CramerVonMisesTest: Cramer-von Mises Test for Normality
Description
Performs the Cramer-von Mises test for the composite hypothesis of normality,
see e.g. Thode (2002, Sec. 5.1.3).
Usage
CramerVonMisesTest(x)
Arguments
x
a numeric vector of data values, the number of
which must be greater than 7. Missing values are allowed.
Value
A list of class htest, containing the following components:
statistic
the value of the Cramer-von Mises statistic.
p.value
the p-value for the test.
method
the character string “Cramer-von Mises normality test”.
data.name
a character string giving the name(s) of the data.
Details
The Cramer-von Mises test is an EDF omnibus test for the composite hypothesis of normality.
The test statistic is
$$
W = \frac{1}{12 n} + \sum_{i=1}^{n} \left (p_{(i)} - \frac{2i-1}{2n} \right),
$$
where \(p_{(i)} = \Phi([x_{(i)} - \overline{x}]/s)\). Here,
\(\Phi\) is the cumulative distribution function
of the standard normal distribution, and \(\overline{x}\) and \(s\)
are mean and standard deviation of the data values.
The p-value is computed from the modified statistic
\(Z=W (1.0 + 0.5/n)\) according to Table 4.9 in
Stephens (1986).
References
Stephens, M.A. (1986) Tests based on EDF statistics In:
D'Agostino, R.B. and Stephens, M.A., eds.: Goodness-of-Fit Techniques.
Marcel Dekker, New York.
Thode Jr., H.C. (2002) Testing for Normality Marcel Dekker, New York.