WAVK {funtimes}

WAVK Statistic

Description

Statistic for testing the parametric form of a regression function, suggested by Wang et al. (2008).

Usage

WAVK(z, kn = NULL)

Arguments

z

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.

kn

length of the local window.

Value

A list with following components:

Tn

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 tth 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 the length and \sigma^2 is the 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.

Author(s)

Yulia R. Gel, Vyacheslav Lyubchich

References

Rice J (1984). “Bandwidth choice for nonparametric regression.” The Annals of Statistics, 12(4), 1215–1230. doi:10.1214/aos/1176346788.

Wang L, Akritas MG, Van Keilegom I (2008). “An ANOVA-type nonparametric diagnostic test for heteroscedastic regression models.” Journal of Nonparametric Statistics, 20(5), 365–382.

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

See Also

wavk_test

Examples

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