trend (version 1.1.1)

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)

Arguments

x

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

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

Details

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

$$T=\frac{\sum _{i=1}^n(r_i-r_{i+1}{)}^2}{\sum _{i=1}^n(r_i-\overline{r}{)}^2}$$

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:

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

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

# NOT RUN {
# 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)
 
# }