Statistic of the BHEP-test

Description

This function returns the value of the statistic of the Baringhaus-Henze-Epps-Pulley (BHEP) test as in Henze and Wagner (1997).

Usage

BHEP(data, a = 1)

Arguments

Details

The test statistic is

BHEP_{n,\beta}=\frac{1}{n} \sum_{j,k=1}^n \exp\left(-\frac{\beta^2\|Y_{n,j}-Y_{n,k}\|^2}{2}\right)- \frac{2}{(1+\beta^2)^{d/2}} \sum_{j=1}^n \exp\left(- \frac{\beta^2\|Y_{n,j}\|^2}{2(1+\beta^2)} \right) + \frac{n}{(1+2\beta^2)^{d/2}}.

Here, Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,n, are the scaled residuals, \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error.

Value

value of the test statistic.

References

Henze, N., and Wagner, T. (1997), A new approach to the class of BHEP tests for multivariate normality, J. Multiv. Anal., 62:1–23, DOI

Epps T.W., Pulley L.B. (1983), A test for normality based on the empirical characteristic function, Biometrika, 70:723-726, DOI

Examples

BHEP(MASS::mvrnorm(50,c(0,1),diag(1,2)))

Statistic of the test of Cox and Small

Description

This function returns the (approximated) value of the test statistic of the test of Cox and Small (1978).

Usage

CS(data, Points = NULL)

Arguments

Details

The test statistic is T_{n,CS}=\max_{b\in\{x\in\mathbf{R}^d:\|x\|=1\}}\eta_n^2(b), where

\eta_n^2(b)=\frac{\left\|n^{-1}\sum_{j=1}^nY_{n,j}(b^\top Y_{n,j})^2\right\|^2-\left(n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^3\right)^2}{n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^4-1-\left(n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^3\right)^2}

. Here, Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,n, are the scaled residuals, \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error. Note that the maximum functional has to be approximated by a discrete version, for details see Ebner (2012).

Value

approximation of the value of the test statistic of the test of Cox and Small (1978).

References

Cox, D.R. and Small, N.J.H. (1978), Testing multivariate normality, Biometrika, 65:263–272.

Ebner, B. (2012), Asymptotic theory for the test for multivariate normality by Cox and Small, Journal of Multivariate Analysis, 111:368–379.

Examples

CS(MASS::mvrnorm(50,c(0,1),diag(1,2)))

Statistic of the DEH test based on harmonic oscillator

Description

Computes the test statistic of the DEH test.

Usage

DEHT(data, a = 1)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

The value of the test statistic.

References

Dörr, P., Ebner, B., Henze, N. (2019) "Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces" arXiv:1909.12624

Examples

DEHT(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=1)

Statistic of the DEH test based on a double estimation in PDE

Description

Computes the test statistic of the DEH based on a double estimation in PDE test.

Usage

DEHU(data, a)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

The value of the test statistic.

References

Dörr, P., Ebner, B., Henze, N. (2019) "A new test of multivariate normality by a double estimation in a characterizing PDE" arXiv:1911.10955

Statistic of the EHS test based on a multivariate Stein equation

Description

Computes the test statistic of the EHS test based on a multivariate Stein equation.

Usage

EHS(data, a = 1)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Note that a=Inf returns the limiting test statistic with value 2*MSkew + MRSSkew and a=0 returns the value of the limit statistic

T_{n,0}=\frac{d}{2}-2^{\frac{d}{2}+1}\frac{1}{n}\sum_{j=1}^n\|Y_{n,j}\|^2\exp(-\frac{\|Y_{n,j}\|^2}{2}).

Value

The value of the test statistic.

References

Ebner, B., Henze, N., Strieder, D. (2020) "Testing normality in any dimension by Fourier methods in a multivariate Stein equation" arXiv:2007.02596

Henze-Jiménes-Gamero test statistic

Description

Computes the test statistic of the Henze-Jimenes-Gamero test.

Usage

HJG(data, a = 5)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error.

Value

The value of the test statistic.

References

Henze, N., Jiménez-Gamero, M.D. (2019) "A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function", TEST, 28, 499-521, DOI

Examples

HJG(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=5)

statistic of the Henze-Jiménes-Gamero-Meintanis test

Description

Computes the test statistic of the Henze-Jiménes-Gamero-Meintanis test.

Usage

HJM(data, a)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error.

Value

The value of the test statistic.

References

Henze, N., Jiménes-Gamero, M.D., Meintanis, S.G. (2019), Characterizations of multinormality and corresponding tests of fit, including for GARCH models, Econometric Th., 35:510–546, DOI.

Examples

HJM(MASS::mvrnorm(20,c(0,1),diag(1,2)),a=2.5)

statistic of the Henze-Visagie test

Description

Computes the test statistic of the Henze-Visagie test.

Usage

HV(data, a = 5)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error.

Note that a=Inf returns the limiting test statistic with value 2*MSkew + MRSSkew.

Value

The value of the test statistic.

References

Henze, N., Visagie, J. (2019) "Testing for normality in any dimension based on a partial differential equation involving the moment generating function", to appear in Ann. Inst. Stat. Math., DOI

Examples

HV(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=5)
HV(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=Inf)

Statistic of the Henze-Zirkler test

Description

This function returns the value of the statistic of the BHEP test as in Henze and Zirkler (1990). The difference to the BHEP test is in the choice of the tuning parameter \beta.

Usage

HZ(data)

Arguments

Details

A BHEP test is performed with tuning parameter \beta chosen in dependence of the sample size n and the dimension d, namely

\beta=\frac{((2d+1)n/4)^(1/(d+4))}{\sqrt{2}}.

Value

value of the test statistic.

References

Henze, N., and Zirkler, B. (1990), A class of invariant consistent tests for multivariate normality, Commun.-Statist. – Th. Meth., 19:3595–3617, DOI

See Also

BHEP

Examples

HZ(MASS::mvrnorm(50,c(0,1),diag(1,2)))

Koziols measure of multivariate sample kurtosis

Description

This function computes the invariant measure of multivariate sample kurtosis due to Koziol (1989).

Usage

KKurt(data)

Arguments

Details

Multivariate sample kurtosis due to Koziol (1989) is defined by

\widetilde{b}_{n,d}^{(2)}=\frac{1}{n^2}\sum_{j,k=1}^n(Y_{n,j}^\top Y_{n,k})^4,

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,n, are the scaled residuals, \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error. Note that for d=1, we have a measure proportional to the squared sample kurtosis.

Value

value of sample kurtosis in the sense of Koziol.

References

Koziol, J.A. (1989), A note on measures of multivariate kurtosis, Biom. J., 31:619–624.

Examples

KKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)))

multivariate kurtosis in the sense of Malkovich and Afifi

Description

This function computes the invariant measure of multivariate sample kurtosis due to Malkovich and Afifi (1973).

Usage

MAKurt(data, Points = NULL)

Arguments

Details

Multivariate sample skewness due to Malkovich and Afifi (1973) is defined by

b_{n,d,M}^{(1)}=\max_{u\in \{x\in\mathbf{R}^d:\|x\|=1\}}\frac{\left(\frac{1}{n}\sum_{j=1}^n(u^\top X_j-u^\top \overline{X}_n )^3\right)^2}{(u^\top S_n u)^3},

where \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error.

Value

value of sample kurtosis in the sense of Malkovich and Afifi.

References

Malkovich, J.F., and Afifi, A.A. (1973), On tests for multivariate normality, J. Amer. Statist. Ass., 68:176–179.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467–506.

Examples

MAKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)))

multivariate skewness in the sense of Malkovich and Afifi

Description

This function computes the invariant measure of multivariate sample skewness due to Malkovich and Afifi (1973).

Usage

MASkew(data, Points = NULL)

Arguments

Details

Multivariate sample skewness due to Malkovich and Afifi (1973) is defined by

b_{n,d,M}^{(1)}=\max_{u\in \{x\in\mathbf{R}^d:\|x\|=1\}}\frac{\left(\frac{1}{n}\sum_{j=1}^n(u^\top X_j-u^\top \overline{X}_n )^3\right)^2}{(u^\top S_n u)^3},

where \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error.

Value

value of sample skewness in the sense of Malkovich and Afifi.

References

Malkovich, J.F., and Afifi, A.A. (1973), On tests for multivariate normality, J. Amer. Statist. Ass., 68:176–179.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467–506.

Examples

MASkew(MASS::mvrnorm(50,c(0,1),diag(1,2)))

Mardias measure of multivariate sample kurtosis

Description

This function computes the classical invariant measure of multivariate sample kurtosis due to Mardia (1970).

Usage

MKurt(data)

Arguments

Details

Multivariate sample kurtosis due to Mardia (1970) is defined by

b_{n,d}^{(2)}=\frac{1}{n}\sum_{j=1}^n\|Y_{n,j}\|^4,

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n.To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error.

Value

value of sample kurtosis in the sense of Mardia.

References

Mardia, K.V. (1970), Measures of multivariate skewness and kurtosis with applications, Biometrika, 57:519–530.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467–506.

Examples

MKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)))

first statistic of Manzotti and Quiroz

Description

This function returns the value of the first statistic of Manzotti and Quiroz (2001).

Usage

MQ1(data)

Arguments

Value

Value of the test statistic

References

Manzotti, A., and Quiroz, A.J. (2001), Spherical harmonics in quadratic forms for testing multivariate normality, Test, 10:87–104, DOI

Examples

MQ1(MASS::mvrnorm(50,c(0,1),diag(1,2)))

second statistic of Manzotti und Quiroz

Description

This function returns the value of the second statistic of Manzotti und Quiroz (2001).

Usage

MQ2(data)

Arguments

Value

Value of the test statistic

References

Manzotti, A., and Quiroz, A.J. (2001), Spherical harmonics in quadratic forms for testing multivariate normality, Test, 10:87–104, DOI

Examples

MQ2(MASS::mvrnorm(50,c(0,1),diag(1,2)))

multivariate skewness of Móri, Rohatgi and Székely

Description

This function computes the invariant measure of multivariate sample skewness due to Móri, Rohatgi and Székely (1993).

Usage

MRSSkew(data)

Arguments

Details

Multivariate sample skewness due to Móri, Rohatgi and Székely (1993) is defined by

\widetilde{b}_{n,d}^{(1)}=\frac{1}{n}\sum_{j=1}^n\|Y_{n,j}\|^2\|Y_{n,k}\|^2Y_{n,j}^\top Y_{n,k},

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error. Note that for d=1, it is equivalent to skewness in the sense of Mardia.

Value

value of sample skewness in the sense of Móri, Rohatgi and Székely.

References

Móri, T. F., Rohatgi, V. K., Székely, G. J. (1993), On multivariate skewness and kurtosis, Theory of Probability and its Applications, 38:547–551.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467–506.

Mardias measure of multivariate sample skewness

Description

This function computes the classical invariant measure of multivariate sample skewness due to Mardia (1970).

Usage

MSkew(data)

Arguments

Details

Multivariate sample skewness due to Mardia (1970) is defined by

b_{n,d}^{(1)}=\frac{1}{n^2}\sum_{j,k=1}^n(Y_{n,j}^\top Y_{n,k})^3,

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the function returns an error. Note that for d=1, we have a measure proportional to the squared sample skewness.

Value

value of sample skewness in the sense of Mardia.

References

Mardia, K.V. (1970), Measures of multivariate skewness and kurtosis with applications, Biometrika, 57:519–530.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467–506.

Examples

MSkew(MASS::mvrnorm(50,c(0,1),diag(1,2)))

Statistic of the Pudelko test

Description

Approximates the test statistic of the Pudelko test.

Usage

PU(data, r = 2)

Arguments

Details

This functions evaluates the test statistic with the given data and the specified parameter r. Since since one has to calculate the supremum of a function inside a d-dimensional Ball of radius r. In this implementation the optim function is used.

Value

approximate Value of the test statistic

References

Pudelko, J. (2005), On a new affine invariant and consistent test for multivariate normality, Probab. Math. Statist., 25:43–54.

Examples

PU(MASS::mvrnorm(20,c(0,1),diag(1,2)),r=2)

Simulated empirical 90% quantiles of the tests contained in package mnt

Description

A dataset containing the empirical 0.9 quantiles of the tests for the dimensions d=2,3,5 and samplesizes n=20,50,100 based on a Monte Carlo Simulation study with 100000 repetitions. The following parameters were used:

• For BHEP the parameter a=1,

• for HV the parameter a=5,

• for HJG the parameter a=1.5,

• for HJM the parameter a=1.5,

• for DEHT the parameter a=0.25,

• for DEHU the parameter a=0.5,

• for CS the parameter Points=NULL,

• for PU the parameter r=2,

• for MASkew the parameter Points=NULL,

• for MAKurt the parameter Points=NULL,

Usage

Quantile09

Format

A data frame with 9 rows and 20 columns.

Simulated empirical 95% quantiles of the tests contained in package mnt

Description

A dataset containing the empirical 0.95 quantiles of the tests for the dimensions d=2,3,5 and samplesizes n=20,50,100 based on a Monte Carlo Simulation study with 100000 repetitions. The following parameters were used:

• For BHEP the parameter a=1,

• for HV the parameter a=5,

• for HJG the parameter a=1.5,

• for HJM the parameter a=1.5,

• for DEHT the parameter a=0.25,

• for DEHU the parameter a=0.5,

• for CS the parameter Points=NULL,

• for PU the parameter r=2,

• for MASkew the parameter Points=NULL,

• for MAKurt the parameter Points=NULL,

Usage

Quantile095

Format

A data frame with 9 rows and 20 columns.

Simulated empirical 99% quantiles of the tests contained in package mnt

Description

A dataset containing the empirical 0.99 quantiles of the tests for the dimensions d=2,3,5 and samplesizes n=20,50,100 based on a Monte Carlo Simulation study with 100000 repetitions. The following parameters were used:

• For BHEP the parameter a=1,

• for HV the parameter a=5,

• for HJG the parameter a=1.5,

• for HJM the parameter a=1.5,

• for DEHT the parameter a=0.25,

• for DEHU the parameter a=0.5,

• for CS the parameter Points=NULL,

• for PU the parameter r=2,

• for MASkew the parameter Points=NULL,

• for MAKurt the parameter Points=NULL,

Usage

Quantile099

Format

A data frame with 9 rows and 20 columns.

statistic of the Székely-Rizzo test

Description

This function returns the value of the statistic of the test of multivariate normality (also called energy test) as in Székely and Rizzo (2005). Note that the scaled residuals use another scaling in the estimator of the covariance matrix as the other functions of the package mnt! It is equivalent to the function mvnorm.e.

Usage

SR(data, abb = 1e-08)

Arguments

Value

value of the test statistic.

References

Székely, G., and Rizzo, M. (2005), A new test for multivariate normality, J. Multiv. Anal., 93:58–80, DOI

See Also

mvnorm.e

Examples

SR(MASS::mvrnorm(50,c(0,1),diag(1,2)))

Monte Carlo simulation of quantiles for normality tests

Description

This function returns the quantiles of a test statistic with optional tuning parameter.

Usage

cv.quan(
  samplesize,
  dimension,
  quantile,
  statistic,
  tuning = NULL,
  repetitions = 1e+05
)

Arguments

Value

empirical quantile of the test statistic.

Examples

cv.quan(samplesize=10, dimension=2,quantile=0.95, statistic=BHEP, tuning=2.5, repetitions=1000)

Print method for tests of multivariate normality

Description

Printing objects of class "mnt".

Usage

## S3 method for class 'mnt'
print(x, ...)

Arguments

Details

A mnt object is a named list of numbers and character string, supplemented with test (the name of the teststatistic). test is displayed as a title. The remaining elements are given in an aligned "name = value" format.

Value

the argument x, invisibly, as for all print methods.

Examples

print(test.DEHU(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=1,MC=500))

Empirical scaled residuals

Description

A function that computes the scaled residuals of the data.

Usage

standard(data)

Arguments

Value

A n x d matrix of the scaled residuals.

Baringhaus-Henze-Epps-Pulley (BHEP) test

Description

Performs the BHEP test of multivariate normality as suggested in Henze and Wagner (1997) using a tuning parameter a.

Usage

test.BHEP(data, a = 1, MC.rep = 10000, alpha = 0.05)

Arguments

Details

The test statistic is

BHEP_{n,\beta}=\frac{1}{n} \sum_{j,k=1}^n \exp\left(-\frac{\beta^2\|Y_{n,j}-Y_{n,k}\|^2}{2}\right)- \frac{2}{(1+\beta^2)^{d/2}} \sum_{j=1}^n \exp\left(- \frac{\beta^2\|Y_{n,j}\|^2}{2(1+\beta^2)} \right) + \frac{n}{(1+2\beta^2)^{d/2}}.

Here, Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,n, are the scaled residuals, \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

#'

References

Henze, N., Wagner, T. (1997), A new approach to the class of BHEP tests for multivariate normality, J. Multiv. Anal., 62:1-23, DOI

See Also

BHEP

Examples

test.BHEP(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

multivariate normality test of Cox and Small

Description

Performs the test of multivariate normality of Cox and Small (1978).

Usage

test.CS(data, MC.rep = 1000, alpha = 0.05, Points = NULL)

Arguments

Details

The test statistic is T_{n,CS}=\max_{b\in\{x\in\mathbf{R}^d:\|x\|=1\}}\eta_n^2(b), where

\eta_n^2(b)=\frac{\left\|n^{-1}\sum_{j=1}^nY_{n,j}(b^\top Y_{n,j})^2\right\|^2-\left(n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^3\right)^2}{n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^4-1-\left(n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^3\right)^2}

. Here, Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,n, are the scaled residuals, \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error. Note that the maximum functional has to be approximated by a discrete version, for details see Ebner (2012).

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

#'

References

Cox, D.R., Small, N.J.H. (1978), Testing multivariate normality, Biometrika, 65:263-272.

Ebner, B. (2012), Asymptotic theory for the test for multivariate normality by Cox and Small, Journal of Multivariate Analysis, 111:368-379.

See Also

CS

Examples

test.CS(MASS::mvrnorm(10,c(0,1),diag(1,2)),MC.rep=100)

Doerr-Ebner-Henze test of multivariate normality based on harmonic oscillator

Description

Computes the multivariate normality test of Doerr, Ebner and Henze (2019) based on zeros of the harmonic oscillator.

Usage

test.DEHT(data, a = 1, MC.rep = 10000, alpha = 0.05)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Doerr, P., Ebner, B., Henze, N. (2019) "Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces" arXiv:1909.12624

See Also

DEHT

Examples

test.DEHT(MASS::mvrnorm(20,c(0,1),diag(1,2)),a=1,MC=500)

Doerr-Ebner-Henze test of multivariate normality based on a double estimation in a PDE

Description

Computes the multivariate normality test of Doerr, Ebner and Henze (2019) based on a double estimation in a PDE.

Usage

test.DEHU(data, a = 0.5, MC.rep = 10000, alpha = 0.05)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Doerr, P., Ebner, B., Henze, N. (2019) "Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces" arXiv:1909.12624

See Also

DEHU

Examples

test.DEHU(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=1,MC=500)

Ebner-Henze-Strieder test of multivariate normality based on Fourier methods in a multivariate Stein equation

Description

Computes the multivariate normality test of Ebner, Henze and Strieder (2020) based on Fourier methods in a multivariate Stein equation.

Usage

test.EHS(data, a = 0.5, MC.rep = 10000, alpha = 0.05)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Ebner, B., Henze, N., Strieder, D. (2020) "Testing normality in any dimension by Fourier methods in a multivariate Stein equation" arXiv:2007.02596

See Also

EHS

Examples

test.EHS(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=1,MC=500)

Henze-Jimenes-Gamero test of multivariate normality

Description

Computes the multivariate normality test of Henze and Jimenes-Gamero (2019) in dependence of a tuning parameter a.

Usage

test.HJG(data, a = 1, MC.rep = 10000, alpha = 0.05)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

#'

References

Henze, N., Jimenez-Gamero, M.D. (2019) "A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function", TEST, 28, 499-521, DOI

See Also

HJG

Examples

test.HJG(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=1.5,MC.rep=500)

Henze-Jimenes-Gamero-Meintanis test of multivariate normality

Description

Computes the test statistic of the Henze-Jimenes-Gamero-Meintanis test.

Usage

test.HJM(data, a = 1.5, MC.rep = 500, alpha = 0.05)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Henze, N., Jimenes-Gamero, M.D., Meintanis, S.G. (2019), Characterizations of multinormality and corresponding tests of fit, including for GARCH models, Econometric Th., 35:510-546, DOI.

See Also

HJM

Examples

test.HJM(MASS::mvrnorm(10,c(0,1),diag(1,2)),a=2.5,MC=100)

The Henze-Visagie test of multivariate normality

Description

Computes the multivariate normality test of Henze and Visagie (2019).

Usage

test.HV(data, a = 5, MC.rep = 10000, alpha = 0.05)

Arguments

Details

This functions evaluates the teststatistic with the given data and the specified tuning parameter a. Each row of the data Matrix contains one of the n (multivariate) sample with dimension d. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Note that a=Inf returns the limiting test statistic with value 2*MSkew + MRSSkew.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Henze, N., Visagie, J. (2019) "Testing for normality in any dimension based on a partial differential equation involving the moment generating function", to appear in Ann. Inst. Stat. Math., DOI

See Also

HV

Examples

test.HV(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=5,MC.rep=500)
test.HV(MASS::mvrnorm(50,c(0,1),diag(1,2)),a=Inf,MC.rep=500)

The Henze-Zirkler test

Description

Performs the test of multivariate normality of Henze and Zirkler (1990).

Usage

test.HZ(data, MC.rep = 10000, alpha = 0.05)

Arguments

Details

A BHEP test is performed with tuning parameter \beta chosen in dependence of the sample size n and the dimension d, namely

\beta=\frac{((2d+1)n/4)^(1/(d+4))}{\sqrt{2}}.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

#'

References

Henze, N., Zirkler, B. (1990), A class of invariant consistent tests for multivariate normality, Commun.-Statist. - Th. Meth., 19:3595-3617, DOI

See Also

HZ

Examples

test.HZ(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

Test of normality based on Koziols measure of multivariate sample kurtosis

Description

Computes the multivariate normality test based on the invariant measure of multivariate sample kurtosis due to Koziol (1989).

Usage

test.KKurt(data, MC.rep = 10000, alpha = 0.05)

Arguments

Details

Multivariate sample kurtosis due to Koziol (1989) is defined by

\widetilde{b}_{n,d}^{(2)}=\frac{1}{n^2}\sum_{j,k=1}^n(Y_{n,j}^\top Y_{n,k})^4,

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,n, are the scaled residuals, \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error. Note that for d=1, we have a measure proportional to the squared sample kurtosis.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Koziol, J.A. (1989), A note on measures of multivariate kurtosis, Biom. J., 31:619-624.

See Also

KKurt

Examples

test.KKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

Test of normality based on multivariate kurtosis in the sense of Malkovich and Afifi

Description

Computes the multivariate normality test based on the invariant measure of multivariate sample kurtosis due to Malkovich and Afifi (1973).

Usage

test.MAKurt(data, MC.rep = 10000, alpha = 0.05, num.points = 1000)

Arguments

Details

Multivariate sample skewness due to Malkovich and Afifi (1973) is defined by

b_{n,d,M}^{(1)}=\max_{u\in \{x\in\mathbf{R}^d:\|x\|=1\}}\frac{\left(\frac{1}{n}\sum_{j=1}^n(u^\top X_j-u^\top \overline{X}_n )^3\right)^2}{(u^\top S_n u)^3},

where \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

number of points used in approximation.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Malkovich, J.F., and Afifi, A.A. (1973), On tests for multivariate normality, J. Amer. Statist. Ass., 68:176-179.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467-506.

See Also

MAKurt

Examples

test.MAKurt(MASS::mvrnorm(10,c(0,1),diag(1,2)),MC.rep=100)

Test of normality based on multivariate skewness in the sense of Malkovich and Afifi

Description

Computes the test of multivariate normality based on skewness in the sense of Malkovich and Afifi (1973).

Usage

test.MASkew(data, MC.rep = 10000, alpha = 0.05, num.points = 1000)

Arguments

Details

Multivariate sample skewness due to Malkovich and Afifi (1973) is defined by

b_{n,d,M}^{(1)}=\max_{u\in \{x\in\mathbf{R}^d:\|x\|=1\}}\frac{\left(\frac{1}{n}\sum_{j=1}^n(u^\top X_j-u^\top \overline{X}_n )^3\right)^2}{(u^\top S_n u)^3},

where \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

number of points used in approximation.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Malkovich, J.F., and Afifi, A.A. (1973), On tests for multivariate normality, J. Amer. Statist. Ass., 68:176-179.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467-506.

See Also

MASkew

Examples

test.MASkew(MASS::mvrnorm(10,c(0,1),diag(1,2)),MC.rep=100)

Test of normality based on Mardias measure of multivariate sample kurtosis

Description

Computes the multivariate normality test based on the classical invariant measure of multivariate sample kurtosis due to Mardia (1970).

Usage

test.MKurt(data, MC.rep = 10000, alpha = 0.05)

Arguments

Details

Multivariate sample kurtosis due to Mardia (1970) is defined by

b_{n,d}^{(2)}=\frac{1}{n}\sum_{j=1}^n\|Y_{n,j}\|^4,

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n.To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Mardia, K.V. (1970), Measures of multivariate skewness and kurtosis with applications, Biometrika, 57:519-530.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467-506.

See Also

MKurt

Examples

test.MKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

Manzotti-Quiroz test 1

Description

Performs the first test of multivariate normality of Manzotti and Quiroz (2001).

Usage

test.MQ1(data, MC.rep = 10000, alpha = 0.05)

Arguments

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

#'

References

Manzotti, A., Quiroz, A.J. (2001), Spherical harmonics in quadratic forms for testing multivariate normality, Test, 10:87-104, DOI

See Also

MQ1

Examples

test.MQ1(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=100)

Manzotti-Quiroz test 2

Description

Performs the second test of multivariate normality of Manzotti and Quiroz (2001).

Usage

test.MQ2(data, MC.rep = 10000, alpha = 0.05)

Arguments

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

#'

References

Manzotti, A., Quiroz, A.J. (2001), Spherical harmonics in quadratic forms for testing multivariate normality, Test, 10:87-104, DOI

See Also

MQ2

Examples

test.MQ2(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

Test of multivariate normality based on the measure of multivariate skewness of Mori, Rohatgi and Szekely

Description

Computes the multivariate normality test based on the invariant measure of multivariate sample skewness due to Mori, Rohatgi and Szekely (1993).

Usage

test.MRSSkew(data, MC.rep = 10000, alpha = 0.05)

Arguments

Details

Multivariate sample skewness due to Mori, Rohatgi and Szekely (1993) is defined by

\widetilde{b}_{n,d}^{(1)}=\frac{1}{n}\sum_{j=1}^n\|Y_{n,j}\|^2\|Y_{n,k}\|^2Y_{n,j}^\top Y_{n,k},

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error. Note that for d=1, it is equivalent to skewness in the sense of Mardia.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Mori, T. F., Rohatgi, V. K., Szekely, G. J. (1993), On multivariate skewness and kurtosis, Theory of Probability and its Applications, 38:547-551.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467-506.

See Also

MRSSkew

Examples

test.MRSSkew(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

Test of normality based on Mardias measure of multivariate sample skewness

Description

Computes the multivariate normality test based on the classical invariant measure of multivariate sample skewness due to Mardia (1970).

Usage

test.MSkew(data, MC.rep = 10000, alpha = 0.05)

Arguments

Details

Multivariate sample skewness due to Mardia (1970) is defined by

b_{n,d}^{(1)}=\frac{1}{n^2}\sum_{j,k=1}^n(Y_{n,j}^\top Y_{n,k})^3,

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error. Note that for d=1, we have a measure proportional to the squared sample skewness.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Mardia, K.V. (1970), Measures of multivariate skewness and kurtosis with applications, Biometrika, 57:519-530.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467-506.

See Also

MSkew

Examples

test.MSkew(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

Pudelko test of multivariate normality

Description

Computes the (approximated) Pudelko test of multivariate normality.

Usage

test.PU(data, MC.rep = 10000, alpha = 0.05, r = 2)

Arguments

Details

This functions evaluates the test statistic with the given data and the specified parameter r. Since since one has to calculate the supremum of a function inside a d-dimensional Ball of radius r. In this implementation the optim function is used.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$param

value tuning parameter.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

References

Pudelko, J. (2005), On a new affine invariant and consistent test for multivariate normality, Probab. Math. Statist., 25:43-54.

See Also

PU

Examples

test.PU(MASS::mvrnorm(20,c(0,1),diag(1,2)),r=2,MC=100)

Szekely-Rizzo (energy) test

Description

Performs the test of multivariate normality of Szekely and Rizzo (2005). Note that the scaled residuals use another scaling in the estimator of the covariance matrix!

Usage

test.SR(data, MC.rep = 10000, alpha = 0.05, abb = 1e-08)

Arguments

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test

name of the test.

$Test.value

the value of the test statistic.

$cv

the approximated critical value.

$Decision

the comparison of the critical value and the value of the test statistic.

#'

References

Szekely, G., Rizzo, M. (2005), A new test for multivariate normality, J. Multiv. Anal., 93:58-80, DOI

See Also

SR

Examples

test.SR(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)