A popular statistic to test for independence is the von Neumann ratio.
VonNeumannTest(x, alternative = c("two.sided", "less", "greater"), unbiased = TRUE)a numeric vector containing the observations
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.
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.
A list with class "htest" containing the components:
the value of the VN statistic and the normalized statistic test.
the size of the data, after the remotion of consecutive duplicate values.
the p-value of the test.
a character string describing the alternative hypothesis.
a character string indicating the test performed.
a character string giving the name of the data.
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.
# NOT RUN {
VonNeumannTest(d.pizza$temperature)
# }
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