scale_cusum: Tests for Scale Changes Based on Pairwise Differences

In robcp: Robust Change-Point Tests

Tests for Scale Changes Based on Pairwise Differences

Description

Performs the CUSUM-based test on changes in the scale.

Usage

scale_cusum(x, version =  c("empVar", "MD", "GMD", "Qalpha"), method = "kernel",
            control = list(), alpha = 0.8, fpc = TRUE, tol,
            plot = FALSE, level = 0.05)

Arguments

x	time series (numeric or ts vector).
version	variance estimation method (see scale_stat).
method	method for estimating the long run variance.
control	a list of control parameters.
alpha	quantile of the distribution function of all absolute pairwise differences used in version = "Qalpha".
fpc	finite population correction (boolean).
tol	tolerance for the computation of the p-value (numeric). Default for kernel-based long run variance estimation: 1e-8. Default for bootstrap: 1e-3.
plot	should the test statistic be plotted (cf. plot.cpStat)? Boolean.
level	significance level of the test (numeric between 0 and 1).

Details

The function performs a CUSUM-type test on changes in the scale. Formally, the hypothesis pair is

H_0: \sigma^2_2 = ... = \sigma_n^2

vs.

H_1: \exists k \in \{2, ..., n-1\}: \sigma^2_k \neq \sigma^2_{k+1}

where \sigma^2_t = Var(X_t) and n is the length of the time series. k is called a 'change point'. The hypotheses do not include \sigma^2_1 since then the variance of one single observation would need to be estimated.

The test statistic is computed using scale_stat and in the case of method = "kernel" asymptotically follows a Kolmogorov distribution. To derive the p-value, the function pKSdist is used.

If method = "bootstrap", a dependent block bootstrap with parameters B = 1/tol and l = control$l is performed in order to derive the p-value of the test. First, select a boostrap sample x_1^*, ..., x_{kl}^*, k = \lfloor n/l \rfloor, the following way: Uniformly draw a random iid sample J_1, ..., J_k from \{1, ..., n-l+1\} and concatenate the blocks x_{J_i}, ..., x_{J_i + l-1} for i = 1, ..., k. Then apply the test statistic \hat{T}_s to the bootstrap sample. Repeat the procedure B times. The p-value is can be obtained as the proportion of the bootstrapped test statistics which is larger than the test statistic on the full sample.

Value

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

statistic	return value of the function scale_stat.
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).

Plus if method = "kernel":

lrv	list containing the compontents method, param and value.

else if method = "bootstrap":

bootstrap

list containing the compontents param (= block length) and crit.value.

Author(s)

Sheila Görz

References

Gerstenberger, C., Vogel, D., and Wendler, M. (2020). Tests for scale changes based on pairwise differences. Journal of the American Statistical Association, 115(531), 1336-1348.

Dürre, A. (2022+). "Finite sample correction for cusum tests", unpublished manuscript

See Also

scale_stat, lrv, pKSdist, Qalpha

Examples

x <- arima.sim(list(ar = 0.5), 100)

# under H_0:
scale_cusum(x)
scale_cusum(x, "MD")

# under the alternative:
x[51:100] <- x[51:100] * 3
scale_cusum(x)
scale_cusum(x, "MD")

robcp documentation built on April 11, 2025, 6:18 p.m.