cvmNormalityTest¶

Performs a Cramer-von Mises test for normality.

Synopsis¶

cvmNormalityTest (x)

Required Arguments¶

float x[] (Input)Vector of length nObs containing the observations.

Return Value¶

The p‑value for the Cramer-von Mises test of normality.

Optional Arguments¶

stat (Output): The Cramer-von Mises statistic.

nMissing (Output)The number of missing observations.

Description¶

Given a data sample \(\{X_i,i=1 .. n\}\), where n = nObs and \(X_i\) = x[i-1], function cvmNormalityTest 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:

\[W = \frac{1}{12n} + \sum_{i=1}^{n} \left[\phi\left(Y_i\right) - \frac{2i-1}{2n}\right]^2\]

Where \(\phi\left( Y_i \right)\) is the cumulative distribution function of standard normal N(0,1) distribution, \(Y_i=\left( X_i-\overline{X} \right)/s\), and \(\overline{X}\) and s are the sample mean and standard deviation respectively. P is calculated by first transforming W to an “n-adjusted” statistic W:

\[W* = W\left(1.0 + \frac{0.5}{n}\right)\]

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 0.05 level of significance.

from __future__ import print_function
from numpy import *
from pyimsl.stat.cvmNormalityTest import cvmNormalityTest

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]
nmiss = []
adstat = []

p_value = cvmNormalityTest(x, stat=adstat, nMissing=nmiss)

print("Cramer-von Mises statistic = %11.4f" % adstat[0])
print("p-value = %11.4f" % p_value)
print("# missing values = %4d" % nmiss[0])

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.