[statistical science and interdisciplinary research] advances in multivariate statistical methods...

September 15, 2009 11:46 World Scientific Review Volume - 9in x 6in AdvancesMultivariate

Chapter 10

Estimation of the Multivariate Box-Cox Transformation Parameters

Mezbahur Rahman1 and Larry M. Pearson2

Minnesota State University, Mankato, MN 56001, USAE-mail:[email protected]; [email protected]

The Box-Cox transformation is a well known family of power transformationsthat brings a set of data closer into agreement with the normality assumption ofthe residuals and, hence, the response variables of a postulated model in regres-sion analysis. This paper implements the Newton-Raphson method in estimatingthe multivariate Box-Cox transformation parameters and gives a new method ofestimation of the parameters by maximizing the multivariate Shapiro-Wilk statis-tic. Simulation is performed to compare the two methods for bivariate transfor-mations.

Contents

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17310.2 Box-Cox Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17410.3 Maximum Likelihood Estimation Using The Newton-Raphson Method . . . . . . . . . . 17510.4 Maximization of the Multivariate Shapiro-Wilk W Statistic . . . . . . . . . . . . . . . . . 17710.5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

10.1. Introduction

In regression analysis, often the key assumption regarding normality of the re-sponse variables is violated. The commonly used remedy is the Box-Cox familyof power transformations (Box and Cox (1964)). The process is to select a param-eter in the Box-Cox transformation which maximizes the normal likelihood usingthe data at hand and then apply regression analysis on the transformed responsevariables. There is no role of the estimates of the location and the scale parameterswhich were derived in the process of estimating the power transformation param-eters in the analysis. The model parameters are usually estimated seperately afterthe necessary Box-Cox power transformation parameters are selected.

173

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174 M. Rahman and L. M. Pearson

In the literature, the estimation procedures of the multivariate Box-Cox powertransformation parameters have not received as much attention as in the univariatecase. The univariate transformation parameter is usually estimated using the max-imization of the normal likelihood function as suggested by Box and Cox (1964),the robustified version of the normal likelihood method of Carroll (1980) and ofBickel and Doksum (1981), the transformation to symmetry method of Hinkley(1975), the quick estimate of Hinkley (1977) and of Taylor (1985). Lin and Vonesh(1989) constructed a nonlinear regression model which is used to estimate thetransformation parameter such that the normal probability plot of the data on thetransformed scale is as close to linearity as possible. Following Box and Cox(1982) and Lin and Vonesh (1989), Halawa (1996) considered the power transfor-mation parameter estimation procedure using an artificial regression model whichgives estimates with very small variabilities compared to the normal likelihoodprocedure. Halawa (1996) conducted an exhaustive comparative study with thenormal likelihood procedure. In that study, he also considered estimation proce-dures of the location and the scale parameters in the likelihood. Most recently,Rahman (1999) introduced a method of estimating the Box-Cox power transfor-mation parameter using maximization of the Shapiro-Wilk W (Shapiro and Wilk(1965)) statistic along with a comparative study of the normal likelihood method(Carroll (1980)), and of the artificial regression model method (Halawa (1996)).

In this paper, the estimation procedure for the multivariate Box-Cox powertransformation parameters is considered using maximization of the normal like-lihood along with the multivariate Newton-Raphson algorithm. In addition,the maximization of the multivariate Shapiro-Wilk W statistic method (Rahman(1999)) is implemented in the multivariate case. Andrews et al. (1971) consideredboth the marginal and the joint transformations and noted that for most purposesthe marginal transformation is sufficient to achieve the goal. Here, we will alsoconsider the marginal transformation.

10.2. Box-Cox Transformation

Let Y1,Y2, · · · ,Yn be a random sample of p-variate vectors from a populationwhose functional form is unknown. The multivariate version of the Box and Cox(1964) transformation suggested by Velilla (1993) is given by

X(Λ) =(

X (λ1)1 ,X (λ2)

2 , . . . ,X (λi)i , . . . ,X (λp)

p

)′∼ Np(µ,Σ) (10.1)

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Estimation of the Multivariate Box-Cox Transformation Parameters 175

where

X (λi)i =

{Y

λii −1λi

, λi 6= 0ln(Yi), λi = 0

,

µ = (µ1,µ2, . . . ,µp)′ and Σ = (σik)p×p.

10.3. Maximum Likelihood Estimation Using The Newton-RaphsonMethod

After applying the transformation identified in equation (10.2.1), the likelihoodfunction of the data can be written as

L(y;µ,Σ,Λ) = |2πΣ|−n/2 etr[− 12 Σ−1V (Λ)] · J, (10.2)

where J = ∏pi=1 ∏

nj=1 yλi−1

i j is the Jacobian of the transformations and V (Λ) =

(vik)p×p, vik = ∑nj=1(x

(λi)i j −µi)(x

(λk)k j −µk).

For a given Λ, the maximum likelihood estimates (MLE’s) of µ and Σ aregiven by µ̂ = X̄(Λ) and Σ̂ = (S(Λ)

ik )p×p where X̄(Λ) = (X̄ (λ1)1 , X̄ (λ2)

2 , . . . , X̄(λp)p )′ and

S(Λ)ik = 1

n ∑nj=1(X

(λi)i j − X̄ (λi)

i )(X (λk)k j − X̄ (λk)

k ), and hence

`max(Λ)≡ Lmax(Λ) =−np2

log(2π)− n2

log|Σ̂|+ np2

+

{p

∑i=1

(λi−1)n

∑j=1

logyi j

}, (10.3)

where `max(Λ) and Lmax(Λ) are the logarithm of the likelihood function 10.2.According to Harville (1999, p.309 (8.6)), the likelihood equations are

∂`max(Λ)∂λi

=∂− n

2 log|Σ̂|∂λi

+n

∑j=1

logYi j =−n2

tr(

Σ̂−1 ∂Σ̂

∂λi

)+

n

∑j=1

logYi j = 0,

(10.4)for i = 1,2, . . . , p , with ∂Σ̂

∂λi= (Dmq)p×p, where, for i′ 6= i,

Dii′ = Di′i =1n

n

∑j=1

(f racλiY

λii j logYi j− (Y λi

i j −1)λ2i −

1n

n

∑l=1

λiYλiil logYil− (Y λi

il −1)λ2

i

)

·

Yλi′i′ j −1

λi′− 1

n

n

∑l=1

Yλi′i′l −1

λi′

(10.5)

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Dii =2n

n

∑j=1

(λiY

λii j logYi j− (Y λi

i j −1)

λ2i

− 1n

n

∑l=1


il −1)λ2

i

)

·

(Y λi

i j −1

λi− 1

n

n

∑l=1

Y λiil −1

λi

)(10.6)

and Dmq = 0 for m 6= i and q 6= i.To obtain Λ̂, we solve the equation (4) using the Newton-Raphson method as

Λ̂(t+1) = Λ̂

(t)−(

H(Λ̂(t)))−1

D(Λ̂(t)) (10.7)

where D(t) is computed using 10.5 and 10.6 and the derivative of D(t), denoted byH(t), is computed using 10.3, 10.3, and 10.3. For the initial value of Λ̂(t), Λ̂(t) = 1is an obvious choice.Note that,

∂2log|Σ̂|∂λi∂λk

= tr(

Σ̂−1 ∂2Σ̂

∂λi∂λk

)− tr

(Σ̂−1 ∂Σ̂

∂λiΣ̂−1 ∂Σ̂

∂λk

),

where for i 6= k,

∂2Σ̂

∂λi∂λk= (Hmq)p×p

with

Hik = Hki

=1n

n

∑j=1

(λiY

λii j log(Yi j)− (Y λi

i j −1)

λ2i

− 1n

n

∑l=1

λiYλiil log(Yil)− (Y λi

il −1)λ2

i

)

·

λkYλkk j log(Yk j)− (Y λk

k j −1)

λ2k

− 1n

n

∑l=1

λkYλkkl log(Ykl)− (Y λk

kl −1)λ2

k

(10.8)

and Hmq = 0 for (m,q) 6= (k, i) or (i,k).For i = k,

∂2Σ̂

∂λ2i

= (Gmq)p×p

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where, for i′ 6= i,

Gii′ = Gi′i =1n

n

∑j=1

(Y λi

i j (logYi j)2

λi−

2Y λii j logYi j

λ2i

+2(Y λi

i j −1)

λ3i

−1n

n

∑l=1

Y λiil (logYil)2

λi−

2Y λiil logYil

λ2i

+2(Y λi

il −1)λ3

i

)Yλi′i′ j −1

λi′− 1

n

n

∑l=1

Yλi′i′l −1

λi′

,

(10.9)

Gii =2n

n

∑j=1

(λiYλii j logYi j− (Y λi

i j −1)

λ2i

− 1n

n

∑l=1


il −1)λ2

i

)2

+

(Y λi

i j (logYi j)2

λi−

2Y λii j logYi j

λ2i

+2(Y λi

i j −1)

λ3i

−1n

n

∑l=1

Y λiil (logYil)2

λi−

2Y λiil logYil

λ2i

+2(Y λi

il −1)λ3

i

)·

(Y λi

i j −1

λi− 1

n

n

∑l=1

Y λiil −1

λi

))(10.10)

and Gmq = 0 for m 6= i and q 6= i.

10.4. Maximization of the Multivariate Shapiro-Wilk W Statistic

Malkovich and Afifi (1973) suggested a test for multivariate normality by intro-ducing the multivariate form of the Shapiro and Wilk (1965) W statistic. Anexhaustive reference of tests for multivariate normality is given by Mecklin andMundfrom (2004).

Rahman (1999) showed that by maximizing the W statistic, the Box-Cox trans-formation parameter also can be estimated successfully with high precision. Herewe maximize the Malkovich and Afifi W ∗ statistic to obtain the multivariate Box-Cox transformation parameters. Now,

W ∗ =1n

[∑nj=1 a jU( j)]2(

X(Λ)m − X̄(Λ)

)′Σ̂−1

(X(Λ)

m − X̄(Λ)) . (10.11)

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where X(Λ), X̄(Λ), and Σ̂ are as defined earlier, the vector a = M′V−1

(M′V−1V−1M)1/2 ,M is the mean vector and V is the variance covariance matrix of the standardnormal order statistics, the U( j) are the ordered

U j =(

X(Λ)m − X̄(Λ)

)′Σ̂−1(

X(Λ)j − X̄(Λ)

)= Z(Λ)′

m Σ̂−1Z(Λ)

j , for

j = 1,2, . . . ,n, and

Z(Λ)′m Σ̂

−1Z(Λ)m = max

1≤ j≤nZ(Λ)′

j Σ̂−1Z(Λ)

j .

Now,

W ∗ =1n

[∑nj=1 a jU( j)]2

Z(Λ)′m Σ̂−1Z(Λ)

m

and U( j) = Z(Λ)′m Σ̂

−1Z(Λ)( j∗),

where Z(Λ)( j∗) corresponds to U( j). Then W ∗ can be maximized by solving the equa-

tions ∂W ∗∂λi

= 0 for i = 1,2, . . . , p where

∂W ∗

∂λi=

1n

Z(Λ)′m Σ̂−1Z(Λ)

m 2∑nj=1 a jU( j) ∑

nj=1 a j

∂U( j)∂λi−(

∑nj=1 a jU( j)

)2∂Z(Λ)′

m Σ̂−1Z(Λ)m

∂λi{Z(Λ)′

m Σ̂−1Z(Λ)m

}2 .

Solving ∂W ∗∂λi

= 0 is equivalent to solving

Z(Λ)′m Σ̂

−1Z(Λ)m 2

n

∑j=1

a jU( j)

n

∑j=1

a j∂U( j)

∂λi−

(n

∑j=1

a jU( j)

)2∂Z(Λ)′

m Σ̂−1Z(Λ)m

∂λi= 0,

(10.12)where

∂U( j)

∂λi=

∂Z(Λ)′m Σ̂−1Z(Λ)

( j∗)∂λi

= Z(Λ)′m Σ̂

−1∂Z(Λ)

( j∗)∂λi

+∂Z(Λ)′

m

∂λiΣ̂−1Z(Λ)

( j∗)−Z(Λ)′m Σ̂

−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

( j∗)

(using Harville (1999, p.307 (8.15)),∂Z(Λ)

( j∗)∂λi

is a vector of zeros except for the ith

element which is

d j∗i =λiY

λii j∗logYi j∗− (Y λi

i j∗−1)

λ2i

− 1n

n

∑l=1


il −1)λ2

i,

∂Z(Λ)′m

∂λiis a vector of zeros except for the ith element which is

dmi =λiY

λiim logYim− (Y λi

im −1)λ2

i− 1

n

n

∑l=1


il −1)λ2

i,

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∂Σ̂

∂λiis defined in Section 10.3, and

∂Z(Λ)′m Σ̂−1Z(Λ)

m

∂λi= 2

∂Z(Λ)′m

∂λiΣ̂−1Z(Λ)

m −Z(Λ)′m Σ̂

−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

m .

To solve (10.12) using the Newton-Raphson method let

Λ̃(t+1) = Λ̃

(t)−(

h(Λ̃(t)))−1

g(Λ̃(t)), (10.13)

where the first derivative of g(Λ) is denoted as

h(Λ) =∂

∂λk

(Z(Λ)′

m Σ̂−1Z(Λ)

m

)2

n

∑j=1

a jU( j)

n

∑j=1

a j∂U( j)

∂λi+Z(Λ)′

m Σ̂−1

Z(Λ)m 2

n

∑j=1

a j∂U( j)

∂λk

n

∑j=1

a j∂U( j)

∂λi+Z(Λ)′

m Σ̂−1Z(Λ)

m 2n

∑j=1

a jU( j)

n

∑j=1

a j∂

∂λk

(∂U( j)

∂λi

)

−2n

∑j=1

a jU( j)

n

∑j=1

a j∂U( j)

∂λk

∂Z(Λ)′m Σ̂−1Z(Λ)

m

∂λi−

(n

∑j=1

a jU( j)

)2∂

∂λk

(∂Z(Λ)′

m Σ̂−1Z(Λ)m

∂λi

)where for k 6= i,

∂

∂λk

(∂U( j)

∂λi

)=

∂

∂λk

∂Z(Λ)′m Σ̂−1∂Z(Λ)

( j∗)∂λi

=

[(∂Z(Λ)m

∂λk

)′−Z(Λ)′

m Σ̂−1(

∂Σ̂

∂λk

)]Σ̂−1

∂Z(Λ)( j∗)

∂λi

+Z(Λ)′m Σ̂

−1 ∂

∂λk

∂Z(Λ)( j∗)

∂λi

+

[∂

∂λk

(∂Z(Λ)m

∂λi

)′− ∂Z(Λ)′

m

∂λi

(Σ̂−1 ∂Σ̂

∂λk

)]Σ̂−1Z(Λ)

( j∗) +∂Z(Λ)′

m

∂λiΣ̂−1

∂Z(Λ)( j∗)

∂λk

−∂Z(Λ)′m

∂λk

(Σ̂−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

( j∗)

)+Z(Λ)′

m Σ̂−1 ∂Σ̂

∂λkΣ̂−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

( j∗)

−Z(Λ)′m Σ̂

−1 ∂

∂λk

(∂Σ̂

∂λi

)Σ̂−1Z(Λ)

( j∗) +Z(Λ)′m Σ̂

−1 ∂Σ̂

∂λiΣ̂−1 ∂Σ̂

∂λkΣ̂−1Z(Λ)

( j∗)

−Z(Λ)′m Σ̂

−1 ∂Σ̂

∂λiΣ̂−1

∂Z(Λ)( j∗)

∂λk,

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for k = i,

∂2U( j)

∂λ2i

= Z(Λ)′m Σ̂

−1∂2Z(Λ)

( j∗)

∂λ2i−2Z(Λ)′

m Σ̂−1 ∂Σ̂

∂λiΣ̂−1

∂Z(Λ)( j∗)

∂λi+2

∂Z(Λ)′m

∂λiΣ̂−1

∂Z(Λ)( j∗)

∂λi

−2∂Z(Λ)′

m

∂λiΣ̂−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

( j∗) +∂2Z(Λ)′

m

∂λ2i

Σ̂−1Z(Λ)

( j∗) +2Z(Λ)′m Σ̂

−1 ∂Σ̂

∂λiΣ̂−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

( j∗)

−Z(Λ)′m Σ̂

−1 ∂2Σ̂

∂λ2i

Σ̂−1Z(Λ)

( j∗),

∂2Z(Λ)′m Σ̂−1Z(Λ)

m

∂λ2i

= 2∂Z(Λ)′

m

∂λiΣ̂−1 ∂Z(Λ)

m

∂λi−4

∂Z(Λ)′m

∂λiΣ̂−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

m

+2∂2Z(Λ)′

m

∂λ2i

Σ̂−1Z(Λ)

m +2Z(Λ)m Σ̂

−1 ∂Σ̂

∂λiΣ̂−1 ∂Σ̂

∂λiΣ̂−1Z(Λ)

( j∗)−Z(Λ)m Σ̂

−1 ∂2Σ̂

∂λ2i

Σ̂−1Z(Λ)

( j∗),

∂2Z(Λ)( j∗)

∂λ2i

is a vector of zeros except for the ith element which is

d(2)j∗i =

(λ3

i (logYi j∗)2Y λii j∗−2λ2

i (logYi j∗)Yλii j∗+2λiY

λii j∗−2λi

λ4i

)

−1n

n

∑l=1

(λ3

i (logYil)2Y λiil −2λ2

i (logYil)Yλiil +2λiY

λiil −2λi

λ4i

)

and ∂2Z(Λ)′m

∂λ2i

is a vector of zeros except for the ith element which is

d(2)mi =

(λ3

i (logYim)2Y λiim −2λ2

i (logYim)Y λiim +2λiY

λiim −2λi

λ4i

)

−1n

n

∑l=1

(λ3

i (logYil)2Y λiil −2λ2

i (logYil)Yλiil +2λiY

λiil −2λi

λ4i

)

and ∂2Σ̂2

∂λ2i

is defined in Section 10.3.

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10.5. Simulation Study

One thousand samples of size 50 were selected from a bivariate normal popula-tion for each parameter combination given in the table below. The means andthe standard deviations were kept fixed. In the table, W0 indicates the multi-variate Shapiro-Wilk statistic for the generated sample, W (Λ̂L) is the multivari-ate Shapiro-Wilk statistic for the transformed data using the maximum likelihoodmethod, and W (Λ̂W ) is the multivariate Shapiro-Wilk statistic for the transformeddata using the maximization of the multivariate Shapiro-Wilk statistic method. Forthe maximum likelihood method, the Newton-Raphson algorithm is implementedas described in section 10.3. When applying the Newton-Raphson algorithm toimplement the maximization of the multivariate Shapiro-Wilk statistic method,different starting values yielded various local maximum values and as a result theNewton-Raphson algorithm was not used for this method. Thus, a grid searchusing a three standard deviation range of the maximum likelihood estimate is im-plemented in the maximization of the multivariate Shapiro-Wilk statistic method.The grid search procedure for the maximization of the likelihood method yieldedsimilar estimates as the Newton-Raphson method.The coefficients ai’s in the Shapiro-Wilk W statistic were obtained from Parish(1992a and 1992b) as the most accurate values available. Means (m) and standarddeviations (s) are given for the estimates of the Box-Cox transformation parame-ters and their corresponding W statistics are displayed in Table 1.

The means of the W statistics are consistently higher with lower standard er-rors for the maximization of the W statistic method in comparison to the maximumlikelihood method.The biases in estimating the Box-Cox parameters are lower for the maximizationof the W statistic method but the standard errors of the estimates are higher whichleads to higher mean squared errors.

References

1. Andrews, D. F., R. Gnanadesikan, and J. L. Warner (1971). Transformations of Multi-variate Data. Biometrics, 27, 825-840.

2. Bickel, P. J. and K. A. Doksum (1981). An Analysis of Transformations Revisited.Journal of the American Statistical Association, 76, 296-311.

3. Box, G. E. P. and D. R. Cox (1964). An Analysis of Transformations. Journal of theRoyal Statistical Society, Series B.,26, 211-252.

4. Box, G. E. P. and D. R. Cox (1982). An Analysis of Transformations Revisited (Rebut-ted). Journal of the American Statistical Association, 77, 209-210.

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Table 10.1. Simulation Results Displaying the Means and the Standard Errors ofthe Estimates

W0 W (Λ̂L) W (Λ̂W ) λ̂1L λ̂2L λ̂1W λ̂2W

Normal: µ1 =−5, µ2 = 10, σ1=1, σ2=2, ρ =−0.7, λ1=-2, λ2=1m 0.9311 0.9775 0.9870 -1.9075 0.9455 -2.1225 1.0231s 0.0555 0.0117 0.0057 1.1517 0.5772 1.8002 0.8968


Normal: µ1 =−5, µ2 = 10, σ1=1, σ2=2, ρ = 0, λ1=-2, λ2=1m 0.9410 0.9785 0.9867 -1.9039 0.9189 -2.0395 1.0229s 0.0464 0.0104 0.0058 1.2688 0.6549 1.9481 0.9905

Normal: µ1 =−5, µ2 = 10, σ1=1, σ2=2, ρ = 0.3, λ1=-2, λ2=1m 0.9402 0.9785 0.9866 -1.8439 0.9734 -1.9180 0.9997s 0.0497 0.0107 0.0059 1.3060 0.6535 2.0447 0.9904




Normal: µ1 =−5, µ2 = 10, σ1=1, σ2=2, ρ = 0, λ1=-1, λ2=2m 0.9273 0.9786 0.9865 -0.9279 1.9002 -1.0350 1.9636s 0.0547 0.0114 0.0055 0.7550 1.2833 1.1268 1.9926



5. Carroll, R. J. (1980). A Robust Method for Testing Transformations to Achieve Ap-proximate Normality. Journal of the Royal Statistical Society, Series B., 42, 71-78.

6. Halawa, Adel M. (1996). Estimating the Box-Cox Transformation via an ArtificialRegression Model. Communications in Statistics Simulation and Computation, 25(2),331-350.

7. Harville, David A. (1999). Matrix Algebra From A Statistician’s Perspective. Springer,New York.

8. Hinkley, D. V. (1975). On Power Transformation to Symmetry. Biometrika, 62, 101-111.

9. Hinkley, D. V. (1977). On Quick Choice of Power Transformation. Applied Statistics,26, 67-68.

10. Lin, L. I. and E. F. Vonesh (1989). An Empirical Nonlinear Data-Fitting Approach forTransforming Data to Normality. American Statistician, 43, 237-243.

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11. Malkovich, J. F. and Afifi, A. A. (1973). On Tests for Multivariate Normality. Journalof the American Statistical Association, 68(341), 176-179.

12. Mecklin, Christopher J. and Mundfrom, Daniel J. (2004). An Appraisal and Bibli-ography of Tests for Multivariate Normality, International Statistical Review, 72(1),123-138.

13. Parish, Rudolph S. (1992a). Computing Expected Values of Normal Order Statistics.Communications in Statistics - Simulation and Computation, 21(1), 57-70.

14. Parish, Rudolph S. (1992b). Computing Variances and Covariances of Normal OrderStatistics. Communications in Statistics - Simulation and Computation, 21(1), 71-101.

15. Rahman, Mezbahur (1999). Estimating the Box-Cox Transformation via Shapiro-WilkW Statistic. Communications in Statistics - Simulation and Computation, 28(1), 223-241.

16. Shapiro, S. S. and M. B. Wilk (1965). An Analysis of Variance Test for Normality.Biometrika, 52, 3 and 4, 591-611.

17. Taylor, J. M. G. (1985). Power Transformations to Symmetry. Annals of MathematicalStatistics, 33, 1-67.

18. Velilla, Santiago (1993). A Note on the Multivariate Box-Cox Transformation. Statis-tics & Probability Letters, 17, 259-263.

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