trend (version 1.1.6)

bartels.test: Bartels Test for Randomness

Description

Performes a rank version of von Neumann's ratio test as proposed by Bartels. The null hypothesis of randomness is tested against the alternative hypothesis

Usage

bartels.test(x)

Value

A list with class "htest"

data.name

character string that denotes the input data

p.value

the p-value

statistic

the test statistic

alternative

the alternative hypothesis

method

character string that denotes the test

Arguments

x

a vector of class "numeric" or a time series object of class "ts"

Details

In this function, the test is implemented as given by Bartels (1982), where the ranks \(r_1, \ldots, r_n\) of the \(X_i, \ldots, X_n\) are used for the statistic:

$$ T = \frac{\sum_{i=1}^n (r_i - r_{i+1})^2}{\sum_{i=1}^n (r_i - \bar{r})^2} $$

As proposed by Bartels (1982), the \(p\)-value is calculated for sample sizes in the range of \((10 \le n < 100)\) with the non-standard beta distribution for the range \(0 \le x \le 4\) with parameters:

$$ a = b = \frac{5 n \left( n + 1\right) \left(n - 1\right)^2} {2 \left(n - 2\right) \left(5n^2 - 2n - 9\right)} - \frac{1}{2} $$

For sample sizes \(n \ge 100\) a normal approximation with \(N(2, 20/(5n + 7))\) is used for \(p\)-value calculation.

References

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

See Also

ww.test, wm.test

Examples

# Example from Schoenwiese (1992, p. 113)
## Number of frost days in April at Munich from 1957 to 1968
## 
frost <- ts(data=c(9,12,4,3,0,4,2,1,4,2,9,7), start=1957)
bartels.test(frost)

## Example from Sachs (1997, p. 486)
x <- c(5,6,2,3,5,6,4,3,7,8,9,7,5,3,4,7,3,5,6,7,8,9)
bartels.test(x)

## Example from Bartels (1982, p. 43)
x <- c(4, 7, 16, 14, 12, 3, 9, 13, 15, 10, 6, 5, 8, 2, 1, 11, 18, 17)
bartels.test(x)