h satisfying (M5) and (M6) is n�1=51 log

�1=5�dn1 or n�1=52 log

�1=5�dn2 for any d >0: Thus, the optimal bandwidth order is under-smoothed by log �1=5�dn1 or log �1=5�dn2,

and the contraction rate of SCB is ðn�11 þ a2n�12 Þ1=2n1=101 log 3=5þ0:5dn1:

3. Error decomposition

An asymptotic SCB for m1ð�Þ �m2ð�Þ is constructed by standardizing and maximizing thedeviation j m̂1ðxÞ � m̂2ðxÞ

� �� m1ðxÞ �m2ðxÞ� �j over the interval In: One can find that

4 J. JIANG ET AL.

m̂sðxÞ �msðxÞ ¼ n�1s f̂�1s ðxÞ

Xnsi¼1

Kh i=ns � xð ÞYs, i �msðxÞ

¼ n�1s f̂�1s ðxÞ

Xnsi¼1

Kh i=ns � xð Þ ms i=nsð Þ �msðxÞ þ rsði=nsÞes, i� �

¼ f̂ �1s ðxÞ As, nsðxÞ þ Bs, nsðxÞ� �

, x 2 Inin which

As, nsðxÞ ¼ n�1sXnsi¼1

Kh i=ns � xð Þ ms i=nsð Þ �msðxÞ� �

, (3)

Bs, nsðxÞ ¼ n�1sXnsi¼1

Kh i=ns � xð Þrsði=nsÞes, i, x 2 In: (4)

The following stochastic processes approximate Bs, nsðxÞ :

Bs, ns, 1ðxÞ ¼ n�1sXnsi¼1

Kh i=ns � xð Þrsði=nsÞZs, ins , (5)

Bs, ns, 2ðxÞ ¼ n�1sXnsi¼1

Kh i=ns � xð ÞrsðxÞZs, ins , (6)

Bs, ns, 3ðxÞ ¼ n�1=2sðKh u� xð ÞrsðxÞdWs, nsðuÞ, x 2 In (7)

where Zs, insf gnsi¼1 are i.i.d. Nð0, 1Þ variables satisfying (M3) and Ws, nsðuÞ is a two-sidedBrownian motion on ð�1, þ1Þ satisfying

Zs, ins ¼ffiffiffin

pWs, ns i=nsð Þ �Ws, ns i� 1ð Þ=ns

� � �, s ¼ 1, 2:

Define a Gaussian process

fðsÞ ¼ n�1=21

ÐK s� rð ÞdW1, n1ðrÞ � an�1=22

ÐK s� rð ÞdW2, n2ðrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n�11 þ a2n�12

� Ð 1

�1 K2ðuÞdu

q , s 2 1, h�1 � 1½ �: (8)The following result of maxima distribution is crucial for proving Theorem 1.

Proposition 1. Under Assumptions (M2) and (M4), for any z 2 R, as n ! 1,P ah sup

s2 1, h�1�1½ �jfðsÞj � bh

� < z

� �! exp �2 exp �zð Þ� �,

with ah and bh given in Theorem 1.

4. Implementation

In this section, we describe detailed procedures for implementing the SCB in Theorem 1

based on data sets ði=ns ,Ys, iÞ� �ns

i¼1, s ¼ 1, 2, that follow model (1). This is used throughoutSection 5 for simulations and data examples.


As the default, we set p ¼ 2, h ¼ 1 in (M1). When constructing the SCB for the func-tion m1ð�Þ �m2ð�Þ in model (1) according to Theorem 1, one chooses a kernel functionK and bandwidth h for computing m̂1ð�Þ, m̂2ð�Þ and estimating the variance functionr21ð�Þ, and then plugs in these estimates, as in Eubank and Speckman (1993), Hall andTitterington (1988), H€ardle (1989) and Xia (1998).

We select the quartic kernel KðuÞ ¼ 15ð1� u2Þ2I juj � 1f g=16 to satisfy (M4), and the band-widths h ¼ minðh1, rot log �1=5�dn1, h2, rot log �1=5�dn2Þ (d > 0) to satisfy (M5), where therule-of-thumb bandwidth hs, rot, s ¼ 1, 2 is from Equation (4.3) of Fan and Gijbels (1996):

hs, rot ¼35Pns

i¼1 Ys, i �P4

k¼0 âk i=nsð Þkn o2

nsPns

i¼1 2â2 þ 6â3 i=nsð Þ þ 12â4 i=nsð Þ2� �2

8><>:

9>=>;

1=5

, (9)

in which ðâkÞ4k¼0 ¼ argminðakÞ4k¼02R5Pns

i¼1 ðYs, i �P4

k¼0 akði=nsÞkÞ2: Here hs, rot has ordern�1=5s and h order n

�1=5s log �1=5�dns, s ¼ 1, 2, satisfying (M5). We have found via exten-

sive simulations that h ¼ minðh1, rot, h2, rotÞ log �1=2ðn1 þ n2Þ=2 works quite well; thus,that is what we recommend.A spline-kernel estimator is used for the variance function r21ð�Þ, which is

r̂21ðxÞ ¼n�11

Pni¼1K~h i=n1 � xð Þê21, i

n�11Pn1

i¼1K~h i=n1 � xð Þ, x 2 In (10)

where ê1, i ¼ Y1, i � m̂1, pði=n1Þ, in which m̂1, pð�Þ is the p-th order spline estimator form1ð�Þ with integer p> 0,

m̂1, p ¼ argming2H p�2ð ÞN

Xn1i¼1

Y1, i � g i=n1ð Þ� �2

, (11)

in which Hðp�2ÞN ¼ Hðp�2ÞN 0, 1½ � is the space of spline functions on interval 0, 1½ � definedbelow. The bandwidth ~h ¼ hrot,r log �1=5�d2n1, in which d2 > 0, hrot,r is defined thesame as in (9), but with Yi replaced by ê

21, i ¼ Y1, i � m̂1, pði=n1Þ

� �2:

Divide the interval 0, 1½ � into ðN þ 1Þ subintervals Jj ¼ vj, vjþ1Þ, j ¼ 0, 1, 2, :::,N

by

equally spaced points vjf gNj¼1 called interior knots,0 ¼ v0 < v1 < � � � < vNþ1 ¼ 1, vj ¼ j= N þ 1ð Þ, j ¼ 0, 1, :::,N þ 1:

Hðp�2ÞN is the space of functions that are polynomials of degree ðp� 1Þ on each Jjwith continuous ðp� 2Þ-th derivative on 0, 1½ �: For instance, Hð�1ÞN consists of functionsthat are constant on each Jj, and Hð0ÞN the space of functions that are linear on each Jjand continuous on 0, 1½ �: Let Bj, pf gNj¼1�p be the basies of space H

ðp�2ÞN , which was intro-

duced in de Boor (2001). The estimator m̂1, pð�Þ can then be expressed as:

m̂1, pð�Þ ¼XNj¼1�p

k̂j, pBj, pð�Þ,

where the vector ðk̂1�p, p , :::, k̂N, pÞT is the solution of the least-squares problem:

6 J. JIANG ET AL.

k̂1�p, p , :::, k̂N, p� �T

¼ argminR

Nþp

Xn1i¼1

Y1, i �XNj¼1�p

kj, pBj, p i=n1ð Þ8<:

9=;

2

: (12)

Cai and Yang (2015) and Cai et al. (2019) have proved that the estimator r̂21ðxÞ isoracally efficient under our current assumptions.The asymptotic 100ð1� aÞ% SCB for m1ð�Þ �m2ð�Þ is

m̂1ðxÞ � m̂2ðxÞ6v̂nðxÞ a�1h qa þ bh�

, x 2 In, (13)

with v̂nðxÞ ¼ h�1=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir̂21ðxÞðn�11 þ a2n�12 Þ

q Ð 1�1 K

2ðuÞdun o1=2

:

5. Empirical studies

5.1. Simulation

To investigate the finite-sample behavior of the proposed SCB in Section 2, the follow-ing four cases were examined.Case 1:

m1ðxÞ ¼ cos 3pxð Þ,m2ðxÞ ¼ 2m1ðxÞ,r1ðxÞ ¼ 0:1 sin 2pxð Þ þ 0:2, r2ðxÞ ¼ 2r1ðxÞ:

Case 2:

m1ðxÞ ¼ x3 þ x2 þ x,m2ðxÞ ¼ m1ðxÞ þ sin ðpxÞ þ cos ðpxÞ � x3 � x2 � x,

r1ðxÞ ¼ exp x=4ð Þ � 0:9exp x=4ð Þ þ 0:9 , r2ðxÞ ¼ 3r1ðxÞ:

Case 3:

m1ðxÞ ¼ sin pxð Þ þ x2 þ 1�

log xþ 1ð Þ,m2ðxÞ ¼ cos2 pxð Þ,r1ðxÞ ¼ 0:1 cos 2pxð Þ þ 0:1x, r2ðxÞ ¼ 4r1ðxÞ:

Case 4:

m1ðxÞ ¼ exp ðxÞ sin ðxÞx2 þ x,m2ðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix3 þ x2 þ x þ 1

psin ðxÞ cos ðxÞ,

r1ðxÞ ¼ log 2x þ 1ð Þ, r2ðxÞ ¼ 2r1ðxÞ:Here, e was either Nð0, 1Þ or the standardized t-distribution with freedom 10, e

0:81=2t10: The sample sizes were n1, n2 ¼ 300, 600, 900, while for the SCB, the confi-dence level was 1� a ¼ 0:95, 0:99: The coverage frequencies by SCB defined in (13) form1ð�Þ �m2ð�Þ are reported in Table 1; these are relative frequencies in 2000 replicationsof coverage of the true curve at equally spaced points xj, j ¼ 1, 2, :::, 400f g on In: In allcases with e Nð0, 1Þ (the left side of the parentheses) and e 0:81=2t10 (inside theparentheses), the coverage frequencies improve and approach the nominal level as thesample size n1 and n2 increases, which supports Theorem 1.To visualize the SCB for m1ð�Þ �m2ð�Þ, Figure 1 was created based on sample size

n1 ¼ n2 ¼ 300, n1 ¼ n2 ¼ 900 in four cases with e Nð0, 1Þ and a confidence level95%. Figures 2 was created based on sample size n1 ¼ 300, n2 ¼ 600, n1 ¼ 300, n2 ¼ 900


and n1 ¼ 600, n2 ¼ 900 in four cases with e Nð0, 1Þ and a confidence level 95%. Eachhas the dashed line as the estimated curve, thick line the true curve and the upper andlower solid lines the SCB. As expected, the SCBs for n1 ¼ n2 ¼ 900 are thinner and fitbetter than those for n1 ¼ n2 ¼ 300:To investigate when the proposed method breaks down and how it compares with

bootstrap method, cases of small sample size (n¼ 50, 100, 200) are implemented. Thebootstrap SCB is constructed by 500 re-samplings. Table 2 shows that when sample sizeis as small as 50, 100, the bootstrap SCB performs better than the asymptotic SCB;when sample size becomes larger, performance of the two is similar. It is in line withthe findings of Claeskens and Van Keilegom (2003) which studied bootstrap SCB forthe non parametric regression function. The asymptotic SCB is much faster to computethan the bootstrap SCB.

5.2. Data examples

Using the two-sample SCB, we have analyzed the data sets provided by Professor JiangYaodong’s research group at China University of Mining and Technology, which areavailable from us upon request. The data are strata pressure records in May 2013 fromthe Bulianta Coal Mine located in Erdos City, Inner Mongolia, China. Information onstrata pressure behavior, range and pressure periodicity in front of a working face isimportant for the coal mine industry to improve underground mining safety and preci-sion, specifically by preparing the roof support design to prevent accidents caused bysudden increase of strata pressure; see Ju and Xu (2013) and Qian, Shi, and Xu (2010).Strata pressure is the vertical stress on the coal seam roof in front of the working

face with unit KN/m2 (working face is the underground location where miners peel

Table 1. Empirical coverage frequencies of the SCB in (13) for m1ðxÞ � m2ðxÞ using 2000 replicationswith e Nð0, 1Þ (the left side of the parentheses) and e 0:81=2t10 (inside the parentheses),respectively.

e Nð0, 1Þðe 0:81=2t10Þn1 n2 1� a Case 1 Case 2 Case 3 Case 4300 300 0.95 0.951(0.927) 0.961(0.952) 0.959(0.929) 0.941(0.960)

0.99 0.993(0.989) 0.993(0.992) 0.995(0.989) 0.986(0.996)600 0.95 0.959(0.949) 0.952(0.953) 0.957(0.949) 0.949(0.962)

0.99 0.995(0.988) 0.996(0.996) 0.994(0.988) 0.990(0.996)900 0.95 0.966(0.941) 0.959(0.951) 0.958(0.947) 0.953(0.963)

0.99 0.995(0.989) 0.994(0.993) 0.993(0.985) 0.991(0.997)

600 300 0.95 0.957(0.949) 0.965(0.963) 0.961(0.953) 0.939(0.966)0.99 0.997(0.993) 0.996(0.995) 0.995(0.992) 0.991(0.995)

600 0.95 0.957(0.961) 0.959(0.964) 0.964(0.955) 0.947(0.966)0.99 0.997(0.997) 0.996(0.997) 0.998(0.992) 0.991(0.997)

900 0.95 0.960(0.956) 0.957(0.965) 0.959(0.958) 0.938(0.972)0.99 0.996(0.992) 0.995(0.995) 0.992(0.988) 0.990(0.996)

900 300 0.95 0.964(0.951) 0.965(0.959) 0.963(0.956) 0.932(0.970)0.99 0.996(0.992) 0.996(0.995) 0.994(0.993) 0.987(0.998)

600 0.95 0.969(0.953) 0.967(0.970) 0.969(0.956) 0.940(0.975)0.99 0.995(0.997) 0.996(0.997) 0.996(0.995) 0.986(0.997)

900 0.95 0.959(0.954) 0.970(0.965) 0.959(0.963) 0.932(0.970)0.99 0.996(0.993) 0.998(0.997) 0.992(0.994) 0.986(0.998)

8 J. JIANG ET AL.

coal from the coal wall mechanically). The pressure sensors are placed at the top ofhydraulic supports in front of the working face, and collect data with a recording 1interval of 0.80m. During the mining process, once the hydraulic support has moved

Figure 1. Plots of 95% SCB (solid) for m1ðxÞ �m2ðxÞ (thick) and the estimator m̂1ðxÞ � m̂2ðxÞ (dashed)in cases 1–4 (first row to fourth row) with e Nð0, 1Þ and n1 ¼ n2 ¼ 300(left), n1 ¼ n2 ¼ 900 (right).


forward 0.80m, a pressure sensor records a mine pressure. The propulsion range of thehydraulic support is from 295.5m to 705.1m, and therefore the sample size n1 ¼ n2 ¼513: For simplicity, we standardized hydraulic support to interval ½0, 1�:Figure 3 shows the plots of the SCB (dashed lines) computed according to (13) for

the function m1ð�Þ �m2ð�Þ, and kernel estimate m̂1ð�Þ � m̂2ð�Þ (solid line) with confi-dence level 95%. In practical applications, engineers are interested in whether two sitesshare the same pressure. We have therefore proposed to test null hypothesis H0 :m1ð�Þ �m2ð�Þ � 0 by the SCB for the difference of mean functions m1ð�Þ �m2ð�Þ: Sincethe lowest confidence levels of SCB containing the horizontal zero curve were99.9999999%, 98.5%, 98.5%, 35%, one retained the null hypothesis with the p values ¼10�7, 0:015, 0:015, 0:65, respectively. Thus, the strata pressures in first three groupsshow significant difference over the whole interval while the fourth group no signifi-cant difference.

Figure 2. Plots of 95% SCB (solid) for m1ðxÞ �m2ðxÞ (thick) and the estimator m̂1ðxÞ � m̂2ðxÞ (dashed)in cases 1-4 (first row to fourth row) with e Nð0, 1Þ: Each column represents different sample size, n1 ¼300, n2 ¼ 600 (left column), n1 ¼ 300, n2 ¼ 900 (middle column), n1 ¼ 600, n2 ¼ 900 (right column).

10 J. JIANG ET AL.

Figure 3. Plots of SCBs (dashed) for m1ðxÞ �m2ðxÞ and the estimator m̂1ðxÞ � m̂2ðxÞ (solid), with95% SCB (first column) and lowest simultaneous confidence band containing null hypothesis (secondcolumn) for groups 1–4.


6. Conclusions

Motivated by the need to compare non parametric regression from two samples, anasymptotically correct simultaneous confidence band (SCB) is proposed for the differ-ence of two-sample non parametric regression functions. An efficient and fast algorithmis proposed that accomodates the theoretical results. Analysis of different samples ofstrata pressure data from the Bullianta Coal Mine in Erdos City, Inner Mongolia, Chinahas illustrated the versatility of the proposed two sample SCB. Further research maylead to similar constructions when one or both samples are based on random designs.

Acknowledgments

The authors are grateful to Professor Jiang Yaodong’s research group at China University ofMining and Technology for providing the strata pressure data, and to two anonymous Reviewersfor many helpful comments.

Funding

This research was supported by National Natural Science Foundation of China award 11771240,11901521 and First Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics).

ORCID

Lijian Yang http://orcid.org/0000-0003-3894-873X

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Table 2. Empirical coverage frequencies and computing time in minutes (inside parentheses) ofthe asymptotic SCB and the bootstrap SCB based on 500 re-samplings with e Nð0, 1Þand 1� a ¼ 0:95:n1 ¼ n2 Method Case 1 Case 2 Case 3 Case 450 proposed 0.822(0.05) 0.866(0.05) 0.81(0.05) 0.89(0.05)

bootstrap 0.948(16.93) 0.956(14.84) 0.94(14.72) 0.95(15.30)

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Appendix

The following Lemmas 1, 2, and 3 are from Cai et al. (2019).

Lemma 1. Under Assumption (M4), for s¼ 1, 2, as ns ! 1,

supx2In

jf̂ sðxÞ � 1j ¼ O n�1s h�2�

:

Lemma 2. Under Assumptions (M1), (M4), and (M5), for s¼ 1, 2, as ns ! 1,supx2In

jAs, nsðxÞj ¼ O hhþp�1 þ n�1s h�1� �

:

Lemma 3. Under Assumptions (M2)–(M4), for s¼ 1, 2, as ns ! 1,

ðaÞ supx2 0, 1½ �

jBs, nsðxÞ � Bs, ns , 1ðxÞj ¼ Op nbs�1s h�1� �

,

ðbÞ supx2 0, 1½ �

jBs, ns , 1ðxÞ � Bs, ns , 2ðxÞj ¼ Op n�1=2s h1=2 log 1=2ns� �

,

ðcÞ supx2In

jBs, ns , 2ðxÞ � Bs, ns , 3ðxÞj ¼ Op n�3=2s h�2 log 1=2ns� �

,

ðdÞ supx2 0, 1½ �

jBs, ns , 3ðxÞj ¼ Op n�1=2s h�1=2 log 1=2ns� �

:

The following is a reformulation of Theorems 11.1.5 and 12.3.5 of Leadbetter, Lindgren, andRootz�en (1983).

Lemma 4. If the Gaussian process fðsÞ, 0 � s � T is stationary with mean zero and variance one,and covariance function satisfying for some constant C> 0

corr fðsÞ, f sþ tð Þ� ¼ EfðsÞf sþ tð Þ ¼ 1� Cjtja þ oðjtjaÞ as t ! 0,then as T ! 1,

P

�aT

�sup

s2½0,T�jfðsÞj � bT

< z

�! exp f�2 exp ð�zÞg, 8z 2 R

in which

aT ¼ 2 logTð Þ1=2, bT ¼ aT þ a�1T 2� a2a

log a2T=2� þ log C1=aHa 2pð Þ�1=22 2�að Þ=2a� �

�

with H1 ¼ 1,H2 ¼ p�1=2:

14 J. JIANG ET AL.

https://doi.org/10.1111/1467-9868.00155https://doi.org/10.1111/1467-9868.00155https://doi.org/10.1080/01621459.2013.866899

Proof of Theorem 1

Notice that

m̂1ðxÞ � m̂2ðxÞ � m1ðxÞ �m2ðxÞ� �

¼ m̂1ðxÞ �m1ðxÞ � m̂2ðxÞ �m2ðxÞ� �

¼ f̂ �11 ðxÞ A1, n1ðxÞ þ B1, n1ðxÞ� �� f̂ �12 ðxÞ A2, n2ðxÞ þ B2, n2ðxÞ� �

¼ f̂ �11 ðxÞA1, n1ðxÞ � f̂�12 ðxÞA2, n2ðxÞ þ f̂

�11 ðxÞB1, n1ðxÞ � f̂

�12 ðxÞB2, n2ðxÞ


�11 ðxÞ � 1

� �B1, n1ðxÞ

� f̂ �12 ðxÞ � 1� �

B2, n2ðxÞ þ B1, n1ðxÞ � B2, n2ðxÞ


�11 ðxÞ � 1

� �B1, n1ðxÞ � f̂

�12 ðxÞ � 1

� �B2, n2ðxÞ

þ B1, n1ðxÞ � B1, n1, 1ðxÞ� �þ B1, n1, 1ðxÞ � B1, n1, 2ðxÞ� �þ B1, n1, 2ðxÞ � B1, n1, 3ðxÞ� �

� B2, n2ðxÞ � B2, n2, 1ðxÞ� �þ B2, n2, 1ðxÞ � B2, n2, 2ðxÞ� �þ B2, n2, 2ðxÞ � B2, n2, 3ðxÞ� �

þ B1, n1, 3ðxÞ � B2, n2, 3ðxÞ� �

� B1, n1, 3ðxÞ � B2, n2, 3ðxÞ� �þ Rn1, n2ðxÞ:

(14)

Applying Lemma 3 to the following

supx2 0, 1½ �

jBs, nsðxÞj � supx2 0, 1½ �

jBs, nsðxÞ � Bs, ns , 1ðxÞj þ supx2 0, 1½ �

jBs, ns , 1ðxÞ � Bs, ns , 2ðxÞj

þ supx2 0, 1½ �

jBs, ns , 2ðxÞ � Bs, ns , 3ðxÞj þ supx2 0, 1½ �

jBs, ns , 3ðxÞj

and Assumptions (M3), (M5) on bs and the bandwidth hns , s ¼ 1, 2, one obtains thatsup

x2 0, 1½ �jBs, nsðxÞj ¼ Op n�1=2s h�1=2 log 1=2ns

� �:

Then Lemma 1 entails that

supx2 0, 1½ �

�� f̂ �1s ðxÞ � 1� �

Bs, nsðxÞ�� ¼ Opðn�3=2s h�5=2Þ, s ¼ 1, 2: (15)

Combining Lemmas 1, 2, and 3, one obtains that

supx2In

jRn1, n2ðxÞj

¼ Op hhþp�1 þX2s¼1

n�1s h�1 þ n�3=2s h�5=2 þ nbs�1s h�1 þ n�1=2s h1=2 log 1=2ns þ n�3=2s h�5=2 log 1=2ns

n o !:

Denote

Bn1, n2, 3ðxÞ ¼ B1, n1, 3ðxÞ � B2, n2, 3ðxÞ, x 2 In:Under assumption (M6), for any x 2 In

EfB2n1, n2, 3ðxÞg ¼ ðn�11 h�1r21ðxÞ þ n�12 h�1a2r21ðxÞÞð1�1

K2ðuÞdu

¼ h�1r21ðxÞ½n�11 þ a2n�12 �ð1�1

K2ðuÞdu:


Standardizing the process Bn1, n2, 3ðxÞ for x 2 In, one obtains a Gaussian processn�1=21

ÐKh x� uð ÞdW1, n1ðuÞ � an�1=22

ÐKh x� uð ÞdW2, n2ðuÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h�1 n�11 þ a2n�12

� Ð 1

�1 K2ðuÞdu

q ,whose absolute maximum follows the same probability law as

L n�1=21 h

�1 Ð K s� u=hð ÞdW1, n1ðuÞ � an�1=22 h�1 Ð K s� u=hð ÞdW2, n2ðuÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih�1 n�11 þ a2n�12

� Ð 1

�1 K2ðuÞdu

q , s 2 1, h�1 � 1½ �8<:

9=;

¼ L n�1=21

ÐK s� rð ÞdW1, n1ðrÞ � an�1=22

ÐK s� rð ÞdW2, n2ðrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n�11 þ a2n�12

� Ð 1

�1 K2ðuÞdu

q , s 2 1, h�1 � 1½ �8<:

9=;,

which is the process fðsÞ defined in (8).The covariance function of fðsÞ is

cov fðsÞ, f s0ð Þ�

¼ EfðsÞf s0ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ef2ðsÞEf2 s0ð Þq ¼

ðn�11 þ a2n�12 ÞðK � KÞðs0 � sÞn�11 þ a2n�12

� Ð 1

�1 K2ðuÞdu

¼ K � Kð Þ s0 � sð ÞÐ 1

�1 K2ðuÞdu

:

Note that, if K 2 C2½�1, 1�, thenðK � KÞðtÞÐ 1�1 K

2ðuÞdu� 1 ¼

ÐKðvÞfKðv� tÞ � KðvÞgdvÐ 1

�1 K2ðuÞdu

¼ �ÐKð1ÞðvÞ2dvÐ 1�1 K

2ðuÞdut2

2þ oðjtj2Þ:

Define next a Gaussian process f1ðtÞ, 0 � t � T ¼ Tn ¼ h�1 � 2,f1ðtÞ ¼ f t þ 1ð Þ,

which is stationary with mean zero and variance one, and covariance function

cov f1ðsÞ, f1 sþ tð Þ� ¼ 1� CK jtja þ oðjtjaÞ as t ! 0,

where CK ¼Ð 1�1 K

ð1ÞðvÞ2dv= Ð 1�1 KðvÞ2dv:As n1, n2 ! 1, h ! 0 so T ! 1, therefore according to Lemma 4,

P

�aT

�sup

s2½0,T�jf1ðsÞj � bT

< z

�! exp f�2 exp ð�zÞg, 8z 2 R

where aT ¼ ð2 logTÞ1=2 and bT ¼ aT þ 12 a�1T log CK2p2� �

:Recall from Theorem 1 that

ah ¼ 2 log h�1ð Þ� �1=2

, bh ¼ ah þ a�1h 2�1 log CK= 2p2ð Þ� n o

:

Note that, under Assumption (M6), as n1, n2 ! 1aha

�1T ! 1, ahðbT � bhÞ ¼ Oð log 1=2n1 h log �1=2n1Þ ! 0:

Hence, applying Slutsky’s Theorem, one obtains that

ah

�sup

s2½0,T�jf1ðsÞj � bh

¼ aha�1T

�aT

�sup

s2½0,T�jf1ðsÞj � bT

�þ ahðbT � bhÞ

converges in distribution to the same limit as aT sups2 0,T½ � jf1ðsÞj � bTn o

:

16 J. JIANG ET AL.

Thus,

P

�ah

�sup

s2½1, h�1�1�jfðsÞj � bh

< z

�! exp f�2 exp ð�zÞg, 8z 2 R: (16)

According to Assumptions (M3) and (M5), for s¼ 1, 2 as n1, n2 ! 1hhþp�1

a�1hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih�1 n�11 þ a2n�12

�q ! 0, n�1s h�1


�q ! 0, n

�3=2s h�5=2


�q ! 0,

nbs�1s h�1


�q ! 0, n

�1=2s h1=2 log 1=2ns


�q ! 0, n

�3=2s h�5=2 log 1=2ns


�q ! 0:

Then

supx2In

jRn1, n2ðxÞj ¼ op h�1=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin�11 þ a2n�12

qa�1h

� �: (17)

Combining (14, 16, 17) and applying Slutsky’s Theorem, one obtains that for any z 2 R

P ah supx2In

jm̂1ðxÞ � m̂2ðxÞ �m1ðxÞ þm2ðxÞjvnðxÞ � bh

" #� z

( )! exp �2 exp �zð Þ� �, (18)

where vnðxÞ ¼ h�1=2r1ðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin�11 þ a2n�12

p Ð 1�1 K

2ðuÞdun o1=2

: By taking 1� a ¼ exp �2 exp ð�zÞ� �for a 2 ð0, 1Þ, the above (18) implies that

P m1ðxÞ �m2ðxÞ 2 m̂1ðxÞ � m̂2ðxÞ6vnðxÞ a�1h qa þ bh�

, 8x 2 In� �! 1� a,

where qa ¼ � log �1=2 log ð1� aÞ� �

: That completes the proof of Theorem 1.


AbstractIntroductionMain resultsError decompositionImplementationEmpirical studiesSimulationData examples

ConclusionsAcknowledgmentsReferences

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Simultaneous confidence band for the difference of ...lijianyang.com/mypapers/TwosampleSCBCIS.pdf · the regression function in multivariate non parametric regression model. To our