Statistic for testing the parametric form of a regression function, suggested by Wang_etal_2008;textualfuntimes.
WAVK(z, kn = NULL)A list with following components:
test statistic based on artificial ANOVA and defined by Wang_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 \(t\)th group and the grand mean, respectively.
standardized version of Tn according to
Theorem 3.1 by Wang_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 Rice_1984funtimes
is used to estimate \(\sigma^2\).
\(p\)-value for Tns based on its
asymptotic \(N(0,1)\) distribution.
filtered univariate time series @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.
length of the local window.
Yulia R. Gel, Vyacheslav Lyubchich
wavk_test
z <- rnorm(300)
WAVK(z, kn = 7)
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