huber_cusum {robcp}R Documentation

Huberized CUSUM test

Description

Performs a CUSUM test on data transformed by psi. Depending in the choosen psi-function different types of changes can be detected.

Usage

huber_cusum(x, fun = "HLm", tol = 1e-8, b_n, k, constant)

Arguments

x

numeric vector containing a single time series or a numeric matrix containing multiple time series (column-wise).

fun

character string specifiyng the transformation function ψ, see details.

tol

tolerance of the distribution function (numeric), which is used do compute p-values.

b_n

bandwidth, which is used to estimate the long run variance, see the help page of sigma2 for details.

k

numeric bound used in psi.

constant

scale factor of the MAD. Default is 1.4826.

Details

The function performs a Huberized CUSUM test. First the data is transformed by a suitable psi-function. To detect changes in location one can apply fun = "HLm", "HLg", "VLm" or "VLg", for changes in scale fun = "HCm" is avaliable and for changes in the dependence respectively covariance structure fun = "HCm", "HCg", "VCm" and "VCg" are possible. The actual definitions of the psi-functions can be found in the help page of psi. Denote Y_1,…,Y_n the transformed time series. If Y_1 is one dimensional, then the teststatistik

V=\max_{k=1,…,n} \frac{1}{√{n}σ} |∑_{i=1}^k Y_i-\frac{k}{n} ∑_{i=1}^n Y_i|

is calculated, where σ^2 is an estimator for the long run variance, see the help function of sigma2 for details. V is asymptoticaly Kolmogorov-Smirnov distributed. We use a finite sample correction V+0.58/√{n} to improve finite sample performance.
If Y[1] is multivariate, then the test statistic

W=\max_{k=1,…,n} \frac{1}{n}(∑_{i=1}^k Y_i-\frac{k}{n} ∑_{i=1}^n Y_i)^TΣ^{-1}(∑_{i=1}^k Y_i-\frac{k}{n} ∑_{i=1}^n Y_i)

is computed, where Σ is the long run covariance, see also sigma2 for details. W is asymptotically distributed like the maximum of a squared Bessel bridge. We use the identity derived in Kiefer to derive p-values. Like in the one dimensional case we use a finite sample correction (√{W}+0.58/√{n})^2.

Value

A list of the class "htest" containing the following compontents:

statistic

value of the test statistic (numeric).

p.value

p-value (numeric).

alternative

alternative hypothesis (character string).

method

name of the performed test(character string).

data.name

name of the data (character string).

Author(s)

Sheila Görz

References

Dürre, A. and Fried, R. (2019). "Robust change point tests by bounded transformations", https://arxiv.org/abs/1905.06201

Kiefer, J. (1959). "K-sample analogues of the Kolmogorov-Smirnov and Cramer-V. Mises tests", The Annals of Mathematical Statistics, 420–447.

See Also

sigma2, psi, h_cumsum, teststat, pKSdist

Examples

set.seed(1895)

#time series with a structural break at t = 20
Z <- c(rnorm(20, 0), rnorm(20, 2))
huber_cusum(Z) 

# two time series with a structural break at t = 20
timeSeries <- matrix(c(rnorm(20, 0), rnorm(20, 2), rnorm(20, 1), rnorm(20, 3), 
                     ncol = 2))
                     
huber_cusum(timeSeries)

[Package robcp version 0.2.4 Index]