randtests (version 1.0)

# bartels.rank.test: Bartels Rank Test

## Description

Performs the Bartels rank test of randomness.

## Usage

bartels.rank.test(x, alternative, pvalue="normal")

## Arguments

x
a numeric vector containing the observations
alternative
a character string with the alternative hypothesis. Must be one of "two.sided" (default), "left.sided" or "right.sided". You can specify just the initial letter.
pvalue
a character string specifying the method used to compute the p-value. Must be one of normal (default), beta, exact or auto.

## Value

A list with class "htest" containing the components:
statistic
the value of 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.
rvn
the value of the RVN statistic (not shown on screen).
nm
the value of the NM statistic, the numerator of RVN (not shown on screen).
mu
the mean value of the RVN statistic (not shown on screen).
var
the variance of the RVN statistic (not shown on screen).

## Details

Missing values are removed.

This is the rank version of von Neumann's Ratio Test for Randomness (von Neumann, 1941).

The test statistic RVN is $$RVN=\frac{\sum_{i=1}^{n-1}(R_i-R_{i+1})^2}{\sum_{i=1}^{n}\left(R_i-(n+1)/2\right)^2}$$ where $R_i=rank(X_i), i=1,...,n$. It is known that $(RVN-2)/\sigma$ is asymptotically standard normal, where $\sigma^2=[4(n-2)(5n^2-2n-9)]/[5n(n+1)(n-1)^2]$.

The possible alternative are "two.sided", "left.sided" and "right.sided". By using the alternative "two.sided" the null hypothesis of randomness is tested against nonrandomness. By using the alternative "left.sided" the null hypothesis of randomness is tested against a trend. By using the alternative "right.sided" the null hypothesis of randomness is tested against a systematic oscillation.

By default (if pvalue is not specified), a normal approximation is used to compute the p-value. With beta, the p-value is computed using an approximation given by the Beta distribution. With exact, the exact p-value is computed. The option exact requires the computation of the exact distribution of the statistic test under the null hypothesis and should only be used for small sample sizes ($n \le 10$).

## References

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

Gibbons, J.D. and Chakraborti, S. (2003). Nonparametric Statistical Inference, 4th ed. (pp. 97--98). URL: http://books.google.pt/books?id=dPhtioXwI9cC&lpg=PA97&ots=ZGaQCmuEUq

von Neumann, J. (1941). Distribution of the Ratio of the Mean Square Successive Difference to the Variance. The Annals of Mathematical Statistics 12(4), 367--395. doi:10.1214/aoms/1177731677. http://projecteuclid.org/euclid.aoms/1177731677

dbartelsrank, pbartelsrank

## Examples

##
## Example 5.1 in Gibbons and Chakraborti (2003), p.98.
## Annual data on total number of tourists to the United States for 1970-1982.
##
years <- 1970:1982
tourists <- c(12362, 12739, 13057, 13955, 14123,  15698, 17523, 18610, 19842,
20310, 22500, 23080, 21916)
plot(years, tourists, pch=20)
bartels.rank.test(tourists, alternative="left.sided", pvalue="beta")
# output
#
#  Bartels Ratio Test
#
#data:  tourists
#statistic = -3.6453, n = 13, p-value = 1.21e-08
#alternative hypothesis: trend

##
## Example in Bartels (1982).
## Changes in stock levels for 1968-1969 to 1977-1978 (in \$A million), deflated by the
## Australian gross domestic product (GDP) price index (base 1966-1967).
x <- c(528, 348, 264, -20, -167, 575, 410, -4, 430, -122)
bartels.rank.test(x, pvalue="beta")