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

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`

.-
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).

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 ($n \le 10$).

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`

## ## 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")