PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Multivar Anal. Author manuscript; available in PMC 2010 August 1.
Published in final edited form as:
J Multivar Anal. 2009 August 1; 100(7): 1432–1439.
doi:  10.1016/j.jmva.2008.12.003
PMCID: PMC2847310
NIHMSID: NIHMS185520

Comparison of Hyperbolic and Constant Width Simultaneous Confidence Bands in Multiple Linear Regression under MVCS Criterion

Summary

A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, it is proposed in Liu and Hayter (2007) to use the area of the confidence set that corresponds to a confidence band as an optimality criterion in comparison of confidence bands; the smaller is the area of the confidence set, the better is the corresponding confidence band. This minimum area confidence set (MACS) criterion can clearly be generalized to the minimum volume confidence set (MVCS) criterion in study of confidence bands for a multiple linear regression model. In this paper the hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and so the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and so the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band.

Keywords: Confidence sets, Linear regression, Simultaneous confidence bands, Statistical inference

1 Introduction

Consider the multiple linear regression model

Y=Xb+e

where Yn×1 is the vector of observed responses, Xn×p is the design matrix with the first column given by (1, (...), 1)T and the jth (2 ≤ jp) column given by (x1,j, (...), xn,j)T, b = (b1, (...), bp)T is the vector of regression coefficients, and en×1 is the error vector with e ~ N(0, σ2I) and σ2 unknown. Assume XTX is non-singular, so the least squares estimator of b is given by b^=(XTX)1XTY. Let σ^2 denote the mean square error with degrees of freedom ν = np. Then σ^2σ2χν2ν and is independent of b^.

Let x = (1, x2, (...), xp)T and x(1) = (x2, (...), xp)T. A simultaneous confidence band for the regression function

xTb=b1+b2x2++bpxp

on a given region X of the p−1 predictor variables x(1) = (x2, (...), xp)T provides useful information on where the true but unknown regression model lies; a linear regression function is a plausible candidate of the unknown regression model if and only if it is contained completely inside the confidence band. There are several recent papers considering various applications of confidence bands; see for example Sun, Raz and Faraway (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu, Jamshidian and Zhang (2004), and Piegorsch et al. (2005).

Construction of simultaneous confidence bands has a history going back to Working and Hotelling (1929). Scheffé (1953) provided the well known two-sided hyperbolic simultaneous confidence band over the whole space X=Rp1 of the p − 1 predictor variables.

For p = 2, that is, when there is only one predictor variable, construction of various exact simultaneous confidence bands has been considered by Gafarian (1964), Bowden and Graybill (1966), Graybill and Bowden (1967), Wynn and Bloomfield (1971), Bohrer and Francis (1972) and Uusipaikka (1983) among others. See a recent review by Liu et al. (2007).

For p > 2, construction of exact confidence bands over a finite region X of the predictor variables is much harder. When p > 2 there are at least two predictor variables and the region X may assume various forms. One useful region X is the rectangular region

Xr={x(1)T:aixibi,i=2,,p},

where −∞ ≤ ai < bi ≤ ∞, i = 2, (...), p are given constants. Construction of two-sided hyperbolic confidence bands over Xr has been considered by Knafl, Sacks and Ylvisaker (1985), Naiman (1987, 1990) and Sun and Loader (1994) among others. All these confidence bands are either conservative or approximate however. A simulation-based method for constructing a two-sided hyperbolic confidence band over Xr for a general p ≥ 2 is given recently in Liu et al. (2005a); the critical constant can be calculated as accurate as one requires if the number of replications in the simulation is set sufficiently large. Construction of a two-sided constant width confidence band over Xr for a general p ≥ 2 is considered in Liu et al. (2005b) by using both numerical integration and simulation.

For p > 2, another useful region X is given by the ellipsoidal region Xe in (1) below. Let X(1) be the n × (p − 1) matrix produced from the design matrix X by deleting the first column of 1’s from X. Let x.j=1ni=1nxij be the mean of the observed values of the jth predictor variable (2 ≤ jp), and let x(1)=(x.2,,x.p)T. Define a (p − 1) × (p − 1) matrix

S=1n(X(1)1x(1)T)T(X(1)1x(1)T)=1n(X(1)TX(1)nx(1)x(1)T)

where 1 is an n-vector of 1’s. Note that matrix S is just the sample variance-covariance matrix of the p − 1 predictor variables, and it is non-singular since X is assumed to be of full column-rank. Now the ellipsoidal region is defined to be

Xe={x(1):(x(1)x(1))TS1(x(1)x(1))a2}
(1)

where a > 0 is a given constant. It is clear that this region is centered at x(1) and has an ellipsoidal shape in x(1) = (x2, (...), xp)T [set membership] Rp−1. One important feature of Xe is that the variance of the fitted regression model at x, Var(xTb^), is a constant for all the x(1) on the surface of the ellipsoid Xe. Construction of an approximate two-sided hyperbolic confidence band over Xe is first considered by by Halperin and Gurian (1968). Construction of exact hyperbolic confidence bands over Xe have been considered by Bohrer (1973), Casella and Strawderman (1980), Seppanen and Uusipaikka (1992) and Liu and Lin (2007) among others.

The purpose of this paper is to compare the hyperbolic and constant width confidence bands over Xe under the minimum area (volume) confidence set optimality criterion proposed in Liu and Hayter (2007). Note that each 1 − α confidence band for the regression model xTb corresponds to a 1 − α confidence set for the regression coefficients b. The minimum area (volume) confidence set optimality prefers a confidence band whose confidence set has a smaller area (volume). For p = 2, various confidence bands for a regression straight line have been assessed and compared under the minimum area confidence set criterion in Liu and Hayter (2007). Before the appearance of the minimum area (volume) confidence set criterion, (weighted) average width of a confidence band has been used exclusively as an optimality criterion in study of confidence bands; see e.g. Naiman (1983, 1884) and Piegorsch (1985a, 1985b). In particular, the hyperbolic and constant width bands over Xe have been compared by Naiman (1983) under the average width criterion.

In Section 2 the hyperbolic and constant width confidence bands and the corresponding confidence sets are presented. In Section 3, comparisons between the two confidence bands under the minimum volume confidence set criterion are given. Much of the notation of Liu and Lin (2007) is adopted in this paper.

2 Confidence bands and confidence sets

In this section the hyperbolic and constant width confidence bands over Xe are given and the corresponding confidence sets are identified. The hyperbolic confidence band is given by

xTbxTb^±chσ^xT(XTX)1xfor allx(1)=(x2,,xp)TXe
(2)

where Xe is defined in (1) and ch is a suitably chosen critical constant so that the simultaneous confidence level of the band is equal to 1 − α.

As in Liu and Lin (2007), let p-vector z=n(1,x(1)T)T, and let p × (p − 1) matrix Z satisfy (z,Z)T(XTX)−1(z,Z) = Ip. It follows therefore that T=(z,Z)1(XTX)(b^b)σ^ is a standard p-dimensional t random vector with ν degrees of freedom; see e.g. Tong (1990) for multivariate t distributions. Note that

1α=P{supx(1)XexT(b^b)σ^xT(XTX)1xch}=P{supx(1)Xe{(z,Z)T(XTX)1x}T{(z,Z)1(XTX)(b^b)σ^}{(z,Z)T(XTX)1x}T{(z,Z)T(XTX)1x}ch}.

Let

Vh={t:supx(1)Xe{(z,Z)T(XTX)1x}Tt(z,Z)T(XTX)1xch}Rp.
(3)

Then the confidence set for the regression coefficients b that corresponds to the hyperbolic band in (2) is given by

Ch(b^,σ^)={b:(z,Z)1(XTX)(bb^)σ^Vh}.
(4)

Let w = (z,Z)T(XTX)−1x = (w1,w(1)T)T where w(1) = (w2, (...), wp)T = ZT(XTX)−1x and w1=zT(XTX)1x=1n. Then it follows from Liu and Lin (2007, expressions (6),(7),(8) and (10)) that Vh can further be expressed as

Vh={t:supwWewTtwch}
(5)

where

We={w:w1=1n,w2(1+a2)n}Rp.
(6)

From (4) and the definition of T, the critical constant ch can be solved from 1 − α = P{T [set membership] Vh} which, from Liu and Lin (2007, expression (28)), is equivalent to

1α=Fp,ν(ch2p)0θ2ksinp2θdθ+0π2θ2ksinp2(θ+θ)Fp,ν{ch2pcos2θ}dθ

where

θ=arccos(11+a2)(0,π2),
(7)

k=1(0πsinp2θdθ), and Fp,ν(·) is the cdf of the F distribution with p and ν degrees of freedom. Note that ch depends on θ*, p, ν and α only.

Now we turn our attention to the constant width band. It is given by

xTbxTb^±cc(1+a2)nσ^for allx(1)=(x2,,xp)TXe,
(8)

where cc is chosen so that the simultaneous confidence level of the band is equal to 1 − α. Hence

1α=P{supx(1)XexT(b^b)σ^cc(1+a2)n}=P{supx(1)Xe{(z,Z)T(XTX)1x}T{(z,Z)1(XTX)(b^b)σ^}cc(1+a2)n}.

Let

Vc={t:supx(1)Xe{(z,Z)T(XTX)1x}Ttcc(1+a2)n}Rp.

Then the confidence set for the regression coefficients b that corresponds to the constant band in (8) is given by

Cc(b^,σ^)={b:(z,Z)1(XTX)(bb^)σ^Vc}.
(9)

Similar to the expression of Vh in (5), Vc can be written as

Vc={t:supwWewTtcc(1+a2)n}
(10)

where We is given in (6). Let t(1) = (t2, (...), tp)T. Note that

supwWewTt=supwWet1n+w(1)Tt(1)t1n+a2nt(1)

and the upper bound above is attained at the wWe with w(1)=sign(t1)a2nt(1)t1. Hence Vc in (10) can further be expressed as

Vc={t:t1n+a2nt(1)cc(1+a2)n}.

Now using the polar coordinates (Rv, θv1, …, θv,p−1)T for a p-dimensional vector v = (v1, (...), vp)T

{v1=Rvcosθv1v2=Rvsinθv1cosθv2v3=Rvsinθv1sinθv2cosθv3vp1=Rvsinθv1sinθv2sinθv,p2cosθv,p1vp=Rvsinθv1sinθv2sinθv,p2sinθv,p1}where{0θv1π0θv2π0θv,p2π0θv,p12πRv0}

the set Vc can be expressed as

Vc={t:Rtcosθt1n+a2nRtsinθt1cc(1+a2)n}=Vc,1+Vc,2
(11)

where

Vc,1={t:0θt1π2,Rtcos(θt1θ)cc}
(12)

Vc,2={t:π2θt1π,Rtcos(πθt1θ)cc}
(13)

with θ* given in (7).

From the definition of T and expressions (9), (11), (12) and (13), the critical constant cc can be solved from

1α=P{TVc}=P{TVc,1}+P{TVc,2}=2P{TVc,1}=0π22ksinp2θFp,ν(cc2pcos2(θθ))dθ

where the last equality follows immediately from the distributions of RT(pFp,ν) and θT1 and the independence of RT and θT1 (see e.g. Liu and Lin, 2007, expressions (11) and (12)). Again cc depends on θ*, p, ν and α only.

3 Comparison under MVCS criterion

In this section we first calculate the volumes of the confidence sets Ch(b^,σ^) and Cc(b^,σ^). We then compare the volumes of the two confidence sets to see which one is smaller and so the corresponding confidence band is better under the MVCS criterion.

Let v(R) denote the volume of a set R [subset or is implied by] Rp, and let Bp(r) denote the ball of radius r in Rp. Note that the Jacobian of the transformation from the Cartesian coordinates to the polar coordinates given in the last section is equal to

J=Rp1sinp2θ1sinp3θ2sinθp2.

Hence it is clear that

v(Bp(r))=R=0rθ1=0πθ2=0πθp2=0πθp1=02πJdRdθ1dθp1=cprp

where cp is a constant depending only on p.

From Liu and Lin (2007, Lemma 4), Vh in (5) can be partitioned into four parts: Vh = Vh,1 + Vh,2 + Vh,3 + Vh,4 where

Vh,1={t:0θt1θ,Rtch},Vh,2={t:θ<θt1π2,Rtcos(θt1θ)ch},Vh,3={t:π2<θt1πθ,Rtcos(πθθt1)ch},Vh,4={t:πθ<θt1π,Rtch}.

Now v(Vh,1) is equal to

R=0chθ1=0θθ2=0πθp2=0πθp1=02πJdRdθ1dθp1=(θ1=0θsinp2θ1dθ1/θ1=0πsinp2θ1dθ1)v(Bp(ch))=kθ1=0θsinp2θ1dθ1v(Bp(ch))

and v(Vh,2) is equal to

Rcos(θ1θ)chθθ1π2θ2=0πθp2=0πθp1=02πJdRdθ1dθp1=(Rcos(θ1θ)chθθ1π2Rp1sinp2θ1dRdθ1/R=0chθ1=0πRp1sinp2θ1dRθ1dθ1)v(Bp(ch))=kθπ2sinp2θ1cosp(θ1θ)dθ1v(Bp(ch)).

Furthermore, we have v(Vh,3) = v(Vh,2) and v(Vh,4) = v(Vh,1). Combining these gives

v(Vh)=2k(θ1=0θsinp2θ1dθ1+θπ2sinp2θ1cosp(θ1θ)dθ1)v(Bp(ch)).
(14)

For the constant width band, similar calculation from expressions (11), (12) and (13) shows that

v(Vc,1)=v(Vc,2)=k0π2sinp2θ1cosp(θ1θ)dθ1v(Bp(cc))

and so

v(Vc)=2k0π2sinp2θ1cosp(θ1θ)dθ1v(Bp(cc)).
(15)

Now note that the confidence sets Ch in (4) and Cc in (9) are of the form

C(b^,σ^)={b:(z,Z)1(XTX)(bb^)σ^V}=σ^(XTX)1(z,Z)V+b^

and so

v(C(b^,σ^))=σ^(XTX)1(z,Z)v(V)=σ^(XTX)12v(V).

Hence from (14) and (15)

effv(Cc(b^,σ^))v(Ch(b^,σ^))=v(Vc)v(Vh)=0π2sinp2θ1cosp(θ1θ)dθ1θ1=0θsinp2θ1dθ1+θπ2sinp2θ1cosp(θ1θ)dθ1(ccpchp).

Under the MVCS criterion, the hyperbolic band is better than the constant width band if and only if eff > 1.

It can be shown that eff does not change if one of the predictor variables has a linear transformation (i.e. eff is location and scale invariant). It is noteworthy that eff depends only on θ*, p, ν and α. The size of the region Xe in (1) is determined by a: the larger is a the bigger is Xe. From the one-to-one relationship (7) between a and θ*, the size of Xe is alternatively determined by θ*; the larger is the angle θ* the bigger is Xe. When θ* → 0 from right, both the hyperbolic and constant width bands approach the two-sided t-confidence interval for xTb at x(1)=x(1). When θ* → π/2, the hyperbolic band approaches the Scheffé band over the whole space of the predictor variables. One the other hand, as θ* → π/2, the critical constant cc of the constant width band approaches a finite constant and so the width of the band 2cc(1+a2)nσ^ and the volume v(Vc) in (15) approach infinity.

We have calculated eff as a function of θ* [set membership] (0, π/2) for given values of p = 3(1)8, ν = 15, 40, ∞ and α = 0.10, 0.05, 0.01. The following patten of the function eff(θ*) has been observed for all the combinations of p, ν and α. The function eff(θ*) first decreases and then increases over θ* [set membership] (0, π/2). When θ* approaches zero from right, eff(θ*) approaches one. When θ* approaches π/2 from left, eff(θ*) approaches infinity. At a certain threshold value θ0=θ0(p,ν,α), eff(θ0) is equal to one. The threshold value θ0(p,ν,α) is pretty stable for different values of ν and α but increases in the value of p. The value of θ0(p,ν,α) is approximately equal to 0.8 for p = 3 and 1.1 for p = 8. Furthermore, minθ*[set membership](0,π/2) eff(θ*) is no less than 0.98 for all the combinations studied. Figure 1 provides a plot of eff(θ*) for p = 3, ν = 15 and α = 0.05, the shape of which is typical for all the combinations of p, ν and α.

Figure 1
Plot of the function eff(θ*) for p = 3; ν = 15 and α = 0:05.

From these observations, the following conclusions can be drawn. When 0<θ<θ0 (i.e. when the value of a in (1) is smaller than a certain threshold value), the constant width band is better than the hyperbolic band. But the advantage of the constant width band over the hyperbolic band in this situation is very limited since minθ*[set membership](0,π/2) eff(θ*) is only very marginally smaller than one. On the other hand, when θ0<θ<π2, the hyperbolic band is better than the constant width band. And the advantage of the hyperbolic band over the constant width band can be enormous especially when θ* is close to the upper limit π/2 since eff(θ*) becomes very large when θ* is close to π/2.

These observations are consistent with those for a simple linear regression made in Liu and Hayter (2007), where the angle θ/2 plays a similar role as the angle θ* here. But in a simple linear regression, the interval over which confidence bands are constructed is not necessary to be symmetric about the mean of the observed values of the predictor variable. It is therefore recommended that the hyperbolic band should be used unless the “constant width” feature of the constant width band is highly desirable for a given problem. One example of constant width shape is more desirable than hyperbolic shape is given in Liu et al. (2007).

Note that, on the set {x(1):(x(1)x(1))TS1(x1x(1))=b2} for a give 0 ≤ ba, the width of the hyperbolic band is a constant. This constant width increases with b and is equal to the width of the constant width band at a particular value of 0 < b < a. So the hyperbolic band is narrower than the constant width band when x(1) is near x(1) and vice-versa when x(1) is near the boundary of Xe. Comparison of the hyperbolic and constant width bands under the average width criterion in Naiman (1983) has made observations that are different from our observations under the MVCS criterion above in two major aspects. First, the average width of the hyperbolic band is always no larger than that of the constant width band. Second, the ratio of the average widths of the two bands is always finite even when θ* → π/2.

Finally, we use a portion of the acetylene data in Snee (1977) to illustrate the methodologies discussed in this paper. The same data set has also been used for illustration by Casella and Strawderman (1980), Naiman (1987) and Liu and Lin (2007) among others. The two predictor variables are reactor temperature (x2) and ratio of H2 to n-Heptane (x3). The response variable (y) is conversion of n-Heptane to Acetylene. There are sixteen data points. So p = 3, n = 16 and ν = 13. The fitted linear regression model is given by y = −130.69 + 0.134x2 + 0.351x3, σ^=3.624, and R2 = 0.92.

The observed values of x2 range from 1100 to 1300 with average x.2 = 1212.5, and the observed values of x3 range from 5.3 to 23 with average x.3 = 12.4. So the ellipsoidal region Xe is centered at (x.2; x.3)T = (1212.5, 12.4)T. The size of Xe increases with the value of a. For a = 1.9 and α = 0.10 considered in Liu and Lin (2007), the region Xe is comparable with the range of observations on the predictor variables and our MATLAB programme calculates θ* = 1.086, ch = 2.723, cc = 2.598 and eff = 1.119. So the hyperbolic band is about 12% more efficient than the constant width band in this case. The smallest value of eff over a [set membership] (0,∞) is equal to 0.987 obtained at a = 0.771.

Contributor Information

W. Liu, S3RI and School of Mathematics University of Southampton, Southampton SO17 1BJ, UK ; ku.ca.notos.shtam@uiL.W.

A. J. Hayter, Department of Statistics and Operations Technology University of Denver, Denver, USA ; ude.ud@retyaH.ynohtnA.

W.W. Piegorsch, Department of Mathematics The University of Arizona, AZ 85721, USA ; ude.anozira.htam@hcsrogeip.

References

  • Al-Saidy OM, Piegorsch WW, West RW, Nitcheva DK. Confidence bands for low-dose risk estimation with quantal response data. Biometrics. 2003;59:1056–1062. [PubMed]
  • Bohrer R. A multivariate t probability integral. Biometrika. 1973;60:647–654.
  • Bohrer R, Francis GK. Sharp one-sided confidence bands for linear regression over intervals. Biometrika. 1972;59:99–107.
  • Bowden DC, Graybill FA. Confidence bands of uniform and proportional width for linear models. J. Amer. Stat. Assoc. 1966;61:182–198.
  • Casella G, Strawderman WE. Confidence bands for linear-regression with restricted predictor variables. J. Amer. Stat. Assoc. 1980;75:862–868.
  • Gafarian AV. Confidence bands in straight line regression. J. Amer. Stat. Assoc. 1964;59:182–213.
  • Graybill FA, Bowden DC. Linear Segment Confidence Bands for Simple Linear Regression Models. J. Amer. Stat. Assoc. 1967;62:403–408.
  • Halperin M, Gurian J. Confidence bands in linear regression with constraints on independent variables. J. Amer. Stat. Assoc. 1968;63:1020–1027.
  • Knafl G, Sacks J, Ylvisaker D. Confidence bands for regression-functions. J. Amer. Stat. Assoc. 1985;80:683–691.
  • Liu W, Jamshidian M, Zhang Y. Multiple comparison of several regression models. J. Amer. Stat. Assoc. 2004;99:395–403.
  • Liu W, Jamshidian M, Zhang Y, Donnelly J. Simulation-based simultaneous confidence bands for a multiple linear regression model when the covariates are constrained. J. of Computational and Graphical Statistics. 2005a;14(2):459–484.
  • Liu W, Jamshidian M, Zhang Y, Bretz F. Constant width simultaneous confidence bands for multiple linear regression when predictor variables are constrained in intervals. J. of Statistical Computation and Simulation. 2005b;75(6):425–436.
  • Liu W, Hayter AJ. Minimum area confidence set optimality for confidence bandsin simple linear regression. J. Amer. Stat. Assoc. 2007;102(477):181–190.
  • Liu W, Jamshidian M, Zhang Y, Bretz F, Han X. Pooling batches in drug stability study by using constant-width simultaneous confidence bands. Statistics in Medicine. 2007;26(14):2759–2771. [PubMed]
  • Liu W, Lin S, Piegorsch WW. Construction of exact simultaneous confidence bands for a simple linear regression model. International Statistical Review. 2007 to appear.
  • Liu W, Lin S. Construction of Exact Simultaneous Confidence Bands in Multiple Linear Regression with Predictor Variables Constrained in an Ellipsoidal Region. Statistica Sinica. 2007 to appear.
  • Naiman DQ. Comparing Scheffé-type to constant-width confidence bands in regression. J. Amer. Stat. Assoc. 1983;78:906–912.
  • Naiman DQ. Average width optimality of simultaneous confidence bands. Annals of Statistics. 1984;12:1199–1214.
  • Naiman DQ. Conservative confidence bands in curvilinear regression. Ann. Statist. 1986;14:896–906.
  • Naiman DQ. Simultaneous confidence-bounds in multiple-regression using predictor variable constraints. J. Amer. Stat. Assoc. 1987;82:214–219.
  • Naiman DQ. On volumes of tubular neighborhoods of spherical polyhedra and statistical inference. Ann. Statist. 1990;18:685–716.
  • Piegorsch WW. Average-width optimality for confidence bands in simple linear regression. J. Amer. Stat. Assoc. 1985a;80:692–697.
  • Piegorsch WW. Admissible and optimal confidence bands in simple linear regression. Annals of Statistics. 1985b;13:801–810.
  • Piegorsch WW, West RW, Pan W, Kodell R. Low dose risk estimation via simultaneous statistical inferences. J. Roy. Stat. Soc. C. 2005:245–258.
  • Scheffé H. A method for judging all contrasts in analysis of variance. Biometrika. 1953;40:87–104.
  • Seppanen E, Uusipaikka E. Confidence bands for linear-regression over restricted regions. Scand. J. Stat. 1992;19:73–81.
  • Snee RD. Validation of regression models: methods and examples. Technometrics. 1977;19:415–428.
  • Spurrier JD. Exact Confidence Bounds for All Contrasts of Three or More Regression Lines. J. Amer. Stat. Assoc. 1999;94:483–88.
  • Sun J, Loader CR. Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 1994;22:1328–1346.
  • Sun J, Raz J, Faraway JJ. Simultaneous confidence bands for growth and response curves. Statistica Sinica. 1999;9(3):679–698.
  • Tong YL. Multivariate Normal Distribution. Springer-Verlag; 1990.
  • Uusipaikka E. Exact confidence bands for linear-regression over intervals. J. Amer. Stat. Assoc. 1983;78:638–644.
  • Working H, Hotelling H. Applications of the theory of error to the interpretation of trends. J. Amer. Stat. Assoc. 1929;24:73–85.
  • Wynn HP, Bloomfield P. Simultaneous confidence bands in regression analysis. J. Roy. Stat. Soc. B. 1971;33:202–217.