VonNeumannTest {DescTools} 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.

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