Cramer-Von Mises Test for Assessing Deviations From Normality

A goodness-of-fit test based on the integrated squared discrepancies between the empirical and theoretical distribution functions. Similar to the Anderson-Darling test, but without increased weighting of the distribution tails.

Usage

cramerVonMisesTest(x)

Arguments

x

a numeric vector of data values, the number of which must be at least 8.

Value

A list with class "htest" containing the following components:

statistic

the value of the Cramer-von Mises statistic.

p.value

the p-value of the test.

method

a character string indicating the test performed.

data.name

a character string giving the name of the data.

Details

Performs the Cramer-von Mises test for the composite hypothesis of normality, see e.g. Thode (2002, Sec. 5.1.3).

The Cramer-von Mises test is an EDF omnibus test for the composite hypothesis of normality. The test statistic is $$W = \frac{1}{12 n} + \sum_{i=1}^{n} \left(p_{(i)} - \frac{2i - 1}{2n}\right)^2,$$ where \(p_{(i)} = \Phi([x_{(i)} - \bar{x}]/s)\). Here, \(\Phi\) is the cumulative distribution function of the standard normal distribution, and \(\bar{x}\) and \(s\) are mean and standard deviation of the data values. The p-value is computed from the modified statistic \(Z = W (1 + 0.5/n)\) according to Table 4.9 in Stephens (1986).

Missing values are silently removed.

Note

Based on code by Juergen Gross previously published in the nortest package, adapted to conform to package standards.

References

Stephens, M.A. (1986) Tests based on EDF statistics. In: D'Agostino, R.B. and Stephens, M.A., eds.: Goodness-of-Fit Techniques. New York: Marcel Dekker.

Thode Jr., H.C. (2002) Testing for Normality. New York: Marcel Dekker.

See also

shapiro.test() for performing the Shapiro-Wilk test for normality, andersonDarlingTest(), pharos::plotQQ() for producing extended normal quantile-quantile plots

Other test.normality: andersonDarlingTest(), jarqueBeraTest(), lillieTest(), pearsonTest(), shapiroFranciaTest()

Examples

set.seed(1)
cramerVonMisesTest(rnorm(100, mean = 5, sd = 3))
#> 
#> 	Cramer-von Mises normality test
#> 
#> data:  rnorm(100, mean = 5, sd = 3)
#> W = 0.026031, p-value = 0.8945
#> 
cramerVonMisesTest(runif(100, min = 2, max = 4))
#> 
#> 	Cramer-von Mises normality test
#> 
#> data:  runif(100, min = 2, max = 4)
#> W = 0.27466, p-value = 0.0006342
#>