kernels {quantspec}

Kernel function.

Description

Implementations of kernel functions

Usage

W0(x)

W1(x)

W2(x)

W3(x)

WDaniell(x, a = (pi/2))

WParzen(u)

Arguments

x

real-valued argument to the function; can be a vector

a

real number between 0 and \pi

u

real number

Details

Daniell kernel function W0:

\frac{1}{2\pi} I\{|x| \leq \pi\}.

Epanechnikov kernel W1 (i. e., variance minimizing kernel function of order 2):

\frac{3}{4\pi} (1-\frac{x}{\pi})^2 I\{|x| \leq \pi\}.

Variance minimizing kernel function W2 of order 4:

\frac{15}{32\pi} (7(x/\pi)^4 -10(x/\pi)^2+3) I\{|x| \leq \pi\}.

Variance minimizing kernel function W3 of order 6:

\frac{35}{256\pi} (-99(x/\pi)^6 + 189(x/\pi)^4 - 105(x/\pi)^2+15) I\{|x| \leq \pi\}.

Kernel yield by convolution of two Daniell kernels:

\frac{1}{\pi+a} \Big(1-\frac{|x|-a}{\pi-a} I\{a \leq |x| \leq \pi\}\Big).

Parzen Window for lagEstimators

Examples

plot(x=seq(-8,8,0.05), y=W0(seq(-8,8,0.05)), type="l")
plot(x=seq(-8,8,0.05), y=W1(seq(-8,8,0.05)), type="l")
plot(x=seq(-8,8,0.05), y=W2(seq(-8,8,0.05)), type="l")
plot(x=seq(-8,8,0.05), y=W3(seq(-8,8,0.05)), type="l")
plot(x=seq(-pi,pi,0.05), y=WDaniell(seq(-pi,pi,0.05),a=(pi/2)), type="l")
plot(x=seq(-2,2,0.05),y=WParzen(seq(-2,2,0.05)),type = "l")