huber_cusum: Huberized CUSUM test

Description

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

Usage

huber_cusum(x, fun = "HLm", k, constant = 1.4826, method = "kernel",
            control = list(), fpc = TRUE, tol = 1e-8, plot = FALSE, ...)

Value

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

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).
cp.location: index of the estimated change point location (integer).
data.name: name of the data (character string).

Arguments

x: numeric vector containing a single time series or a numeric matrix containing multiple time series (column-wise).
fun: character string specifying the transformation function $\psi$, see details. For the ordinary CUSUM test, use fun = "none".
k: numeric bound used in psi.
constant: scale factor of the MAD. Default is 1.4826.
method: method for estimating the long run variance.
control: a list of control parameters for the estimation of the long run variance (cf. lrv).
fpc: finite population correction (boolean).
tol: tolerance of the distribution function (numeric), which is used to compute p-values.
plot: should the test statistic be plotted (cf. plot.cpStat). Boolean.
...: further arguments to be passed to CUSUM.

Author

Sheila Görz

Details

The function performs a Huberized CUSUM test. It tests the null hypothesis $H_0: \boldsymbol{\theta}$ does not change for $x$ against the alternative of a change, where $\boldsymbol{\theta}$ is the parameter vector of interest. $k$ is called a 'change point'. First the data is transformed by a suitable psi-function. To detect changes in location one can apply fun = "HLm", "HLg", "SLm" or "SLg" and the hypothesis pair is $$H_0: \mu_1 = ... = \mu_n$$ $$vs.$$ $$H_1: \exists k \in \{1, ..., n-1\}: \mu_k \neq \mu_{k+1}$$ where $\mu_t = E(X_t)$ and $n$ is the length of the time series. For changes in scale fun = "HCm" is available and for changes in the dependence respectively covariance structure fun = "HCm", "HCg", "SCm" and "SCg" are possible. The hypothesis pair is the same as in the location case, only with $\mu_i$ being replaced by $\Sigma_i$, $\Sigma_i = Cov(X_i)$. Exact definitions of the psi-functions can be found on the help page of psi.

Denote by $Y_1,\ldots,Y_n$ the transformed time series. If $Y_1$ is one-dimensional, then the test statistic $$V_n = \max_{k=1,\ldots,n} \frac{1}{\sqrt{n}\sigma} \left|\sum_{i=1}^k Y_i-\frac{k}{n} \sum_{i=1}^n Y_i\right| $$ is calculated, where $\sigma^2$ is an estimator for the long run variance, see the help function of lrv for details. $V$ is asymptotically Kolmogorov-Smirnov distributed. If fpc is TRUE we use a finite population correction $V+0.58/\sqrt{n}$ to improve finite sample performance (Dürre, 2021+).
If $Y_1$ is multivariate, then the test statistic $$W_n=\max_{k=1,\ldots,n} \frac{1}{n}\left(\sum_{i=1}^k Y_i-\frac{k}{n} \sum_{i=1}^n Y_i\right)' \Sigma^{-1}\left(\sum_{i=1}^k Y_i-\frac{k}{n} \sum_{i=1}^n Y_i\right) $$ is computed, where $\Sigma$ is the long run covariance, see also lrv for details. $W$ is asymptotically distributed like the maximum of a squared Bessel bridge. We use the identity derived by Kiefer (1959) to derive p-values. Like in the one dimensional case if fpc is TRUE we use a finite sample correction $(\sqrt{W}+0.58/\sqrt{n})^2$.

The change point location is estimated as the time point $k$ for which the CUSUM process takes its maximum.

References

Hušková, M., & Marušiaková, M. (2012). M-procedures for detection of changes for dependent observations. Communications in Statistics-Simulation and Computation, 41(7), 1032-1050.

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

Dürre, A. (2021+). "Finite sample correction for cusum tests", unpublished manuscript

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

Examples

Run this code

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)

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