cor_stat: Test statistic to detect Fluctuation Changes

In robcp: Robust Change-Point Tests

Test statistic to detect Fluctuation Changes

Description

Computes the test statistic for a CUSUM-based tests on changes in Spearman's rho or Kendall's tau.

Usage

cor_stat(x, version = c("tau", "rho"), method = "kernel", control = list())

Arguments

x	time series (numeric or ts vector).
version	version of the test. Either "rho" or "tau".
method	methods of long run variance estimation. Options are "kernel" and "none".
control	a list of control parameters.

Details

Let n be the length of the time series, i.e. the number of rows in x. In general, the (scaled) CUSUM test statistic is defined as

\hat{T}_{ξ; n} = \max_{k = 1, ..., n} \frac{k}{2√{n}\hat{σ}} | \hat{ξ}_k - \hat{ξ}_n |,

where \hat{ξ} is an estimator for the property on which to test, and \hat{σ} is an estimator for the square root of the corresponding long run variance (cf. lrv).

If version = "tau", the function tests if the correlation between x_i and x_i of the bivariate time series (x_i, x_i)_{i = 1, ..., n} stays constant for all i = 1, ..., n by considering Kendall's tau. Therefore, \hat{ξ} = \hat{τ} is the the sample version of Kendall's tau:

\hat{τ}_k = \frac{2}{k(k-1)} ∑_{1 ≤q i < j ≤q k} sign≤ft((x_j - x_i)(y_j - y_i)\right).

The default bandwidth for the kernel-based long run variance estimation is b_n = \lfloor 2n^{1/3} \rfloor and the default kernel function is the quatratic kernel.

If version = "rho", the function tests if the correlation of a time series of an arbitrary dimension d (>= 2) stays constant by considering Spearman's rho. Therefore, \hat{ξ} = \hat{ρ} is the sample version of Spearman's rho:

\hat{ρ}_k = a(d) ≤ft( \frac{2^d}{k} ∑_{j = 1}^k ∏_{i = 1}^d (1 - U_{i, j; n}) - 1 \right)

where U_{i, j; n} = n^{-1} (rank of x_{i,j} in x_{i,1}, ..., x_{i,n}) and a(d) = (d+1) / (2^d - d - 1). Here it is essential to use \hat{U}_{i, j; n} instead of \hat{U}_{i, j; k}. The default bandwidth for the kernel-based long run variance estimation is √{n} and the default kernel function is the Bartlett kernel.

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

Wied, D., Dehling, H., Van Kampen, M., and Vogel, D. (2014). A fluctuation test for constant Spearman’s rho with nuisance-free limit distribution. Computational Statistics & Data Analysis, 76, 723-736.

See Also

lrv, cor_cusum

robcp documentation built on Sept. 16, 2022, 5:05 p.m.