WAVK: WAVK Statistic

Description

Computes statistic for testing the parametric form of a regression function, suggested by Wang, Akritas and Van Keilegom (2008).

Usage

WAVK(z, kn = NULL)

Arguments

pre-filtered univariate time series (see formula (2.1) by Wang and Van Keilegom, 2007): $$Z_i=\left(Y_{i+p}-\sum_{j=1}^p{\hat{\phi}_{j,n}Y_{i+p-j}} \right)-\left( f(\hat{\theta},t_{i+p})-\sum_{j=1}^p{\hat{\phi}_{j,n}f(\hat{\theta},t_{i+p-j})} \right) ,$$ 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 and Van Keilegom (2007) 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 and Van Keilegom (2007): $$T_{ns} = \left( \frac{n}{k_n} \right)^{\frac{1}{2}}T_n \bigg/ \left(\frac{4}{3}\right)^{\frac{1}{2}} \sigma^2,$$ where $n$ is length and $\sigma^2$ is variance of the time series. Robust difference-based Rice's estimator (Rice, 1984) is used to estimate $\sigma^2$.

p.value

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

References

Rice, J. (1984). Bandwidth choice for nonparametric regression. The Annals of Statistics 12: 1215--1230. DOI: 10.1214/aos/1176346788

Wang, L., Akritas, M. G., and Van Keilegom, I. (2008). An ANOVA-type nonparametric diagnostic test for heteroscedastic regression models. Journal of Nonparametric Statistics 20(5): 365--382. DOI: 10.1080/10485250802066112

Wang, L. and Van Keilegom, I. (2007). Nonparametric test for the form of parametric regression with time series errors. Statistica Sinica 17: 369--386.

Examples

Run this code

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