VonNeumannTest: Von Neumann's Successive Difference Test
In AndriSignorell/DescTools: Tools for Descriptive Statistics

VonNeumannTest

R Documentation

Von Neumann's Successive Difference Test

Description

A popular statistic to test for independence is the von Neumann ratio.

Usage

VonNeumannTest(x, alternative = c("two.sided", "less", "greater"), unbiased = TRUE)

Arguments

`x`	a numeric vector containing the observations
`alternative`	a character string specifying the alternative hypothesis, must be one of `"two.sided"` (default), `"greater"` or `"less"`. You can specify just the initial letter.
`unbiased`	logical. In order for VN to be an unbiased estimate of the true population value, the calculated value is multiplied by `n/(n-1)`. Default is TRUE.

Details

The VN test statistic is in the unbiased case

VN=\frac{\sum_{i=1}^{n-1}(x_i-x_{i+1})^2 \cdot n}{\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2 \cdot (n-1)}

It is known that (VN-\mu)/\sigma is asymptotically standard normal, where \mu=\frac{2n}{n-1} and \sigma^2=4\cdot n^2 \frac{(n-2)}{(n+1)(n-1)^3}.

The VN test statistic is in the original (biased) case

VN=\frac{\sum_{i=1}^{n-1}(x_i-x_{i+1})^2}{\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2}

The test statistic (VN-2)/\sigma is asymptotically standard normal, where \sigma^2=\frac{4\cdot(n-2)}{(n+1)(n-1)}.

Missing values are silently removed.

Value

A list with class "htest" containing the components:

`statistic`	the value of the VN statistic and the normalized statistic test.
`parameter`, `n`	the size of the data, after the remotion of consecutive duplicate values.
`p.value`	the p-value of the test.
`alternative`	a character string describing the alternative hypothesis.
`method`	a character string indicating the test performed.
`data.name`	a character string giving the name of the data.

Author(s)

Andri Signorell <andri@signorell.net>

References

von Neumann, J. (1941) Distribution of the ratio of the mean square successive difference to the variance. Annals of Mathematical Statistics 12, 367-395.