multivar_normality_test
Computes Mardia’s multivariate measures of skewness and kurtosis and tests for multivariate normality.
Synopsis
#include <imsls.h>
float *imsls_f_multivar_normality_test (int n_observations, int n_variables, float x[], ..., 0)
The type double function is imsls_d_multivar_normality_test.
Required Arguments
int n_observations (Input)
Number of observations (number of rows of data) x.
int n_variables (Input)
Dimensionality of the multivariate space for which the skewness and kurtosis are to be computed. Number of variables in x.
float x[] (Input)
Array of size n_observations by n_variables containing the data.
Return Value
A pointer to an array of dimension 13 containing output statistics
i | stat[i] |
|---|
0 | Estimated skewness. |
1 | Expected skewness assuming a multivariate normal distribution. |
2 | Asymptotic chi-squared statistic assuming a multivariate normal distribution. |
3 | Probability of a greater chi-squared. |
4 | Mardia and Foster's standard normal score for skewness. |
5 | Estimated kurtosis. |
6 | Expected kurtosis assuming a multivariate normal distribution. |
7 | Asymptotic standard error of the estimated kurtosis. |
8 | Standard normal score obtained from stat[5] through stat[7]. |
9 | p-value corresponding to stat[8]. |
10 | Mardia and Foster's standard normal score for kurtosis. |
11 | Mardia's SW statistic based upon stat[4] and stat[10]. |
12 | p-value for stat[11]. |
Synopsis with Optional Arguments
#include <imsls.h>
float imsls_f_multivar_normality_test (int n_observations, int n_variables, float x[],
IMSLS_FREQUENCIES, float frequencies[],
IMSLS_WEIGHTS, float weights[],
IMSLS_SUM_FREQ, int *sum_frequencies,
IMSLS_SUM_WEIGHTS, float *sum_weights,
IMSLS_N_ROWS_MISSING, int *nrmiss,
IMSLS_MEANS, float **means,
IMSLS_MEANS_USER, float means[],
IMSLS_R, float **R_matrix,
IMSLS_R_USER, float R_matrix[],
IMSLS_RETURN_USER, float test_statistics[],
0)
Optional Arguments
IMSLS_FREQUENCIES, float frequencies[] (Input)
Array of size n_observations containing the frequencies. Frequencies must be integer valued. Default assumes all frequencies equal one.
IMSLS_WEIGHTS, float weights[] (Input)
Array of size n_observations containing the weights. Weights must be greater than non-negative. Default assumes all weights equal one.
IMSLS_SUM_FREQ, int *sum_frequencies (Output)
The sum of the frequencies of all observations used in the computations.
IMSLS_SUM_WEIGHTS, float *weights[] (Output)
The sum of the weights times the frequencies for all observations used in the computations.
IMSLS_N_ROWS_MISSING, int *nrmiss (Output)
Number of rows of data in x[] containing any missing values (NaN).
IMSLS_MEANS, float **means (Output)
The address of a pointer to an array of length n_variables containing the sample means.
IMSLS_MEANS_USER, float means[] (Output)
Storage for array means is provided by user. See IMSLS_MEANS.
IMSLS_R, float **R_matrix (Output)
The address of a pointer to an n_variables by n_variables upper triangular matrix containing the Cholesky RTR factorization of the covariance matrix.
IMSLS_R_USER, float R_matrix[] (Output)
Storage for array R_matrix is provided by user. See IMSLS_R.
IMSLS_RETURN_USER, float stat[] (Output)
User supplied array of dimension 13 containing the estimates and their associated test statistics.
Description
Function imsls_f_multivar_normality_test computes Mardia’s (1970) measures b1,p and b2,p of multivariate skewness and kurtosis, respectfully, for p = n_variables. These measures are then used in computing tests for multivariate normality. Three test statistics, one based upon b1,p alone, one based upon b2,p alone, and an omnibus test statistic formed by combining normal scores obtained from b1,p and b2,p are computed. On the order of np3, operations are required in computing b1,p when the method of Isogai (1983) is used, where n = n_observations. On the order of np2, operations are required in computing b2,p.
Let
where
fi is the frequency of the i-th observation, and wi is the weight for this observation. (Weights wi are defined such that xi is distributed according to a multivariate normal, N(μ, Σ/wi) distribution, where Σ is the covariance matrix.) Mardia’s multivariate skewness statistic is defined as:
while Mardia’s kurtosis is given as:
Both measures are invariant under the affine (matrix) transformation AX + D, and reduce to the univariate measures when p = n_variables = 1. Using formulas given in Mardia and Foster (1983), the approximate expected value, asymptotic standard error, and asymptotic p‑value for b2,p, and the approximate expected value, an asymptotic chi-squared statistic, and p‑value for the b1,p statistic are computed. These statistics are all computed under the null hypothesis of a multivariate normal distribution. In addition, standard normal scores W1(b1,p) and W2(b2,p) (different from but similar to the asymptotic normal and chi-squared statistics above) are computed. These scores are combined into an asymptotic chi-squared statistic with two degrees of freedom:
This chi-squared statistic may be used to test for multivariate normality. A p‑value for the chi-squared statistic is also computed.
Example
In this example, 150 observations from a 5 dimensional standard normal distribution are generated via routine
imsls_f_random_normal (Chapter 12,
Random Number Generation). The skewness and kurtosis statistics are then computed for these observations.
#include <imsls.h>
#include <stdio.h>
int main()
{
float *x, swt, *xmean, *r, *stats;
int nobs = 150, ncol = 5, nvar = 5, izero = 0, ni, nrmiss;
imsls_random_seed_set(123457);
x = imsls_f_random_normal(nobs*nvar, 0);
stats = imsls_f_multivar_normality_test(nobs, nvar, x,
IMSLS_SUM_FREQ, &ni,
IMSLS_SUM_WEIGHTS, &swt,
IMSLS_N_ROWS_MISSING, &nrmiss,
IMSLS_R, &r,IMSLS_MEANS, &xmean,
0);
printf("Sum of frequencies = %d\nSum of the weights =%8.3f\n",
ni, swt);
printf(" Number rows missing = %3d\n", nrmiss);
imsls_f_write_matrix("stat", 13, 1, stats,
IMSLS_ROW_NUMBER_ZERO, 0);
imsls_f_write_matrix("means", 1, nvar, xmean, 0);
imsls_f_write_matrix("R", nvar, nvar, r, 0);
}
Output
Sum of frequencies = 150
Sum of the weights = 150.000
Number rows missing = 0
stat
0 0.73
1 1.36
2 18.62
3 0.99
4 -2.37
5 32.67
6 34.54
7 1.27
8 -1.48
9 0.14
10 1.62
11 8.24
12 0.02
means
1 2 3 4 5
0.02623 0.09238 0.06536 0.09819 0.05639
R
1 2 3 4 5
1 1.033 -0.084 -0.065 0.108 0.067
2 0.000 1.049 -0.097 -0.042 -0.021
3 0.000 0.000 1.063 0.006 -0.145
4 0.000 0.000 0.000 0.942 -0.084
5 0.000 0.000 0.000 0.000 0.949
