DescTools (version 0.99.10)

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 with 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 $$W=\frac{1}{12n}+\sum _{i=1}^n(p_{(i)}-\frac{2i-1}{2n}),$$ 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.

See Also

shapiro.test for performing the Shapiro-Wilk test for normality. AndersonDarlingTest, LillieTest, PearsonTest, ShapiroFranciaTest for performing further tests for normality. qqnorm for producing a normal quantile-quantile plot.

Examples

CramerVonMisesTest(rnorm(100, mean = 5, sd = 3))
CramerVonMisesTest(runif(100, min = 2, max = 4))