Wald-Wolfowitz Test for Independence and Stationarity
Performes the non-parametric Wald-Wolfowitz test for independence and stationarity.
ww.test(x)
x |
a vector or a time series object of class "ts" |
Let x_1, x_2, ..., x_n denote the sampled data, then the test statistic of the Wald-Wolfowitz test is calculated as:
R = sum(x[1:(n-1)] * x[2:n]) + x[1] * x[n]
The expected value of R is:
E(R) = (s1^2 - s2) / (n - 1)
The expected variance is:
V(R) = (s2^2 - s4) / (n - 1) - er^2 + (s1^4 - 4 * s1^2 * s2 + 4 * s1 * s3 + s2^2 - 2 * s4) / ((n - 1) * (n - 2))
with:
st = sum(x^t), ~~ t = 1, 2, 3, 4
For n > 10 the test statistic is normally distributed, with:
z = (R - E(R)) / V(R)^0.5
ww.test calculates p-values from the standard normal distribution for the two-sided case.
An object of class "htest"
method |
a character string indicating the chosen test |
data.name |
a character string giving the name(s) of the data |
statistic |
the Wald-Wolfowitz z-value |
alternative |
a character string describing the alternative hypothesis |
p.value |
the p-value for the test |
NA values are omitted.
R. K. Rai, A. Upadhyay, C. S. P. Ojha and L. M. Lye (2013), Statistical analysis of hydro-climatic variables. In: R. Y. Surampalli, T. C. Zhang, C. S. P. Ojha, B. R. Gurjar, R. D. Tyagi and C. M. Kao (ed. 2013), Climate change modelling, mitigation, and adaptation. Reston, VA: ASCE. doi = 10.1061/9780784412718.
A. Wald and J. Wolfowitz (1943), An exact test for randomness in the non-parametric case based on serial correlation. Annual Mathematical Statistics 14, 378–388.
WMO (2009), Guide to Hydrological Practices. Volume II, Management of Water Resources and Application of Hydrological Practices, WMO-No. 168.
ww.test(nottem) ww.test(Nile) set.seed(200) x <- rnorm(100) ww.test(x)
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