Package 'mnt'

mnt: Affine Invariant Tests of Multivariate Normality

Description

Various affine invariant multivariate normality tests are provided. It is designed to accompany the survey article Ebner, B. and Henze, N. (2020) doi:10.1007/s11749-020-00740-0 titled "Tests for multivariate normality – a critical review with emphasis on weighted L^2-statistics". We implement new and time honoured L^2-type tests of multivariate normality, such as the Baringhaus-Henze-Epps-Pulley (BHEP) test, the Henze-Zirkler test, the test of Henze-Jiménes-Gamero, the test of Henze-Jiménes-Gamero-Meintanis, the test of Henze-Visage, the Dörr-Ebner-Henze test based on harmonic oscillator and the Dörr-Ebner-Henze test based on a double estimation in a PDE. Secondly, we include the measures of multivariate skewness and kurtosis by Mardia, Koziol, Malkovich and Afifi and Móri, Rohatgi and Székely, as well as the associated tests. Thirdly, we include the tests of multivariate normality by Cox and Small, the 'energy' test of Székely and Rizzo, the tests based on spherical harmonics by Manzotti and Quiroz and the test of Pudelko. All the functions and tests need the data to be a n x d matrix where n is the samplesize (number of rows) and d is the dimension (number of columns).

Use with model residuals

The tests implemented in mnt are calibrated for independent and identically distributed observations. Applying them to residuals from fitted models requires additional justification, because parameter estimation may change the null distribution and invalidate the usual critical values. As discussed by Jimenez-Gamero (2020) and acknowledged by Ebner and Henze (2020), validity depends on both the test and the underlying model.

Author(s)

Maintainer: Bruno Ebner [email protected]

Authors:

• Bruno Ebner [email protected]

• Lucas Butsch [email protected]

Other contributors:

• Jaco Visagie [email protected] [contributor]

• Johann Siemens [email protected] [contributor]

References

Jimenez-Gamero, M. D. (2020). Comments on: Tests for multivariate normality—a critical review with emphasis on weighted L2-statistics. TEST, 29, 893–897. doi:10.1007/s11749-020-00742-y.

Ebner, B. and Henze, N. (2020). Rejoinder on: Tests for multivariate normality—a critical review with emphasis on weighted L2-statistics. TEST, 29, 911–913. doi:10.1007/s11749-020-00744-w.

---

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

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).

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:10.1006/jmva.1997.1684

Epps T.W., Pulley L.B. (1983), A test for normality based on the empirical characteristic function, Biometrika, 70:723-726, doi:10.1093/biomet/70.3.723

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

data	a n x d matrix of d dimensional data vectors.
Points	points for approximation of the maximum on the sphere. Points=NULL generates 5000 uniformly distributed Points on the d dimensional unit sphere.

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)))

---

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

samplesize	samplesize for which the empirical quantile should be calculated.
dimension	a natural number to specify the dimension of the multivariate normal distribution
quantile	a number between 0 and 1 to specify the quantile of the empirical distribution of the considered test
statistic	a function specifying the test statistic.
tuning	the tuning parameter of the test statistic.
repetitions	number of Monte Carlo runs.

Value

empirical quantile of the test statistic.

Examples

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

---

Statistic of the DEH test based on harmonic oscillator

Description

Computes the test statistic of the DEH test.

Usage

DEHT(data, a = 1)

Arguments

data	a n x d numeric matrix of data values.
a	positive numeric number (tuning parameter).

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. (2020) "Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces", Scandinavian Journal of Statistics, 48(2): 456-501, doi:10.1111/sjos.12477.

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

data	a (d,n) numeric matrix containing the data.
a	positive numeric number (tuning parameter).

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. (2021) "A new test of multivariate normality by a double estimation in a characterizing PDE" Metrika 84: 401-427, doi:10.1007/s00184-020-00795-x.

---

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

data	a (d,n) numeric matrix containing the data.
a	positive numeric number (tuning parameter).

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. (2022) "Testing normality in any dimension by Fourier methods in a multivariate Stein equation", The Canadian Journal of Statistics, 50(3): 992-1033, doi:10.1002/cjs.11670.

---

Henze-Jiménes-Gamero test statistic

Description

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

Usage

HJG(data, a = 5)

Arguments

data	a n x d numeric matrix of data values.
a	positive numeric number (tuning parameter).

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:10.1007/s11749-018-0589-z

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

data	a n x d numeric matrix of data values.
a	positive numeric number (tuning parameter).

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:10.1017/S0266466618000154.

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

data	a n x d numeric matrix of data values.
a	numeric number greater than 1 (tuning parameter).

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", Ann. Inst. Stat. Math., 72:1109–1136, doi:10.1007/s10463-019-00720-8

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

data	a n x d matrix of d dimensional data vectors.

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:10.1080/03610929008830400

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

data	a n x d matrix of d dimensional data vectors.

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

data	a n x d matrix of d dimensional data vectors.
Points	points for approximation of the maximum on the sphere. Points=NULL generates 1000 uniformly distributed Points on the d dimensional unit sphere.

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

data	a n x d matrix of d dimensional data vectors.
Points	points for approximation of the maximum on the sphere. Points=NULL generates 1000 uniformly distributed Points on the d dimensional unit sphere.

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

data	a n x d matrix of d dimensional data vectors.

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

data	a n x d matrix of d dimensional data vectors.

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:10.1007/BF02595825

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

data	a n x d matrix of d dimensional data vectors.

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:10.1007/BF02595825

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

data	a n x d matrix of d dimensional data vectors.

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

data	a n x d matrix of d dimensional data vectors.

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)))

---

Print method for tests of multivariate normality

Description

Printing objects of class "mnt".

Usage

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

Arguments

x	object of class "mnt".
...	further arguments to be passed to or from methods.

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))

---

Statistic of the Pudelko test

Description

Approximates the test statistic of the Pudelko test.

Usage

PU(data, r = 2)

Arguments

data	a n x d numeric matrix of data values.
r	a positive number (radius of Ball)

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

data	a n x d matrix of d dimensional data vectors.
abb	Stop criterium.

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:10.1016/j.jmva.2003.12.002

See Also

mvnorm.e

Examples

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

---

Empirical scaled residuals

Description

A function that computes the scaled residuals of the data.

Usage

standard(data)

Arguments

data	a n x d matrix of d dimensional data vectors..

Value

A n x d matrix of the scaled residuals.

---

Baringhaus-Henze-Epps-Pulley (BHEP) test

Description

Performs the BHEP test of multivariate normality of Henze and Wagner (1997), with tuning parameter a. Critical values and p-values are obtained by Monte Carlo simulation or by fitting a Pearson or Johnson distribution to the first four moments of the limiting null distribution.

Usage

test.BHEP(
  data,
  a = 1,
  MC.rep = 10000,
  alpha = 0.05,
  cv.method = c("MC", "Pearson", "Johnson")
)

Arguments

data	A numeric $n \times d$ matrix with observations in rows. A numeric vector is treated as a univariate sample. All entries must be finite, $n \ge d+1$, and the sample covariance must be nonsingular.
a	Positive, finite tuning parameter; corresponds to $\beta$ in the formulas below.
MC.rep	Positive integer giving the number of Monte Carlo replications. Used only when cv.method = "MC".
alpha	Significance level, strictly between zero and one.
cv.method	Method used to approximate the critical value: "MC" (default), "Pearson", or "Johnson". Partial matching is supported.

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 $\beta=a$, $Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n)$, and $S_n=n^{-1}\sum_{j=1}^n (X_j-\overline{X}_n)(X_j-\overline{X}_n)^\top$.

With cv.method = "MC", cv.quan simulates the null distribution at the observed sample size $n$ and dimension $d$, using MC.rep replications. The same simulated statistics are used for the critical value and the Monte Carlo p-value

$p_{MC}=\frac{1+\sum_{b=1}^{B}\mathbf{1}\{T_b\ge T_{obs}\}}{B+1},$

where $B$ is MC.rep. The critical value retains the order-statistic convention of cv.quan, and Decision remains Test.value > cv. Because the Monte Carlo p-value is discrete and uses the plus-one correction, this decision can differ from pvalue <= alpha near the cutoff.

With cv.method = "Pearson" or "Johnson", the critical value is obtained from a moment-based approximation to the limiting null distribution $T_\beta(d)$. Let $\kappa_r$ denote its cumulants. The first three cumulants are given by Henze and Wagner (1997), and the fourth by Theorem 3.2 and Appendix A of Ebner and Henze's announced manuscript. The mean, variance, skewness and ordinary kurtosis are

$\mu=\kappa_1,\quad \sigma^2=\kappa_2,\quad \gamma_1=\kappa_3/\kappa_2^{3/2},\quad \beta_2=3+\kappa_4/\kappa_2^2.$

In particular, the fourth central moment is $\mu_4=\kappa_4+3\kappa_2^2$; it is not the fourth cumulant.

Value

An object of class "mnt", with components:

Test

Name of the test.

param

Value of the tuning parameter.

Test.value

Value of the test statistic.

cv

Approximated upper $1-\alpha$ critical value.

Decision

TRUE if Test.value > cv, so the null hypothesis is rejected at the chosen significance level.

cv.method

Critical-value method actually used.

pvalue

Upper-tail p-value computed using the selected method. The Pearson and Johnson p-values are asymptotic approximations.

References

Henze, N. and Wagner, T. (1997). A new approach to the BHEP tests for multivariate normality. Journal of Multivariate Analysis, 62, 1–23. doi:10.1006/jmva.1997.1684.

Ebner, B. and Henze, N. Four-Moment Approximations and Fast p-Values for BHEP Tests of Multivariate Normality. Supplied manuscript ‘BHEP_fourth_moment.pdf’, Theorem 3.2 and Appendix A.

See Also

BHEP, cv.quan

Examples

set.seed(123)
x <- matrix(rnorm(100), ncol = 2)
test.BHEP(x, MC.rep = 500)
if (requireNamespace("PearsonDS", quietly = TRUE)) {
  test.BHEP(x, cv.method = "Pearson")
}
if (requireNamespace("SuppDists", quietly = TRUE)) {
  test.BHEP(x, cv.method = "Johnson")
}

---

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.
Points	number of points to approximate the maximum functional on the unit sphere.

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

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.

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

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.

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

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.

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

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.

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:10.1007/s11749-018-0589-z

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

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.

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:10.1017/S0266466618000154.

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

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.

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:10.1007/s10463-019-00720-8

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test

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:10.1080/03610929008830400

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test
num.points	number of points distributed uniformly over the sphere for approximation of the maximum on the sphere.

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test
num.points	number of points distributed uniformly over the sphere for approximation of the maximum on the sphere.

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test

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:10.1007/BF02595825

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test

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:10.1007/BF02595825

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test

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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test

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)

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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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value.
alpha	level of significance of the test.
r	a positive number (radius of Ball)

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)

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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

data	a n x d matrix of d dimensional data vectors.
MC.rep	number of repetitions for the Monte Carlo simulation of the critical value
alpha	level of significance of the test
abb	Stop criterium.

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:10.1016/j.jmva.2003.12.002

See Also

SR

Examples

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

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