dCovTS (version 1.0)

kernelFun: Several kernel functions

Description

Computes several kernel functions(truncated, Bartlett, Daniell, QS, Parzen). These kernels are for constructing test statistics for testing pairwise independence.

Usage

kernelFun(type, z)

Arguments

type

character string which indicates the name of the smoothing kernel. kernelFun can be: 'truncated', 'bartlett', 'daniell', 'QS', 'parzen'. No default is given.

z

real number.

Value

• A value that lies in the interval $[-1,1]$.

Details

kernelFun computes several kernel functions including truncated, Bartlett, Daniell, QS and Parzen. The exact definition of each of the above functions are given below:

• Truncated$$k(z) = \left{ \begin{array}{ll} 1, & |z| \leq 1, \[1ex] 0, & \mbox{otherwise}. \end{array} \right.$$
• Bartlett$$k(z) = \left{ \begin{array}{ll} 1 - |z|, & |z| \leq 1, \[1ex] 0, & \mbox{otherwise}. \end{array} \right.$$
• Daniell$$k(z) = \frac{\mbox{sin}(\pi z)}{\pi z}, z \in \Re - {0}$$
• QS$$k(z)=(9/5\pi^2z^2){\mbox{sin}(\sqrt{5/3}\pi z)/\sqrt{5/3}\pi z-\mbox{cos}(\sqrt{5/3}\pi z)}, z \in \Re$$
• Parzen$$k(z) = \left{ \begin{array}{ll} 1-6(\pi z/6)^2 + 6|\pi z/6|^3, & |z| \leq 3/\pi, \[1ex] 2(1-|\pi z/6|)^3, & 3/\pi \leq |z| \leq 6/\pi, \[1ex] 0, & \mbox{otherwise} \end{array} \right.$$

All these kernel functions are mainly used to smooth the generalized spectral density function, firstly introduced by Hong (1999). Assumptions and theoretical properties of these functions can be found in Hong (1996;1999) and Fokianos and Pitsillou (2016).

References

Fokianos K. and M. Pitsillou (2016). Consistent testing for pairwise dependence in time series. Technometrics, http://dx.doi.org/10.1080/00401706.2016.1156024. Hong, Y. (1996). Consistent testing for serial correlation of unknown form. Econometrica $\textbf{64}$, 837-864, http://dx.doi.org/10.2307/2171847. Hong, Y. (1999). Hypothesis testing in time series via the empirical characteristic function: A generalized spectral density approach. Journal of the American Statistical Association $\textbf{94}$, 1201-1220, http://dx.doi.org/10.1080/01621459.1999.10473874.

Examples

k1 <- kernelFun("bartlett",z=1/3)
k2 <- kernelFun("bar",z=1/5)
k3 <- kernelFun("dan",z=0.5)