cvm_normality

Given a data sample {Xi, i=1 .. n}, where n = nobs and Xi = x[i-1], function imsls_f_cvm_normality_test computes the Cramer-von Mises (CvM) normality statistic W = cvmstat and the corresponding Return Value (p-value) P = P == {probability that a normally distributed n element sample would have a CvM statistic > W}. If P is sufficiently small (e.g. P < .05), then the CvM test indicates that the null hypothesis that the data sample is normally-distributed should be rejected. W is calculated:

Where

is the cumulative distribution function of standard normal N(0,1) distribution,

, and

and s are the sample mean and standard deviation respectively. P is calculated by first transforming W to an “n-adjusted” statistic W*:

and then calculating P in terms of W* using a parabolic approximation taken from Table 4.9 in Stephens (1986).

Example

This example is taken from Conover (1980, pages 364 and 195). The data consists of 50 two digit numbers taken from a telephone book. The CvM test fails to reject the null hypothesis of normality at the .05 level of significance.

#include <imsls.h>

#include <stdio.h>

int main()

{

int nobs = 50, nmiss;

float p_value, cvmstat;

float x[] = {

23.0, 36.0, 54.0, 61.0, 73.0, 23.0, 37.0, 54.0, 61.0, 73.0,

24.0, 40.0, 56.0, 62.0, 74.0, 27.0, 42.0, 57.0, 63.0, 75.0,

29.0, 43.0, 57.0, 64.0, 77.0, 31.0, 43.0, 58.0, 65.0, 81.0,

32.0, 44.0, 58.0, 66.0, 87.0, 33.0, 45.0, 58.0, 68.0, 89.0,

33.0, 48.0, 58.0, 68.0, 93.0, 35.0, 48.0, 59.0, 70.0, 97.0};

p_value = imsls_f_cvm_normality_test (nobs, x,

IMSLS_STAT, &cvmstat,

IMSLS_N_MISSING, &nmiss,

0);

printf ("Cramer-von Mises statistic = %11.4f \n", cvmstat);

printf ("p-value = %11.4f\n", p_value);

printf ("# missing values = %4d\n", nmiss);

}

Output

Cramer-von Mises statistic = 0.0520

p-value = 0.4747

# missing values = 0

Informational Errors

IMSLS_PVAL_UNDERFLOW

The p‑value has fallen below the minimum value of # for which its calculation has any accuracy; ZERO is returned.

Fatal Errors

IMSLS_TOO_MANY_MISSING

After removing the missing observations only 2 observations remain. The test cannot proceed.