Bartels Test for Randomness
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
bartels.test(x)
x |
a vector of class "numeric" or a time series object of class "ts" |
In this function, the test is implemented as given by Bartels (1982), where the ranks r[1], ..., r[n] of the X[i], ..., X[n] are used for the statistic:
T = ∑_{i=1}^n (r[i] - r[i+1])^2 / ∑(r[i] - meanr)^2
As proposed by Bartels (1982), the p-value is calculated for sample sizes in the range of (10 <= n < 100) with the non-standard beta distribution for the range 0 <= x <= 4 with parameters:
a = b = 5 * n * ( n + 1) * (n - 1)^2 / (2 * ( n - 2) * (5 * n^2 - 2 * n - 9)) - 1/2
For sample sizes n >= 100 a normal approximation with N(2, 20/(5n + 7)) is used for p-value calculation.
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 |
The current function is for complete observations only.
R. Bartels (1982), The Rank Version of von Neumann's Ratio Test for Randomness, Journal of the American Statistical Association 77, 40–46.
# 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)
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