Computes the test statistic for CUSUM-based tests on scale changes.
scale_stat(x, version = c("empVar", "MD", "GMD", "Qalpha"), method = "kernel",
control = list(), alpha = 0.8)Test statistic (numeric value) with the following attributes:
indicating at which index a change point is most likely.
test process (before taking the maximum).
long run variance estimation method.
If method = "kernel" the following attributes are also included:
estimated long run variance.
parameter used for the lrv estimation.
kernel function used for the lrv estimation.
Is an S3 object of the class "cpStat".
time series (numeric or ts vector).
variance estimation method.
either "kernel" for performing a kernel-based long run variance estimation, or "bootstrap" for performing a dependent wild bootstrap. See 'Details' below.
a list of control parameters.
quantile of the distribution function of all absolute pairwise differences used in version = "Qalpha".
Sheila Görz
Let \(n\) be the length of the time series. The CUSUM test statistic for testing on a change in the scale is then defined as $$\hat{T}_{s} = \max_{1 < k \leq n} \frac{k}{\sqrt{n}} |\hat{s}_{1:k} - \hat{s}_{1:n}|,$$ where \(\hat{s}_{1:k}\) is a scale estimator computed using only the first \(k\) elements of the time series \(x\).
If method = "kernel", the test statistic \(\hat{T}_s\) is divided by the estimated long run variance \(\hat{D}_s\) so that it asymptotically follows a Kolmogorov distribution. \(\hat{D}_s\) is computed by the function lrv using kernel-based estimation.
For the scale estimator \(\hat{s}_{1:k}\), there are five different options which can be set via the version parameter:
Empirical variance (empVar)
$$\hat{\sigma}^2_{1:k} = \frac{1}{k-1} \sum_{i = 1}^k (x_i - \bar{x}_{1:k})^2; \; \bar{x}_{1:k} = k^{-1} \sum_{i = 1}^k x_i.$$
Mean deviation (MD)
$$\hat{d}_{1:k}= \frac{1}{k-1} \sum_{i = 1}^k |x_i - med_{1:k}|,$$
where \(med_{1:k}\) is the median of \(x_1, ..., x_k\).
Gini's mean difference (GMD)
$$\hat{g}_{1:k} = \frac{2}{k(k-1)} \sum_{1 \leq i < j \leq k} (x_i - x_j).$$
\(Q^{\alpha}\) (Qalpha)
$$\hat{Q}^{\alpha}_{1:k} = U_{1:k}^{-1}(\alpha) = \inf\{x | \alpha \leq U_{1:k}(x)\},$$
where \(U_{1:k}\) is the empirical distribtion function of \(|x_i - x_j|, \, 1 \leq i < j \leq k\) (cf. Qalpha).
For the kernel-based long run variance estimation, the default bandwidth \(b_n\) is determined as follows:
If \(\hat{\rho}_j\) is the estimated autocorrelation to lag \(j\), a maximal lag \(l\) is selected to be the smallest integer \(k\) so that $$\max \{|\hat{\rho}_k|, ..., |\hat{\rho}_{k + \kappa_n}|\} \leq 2 \sqrt(\log_{10}(n) / n), $$ \(\kappa_n = \max \{5, \sqrt{\log_{10}(n)}\}\). This \(l\) is determined for both the original data \(x\) and the squared data \(x^2\) and the maximum \(l_{max}\) is taken. Then the bandwidth \(b_n\) is the minimum of \(l_{max}\) and \(n^{1/3}\).
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.
lrv, Qalpha
x <- arima.sim(list(ar = 0.5), 100)
# under H_0:
scale_stat(x, "GMD")
scale_stat(x, "Qalpha", method = "bootstrap")
# under the alternative:
x[51:100] <- x[51:100] * 3
scale_stat(x)
scale_stat(x, "Qalpha", method = "bootstrap")
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