Von Neumann's Successive Difference Test for Detecting Serial Dependence in Continuous Sequences

A test for randomness or autocorrelation in a sequence, based on the mean square successive difference relative to the sample variance, closely related to the Durbin-Watson test.

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".
unbiased: logical. If TRUE (default), applies the finite-sample correction $n/(n-1)$ so that VN is an unbiased estimate of the population value.

Value

A list with class "htest" containing:

statistic: the normalized z-statistic.
parameter: named vector with n, the sample size after removal of NAs.
p.value: the p-value of the test.
alternative: a character string describing the alternative hypothesis.
data.name: a character string giving the name of the data.
vn: the value of the VN statistic (not printed).

Details

The test is based on the von Neumann ratio statistic.

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 $\mu = 2n/(n-1)$ and $\sigma^2 = 4 n^2 (n-2) / [(n+1)(n-1)^3]$.

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 $\sigma^2 = 4(n-2) / [(n+1)(n-1)]$.

Missing values are silently removed.

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.

Young, L. C. (1941) On randomness in ordered sequences. Annals of Mathematical Statistics 12, 293–300.

Bartels, R. (1982) The Rank Version of von Neumann's Ratio Test for Randomness. Journal of the American Statistical Association, 77(377), 40–46.

Examples

set.seed(2)
vonNeumannTest(runif(20))
#> 
#> 	Von Neumann Successive Difference Test
#> 
#> data:  runif(20)
#> z = 0.73709, n = 20, p-value = 0.4611
#> alternative hypothesis: two.sided
#> 

# trend: small VN expected
vonNeumannTest(cumsum(rnorm(30)), alternative = "less")
#> 
#> 	Von Neumann Successive Difference Test
#> 
#> data:  cumsum(rnorm(30))
#> z = -4.8849, n = 30, p-value = 5.174e-07
#> alternative hypothesis: less
#>