WAVK: WAVK Statistic

Description

Statistic for testing the parametric form of a regression function, suggested by Wang_etal_2008;textualfuntimes.

Usage

WAVK(z, kn = NULL)

Arguments

pre-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.

Value

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.

Tns

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 length and $\sigma^2$ is variance of the time series. Robust difference-based Rice's estimator Rice_1984funtimes is used to estimate $\sigma^2$.

p.value

$p$-value for Tns based on its asymptotic $N(0,1)$ distribution.

References

Examples

Run this code

# NOT RUN {
z <- rnorm(300)
WAVK(z, kn = 7)

# }