Test for Multivariate Normal Distribution Based on a Characterization

Description

Provides a test of multivariate normality of an unknown sample that does not require estimation of the nuisance parameters, the mean and covariance matrix. Rather, a sequence of transformations removes these nuisance parameters and results in a set of sample matrices that are positive definite. These matrices are uniformly distributed on the space of positive definite matrices in the unit hyper-rectangle if and only if the original data is multivariate normal (Fairweather, 1973, Doctoral dissertation, University of Washington). The package performs a goodness of fit test of this hypothesis. In addition to the test, functions in the package give visualizations of the support region of positive definite matrices for bivariate samples.

Details

The DESCRIPTION file:

Index of help topics:

MVNtestchar-package     Test for Multivariate Normal Distribution Based
                        on a Characterization
maxv12                  Rotatable Plot of Surface of Possible Maximum
                        Values of Off-diagonal Variable
slice.v1                Rotatable Plot of Slice Through Support Region
                        in Positive Definite 2 x 2 Matrix
slice.v12               Rotatable Plot of Slice Through Support Region
                        in Positive Definite 2 x 2 Matrix
support.p2              Show Support Region of Positive Definite
                        Matrices with Rank 2
testunknown             Process the Samples Whose Distribution is to be
                        Tested
unknown.Bp2             A Sample From an Unknown Bivariate Distribution
unknown.Bp4             A Sample From an Unknown Four-variate
                        Distribution
unknown.Np2             A Sample From an Unknown Bivariate Distribution
unknown.Np4             A Sample From an Unknown Four-variate
                        Distribution

Provides a test of multivariate normality of a sample which does not require estimation of the nuisance parameters, the mean vector and covariance matrix. Rather, a sequence of transformations removes these nuisance parameters, resulting in a set of sample matrices that are positive definite. If, and only if the original data is multivariate normal, these matrices are uniformly distributed on the space of positive definite matrices in the unit hyper-rectangle. The package performs a goodness of fit test of this hypothesis. In addition to the test, functions in the package give visualizations of the support region of positive definite matrices for p equals 2.

Author(s)

person("Fairweather", "William", email = "wrf343@flowervalleyconsulting.com", role = c("aut", "cre"))

References

Anderson, TW. (1958), An Introduction to Multivariate Statistical Analysis, John Wiley, New York.

Cramer, H (1962). Random Variables and Probability Distributions, Cambridge University Press, London.

Csorgo M and Seshadri V (1970). On the problem of replacing composite hypotheses by equivalent simple ones, Rev. Int. Statist. Instit., 38, 351-368

Csorgo M and Seshadri V (1971). Characterizing the Gaussian and exponential laws by mappings onto the unit interval, Z. Wahrscheinlickhkeitstheorie verw. Geb., 18, 333-339

Deemer,WL and Olkin,I (1951). The Jacobians of certain matrix transformations useful in multivariate analysis, *Biometrika*, **58**, 345 367.

Fairweather WR (1973). A test for multivariate normality based on a characterization. Dissertation submitted in partial fulfillment of the requirements for the Doctor of Philosophy, University of Washington, Seattle WA

Rotatable Plot of Surface of Possible Maximum Values of Off-diagonal Variable

Description

Rotatable plot of surface of possible maximum values of off-diagonal variable v12 in positive definite 2 x 2 matrix

Usage

maxv12(theta = 30, phi = 30, inc = 25, lseq = 200, ticktype="detailed", 
    diagnose = FALSE, verbose = TRUE)

Arguments

Value

Output is a plot that is rotatable via keyboard input. Upon exit, the latest values of the rotation parameters is listed to facilitate return to the latest plot

Author(s)

William R. Fairweather

See Also

support.p2()

Examples

## Not run: maxv12(theta = 30, phi = 30, inc = 25, lseq = 200, 
    ticktype = "detailed", diagnose = FALSE, verbose = TRUE)

## End(Not run)

Rotatable Plot of Slice Through Support Region in Positive Definite 2 x 2 Matrix

Description

Rotatable plot of slice through support region in positive definite 2 x 2 matrix at fixed value of diagonal variable v1

Usage

slice.v1(level3 = 0.6, theta = 0, phi = 60, inc = 25, lseq = 100, ticktype="detailed",
    diagnose = FALSE, verbose = TRUE)

Arguments

Value

Output is a plot that is rotatable via keyboard input. Upon exit, the latest values of the rotation parameters is listed to facilitate return to the latest plot

Author(s)

William R. Fairweather

See Also

support.p2()

Examples

## Not run: slice.v1(level3 = 0.6, theta = 0, phi = 60, inc = 25, lseq = 100, 
    ticktype = "detailed")

## End(Not run)

Rotatable Plot of Slice Through Support Region in Positive Definite 2 x 2 Matrix

Description

Rotatable plot of slice through support region in positive definite 2 x 2 matrix at fixed value of off-diagonal variable v12

Usage

slice.v12(level3 = 0.3, theta = 30, phi = 10, inc = 25, lseq = 100, ticktype="detailed",
    diagnose = FALSE, verbose = TRUE)

Arguments

Value

Output is a plot that is rotatable via keyboard input. Upon exit, the latest values of the rotation parameters is listed to facilitate return to the latest plot

Author(s)

William R. Fairweather

See Also

support.p2()

Examples

## Not run: slice.v12(level3 = 0.3, theta = 30, phi = 10, inc = 25, lseq = 100, 
   ticktype = "detailed") 

## End(Not run)

Show Support Region of Positive Definite Matrices with Rank 2

Description

Rotatable plot of support region for positive definite matrix with p=2

Usage

support.p2(theta = 110, phi = 10, lseq = 150, inc = 25, ticktype="detailed", 
    diagnose = FALSE, verbose = TRUE)

Arguments

Details

Support region for p-variate positive definite matrix distributions is difficult to envision except for p=2. The diagonals of the matrix are V1 and V2 and the off-diagonal variable is V12. In our application 0<=V1,V2<=1, and -1<=V12<=1, so the bounded space is a hyper-rectangle. Each point in this region represents a symmetric pxp matrix, but not all of these are positive definite. This function shades the region of positive definite matrices.

Value

Output is a plot that is rotatable via keyboard input. Upon exit, the latest values of the rotation parameters is listed to facilitate return to the latest plot

Author(s)

William R. Fairweather

Examples

## Not run: support.p2(theta = 110, phi = 10, lseq = 150, inc = 25, 
    ticktype = "detailed")

## End(Not run)

Process the Samples Whose Distribution is to be Tested

Description

Create positive definite matrices without nuisance parameters. Tabulate distribution. Calculate goodness of fit

Usage

testunknown(x, pvector, k, diagnose.s = FALSE, diagnose = FALSE, 
    verbose = TRUE) 

Arguments

Value

a list including elements

Author(s)

William R. Fairweather

References

Csorgo, M and Seshadri, V (1970). On the problem of replacing composite hypotheses by equivalent simple ones, Rev. Int. Statist. Instit., 38, 351-368 Csorgo,M and Seshadri,V (1971). Characterizing the Gaussian and exponential laws by mappings onto the unit interval, Z. Wahrscheinlickhkeitstheorie verw. Geb., 18, 333-339. Fairweather, WR (1973). A test for multivariate normality based on a characterization. Dissertation submitted in partial fulfillment of the requirements for the Doctor of Philosophy, University of Washington, Seattle WA.

Examples

data(unknown.Np2)
testunknown(x=unknown.Np2, pvector=2, k=20, 
     diagnose.s = FALSE, diagnose = FALSE, verbose = TRUE)

A Sample From an Unknown Bivariate Distribution

Description

A 3600 x 2 x 1 array generated from 7200 modified Bernoulli(0,1) variables.

Usage

data("unknown.Bp2")

Format

3600 x 2 x 1 array

Source

Generated by the author

Examples

data("unknown.Bp2")

A Sample From an Unknown Four-variate Distribution

Description

A 6000 x 4 matrix generated from 24,000 Bernoulli(0,1) variables

Usage

data("unknown.Bp4")

Format

6000 x 4 x 1 array

Source

Generated by the author

Examples

data("unknown.Bp4")

A Sample From an Unknown Bivariate Distribution

Description

A 2500 x 2 matrix generated from 5000 normal(0,1) variables

Usage

data("unknown.Np2")

Format

2500 x 2 matrix

Source

Generated by the author

Examples

data("unknown.Np2")

A Sample From an Unknown Four-variate Distribution

Description

A 6000 x 4 x 1 array generated from 24000 normal(0,1) variables

Usage

data("unknown.Np4")

Format

6000 x 4 x 1 array

Source

Generated by the author

Examples

data("unknown.Np4")