wilcox_stat: Wilcoxon-Mann-Whitney Test Statistic for Change Points
| wilcox_stat | R Documentation |
Wilcoxon-Mann-Whitney Test Statistic for Change Points
Description
Computes the test statistic for the Wilcoxon-Mann-Whitney change point test
Usage
wilcox_stat(x, h = 1L, method = "kernel", control = list())
Arguments
x |
Time series (numeric or ts vector). |
h |
Kernel function of the U statistic (1L or 2L, or a function with two parameters). |
method |
Method for estimating the long run variance. Options are |
control |
A list of control parameters for the estimation of the long run variance (cf. |
Details
Let n be the length of x, i.e. the number of observations.
h = 1L:
T_n = \frac{1}{\hat{\sigma}} \max_{1 \leq k \leq n} \left| \frac{1}{n^{3/2}} \sum_{i = 1}^k \sum_{j = k+1}^n (1_{\{x_i < x_j\}} - 0.5) \right|
h = 2L:
T_n = \frac{1}{\hat{\sigma}} \max_{1 \leq k \leq n} \left| \frac{1}{n^{3/2}} \sum_{i = 1}^k \sum_{j = k+1}^n (x_i - x_j) \right|
\hat{\sigma} is estimated by the square root of lrv. The denominator corresponds to that in the ordinary CUSUM change point test.
By default, kernel-based estimation is used.
If h = 1L, the default for distr is TRUE. If no block length is supplied, first the time series x is corrected for the estimated change point and Spearman's autocorrelation to lag 1 (\rho) is computed. Then the default bandwidth follows as
b_n = \max\left\{\left\lceil n^{0.25} \left( \frac{2\rho}{1 - \rho^2}\right)^{0.8} \right\rceil, 1\right\}.
Otherwise, the default for distr is FALSE and the default bandwidth is
b_n = \max\left\{\left\lceil n^{0.4} \left( \frac{2\rho}{1 - \rho^2}\right)^{1/3} \right\rceil, 1\right\}.
Value
Test statistic (numeric value) with the following attributes:
cp-location |
indicating at which index a change point is most likely. |
teststat |
test process (before taking the maximum). |
lrv-estimation |
long run variance estimation method. |
sigma |
estimated long run variance. |
param |
parameter used for the lrv estimation. |
kFun |
kernel function used for the lrv estimation. |
Is an S3 object of the class "cpStat".
Author(s)
Sheila Görz
References
Dehling, H., et al. "Change-point detection under dependence based on two-sample U-statistics." Asymptotic laws and methods in stochastics. Springer, New York, NY, 2015. 195-220.
See Also
lrv
Examples
# time series with a location change at t = 20
x <- c(rnorm(20, 0), rnorm(20, 2))
# Wilcoxon-Mann-Whitney change point test statistic
wilcox_stat(x, h = 1L, control = list(b_n = length(x)^(1/3)))
