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

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

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.

- 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.

Andri Signorell <andri@signorell.net>

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.

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

`BartelsRankTest`

```
VonNeumannTest(d.pizza$temperature)
```

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