WAVK: WAVK Statistic

In funtimes: Functions for Time Series Analysis

WAVK Statistic

Description

Statistic for testing the parametric form of a regression function, suggested by \insertCiteWang_etal_2008;textualfuntimes.

Usage

WAVK(z, kn = NULL)

Arguments

z

filtered univariate time series \insertCite@see formula (2.1) by @Wang_VanKeilegom_2007funtimes: Z_i=\left(Y_{i+p}-\sum_{j=1}^p{\hat{\phi}_{j,n}Y_{i+p-j}} \right)- \left( f(\hat{\theta},t_{i+p})- \sum_{j=1}^p{\hat{\phi}_{j,n}f(\hat{\theta},t_{i+p-j})} \right), where Y_i is observed time series of length n, \hat{\theta} is an estimator of hypothesized parametric trend f(\theta, t), and \hat{\phi}_p=(\hat{\phi}_{1,n}, \ldots, \hat{\phi}_{p,n})' are estimated coefficients of an autoregressive filter of order p. Missing values are not allowed.

kn

length of the local window.

Value

A list with following components:

Tn	test statistic based on artificial ANOVA and defined by \insertCiteWang_VanKeilegom_2007;textualfuntimes as a difference of mean square for treatments (MST) and mean square for errors (MSE): T_n= MST - MSE =\frac{k_{n}}{n-1} \sum_{t=1}^T \biggl(\overline{V}_{t.}-\overline{V}_{..}\biggr)^2 - \frac{1}{n(k_{n}-1)} \sum_{t=1}^n \sum_{j=1}^{k_{n}}\biggl(V_{tj}-\overline{V}_{t.}\biggr)^2, where \{V_{t1}, \ldots, V_{tk_n}\}=\{Z_j: j\in W_{t}\}, W_t is a local window, \overline{V}_{t.} and \overline{V}_{..} are the mean of the tth group and the grand mean, respectively.
Tns	standardized version of Tn according to Theorem 3.1 by \insertCiteWang_VanKeilegom_2007;textualfuntimes: T_{ns} = \left( \frac{n}{k_n} \right)^{\frac{1}{2}}T_n \bigg/ \left(\frac{4}{3}\right)^{\frac{1}{2}} \sigma^2, where n is the length and \sigma^2 is the variance of the time series. Robust difference-based Rice's estimator \insertCiteRice_1984funtimes is used to estimate \sigma^2.
p.value	p-value for Tns based on its asymptotic N(0,1) distribution.

Author(s)

Yulia R. Gel, Vyacheslav Lyubchich

References

\insertAllCited

See Also

wavk_test

Examples

z <- rnorm(300)
WAVK(z, kn = 7)

funtimes documentation built on March 31, 2023, 7:35 p.m.