scale_stat: Test statistic to detect Scale Changes

In robcp: Robust Change-Point Tests

Test statistic to detect Scale Changes

Description

Computes the test statistic for CUSUM-based tests on scale changes.

Usage

scale_stat(x, version =  c("empVar", "MD", "GMD", "Qalpha"), method = "kernel",
           control = list(), alpha = 0.8)

Arguments

x	time series (numeric or ts vector).
version	variance estimation method.
method	either "kernel" for performing a kernel-based long run variance estimation, or "bootstrap" for performing a dependent wild bootstrap. See 'Details' below.
control	a list of control parameters.
alpha	quantile of the distribution function of all absolute pairwise differences used in version = "Qalpha".

Details

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}.

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.

If method = "kernel" the following attributes are also included:

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

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.

See Also

lrv, Qalpha

Examples

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")

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