Kernel function.
Implementations of kernel functions
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
:
2π1I{∣x∣≤π}.1/(2pi)I∣x∣<=pi.
Epanechnikov kernel W1
(i. e., variance minimizing kernel function of order 2):
4π3(1−πx)2I{∣x∣≤π}.3/(4pi)(1−x/pi)2I∣x∣<=pi.
Variance minimizing kernel function W2
of order 4:
32π15(7(x/π)4−10(x/π)2+3)I{∣x∣≤π}.(15/(32pi)(7(x/pi)4−10(x/pi)2+3)I∣x∣<=pi.
Variance minimizing kernel function W3
of order 6:
256π35(−99(x/π)6+189(x/π)4−105(x/π)2+15)I{∣x∣≤π}.(35/(256pi)(−99(x/pi)6+189(x/pi)4−105(x/pi)2+15)I∣x∣<=pi.
Kernel yield by convolution of two Daniell kernels:
π+a1(1−π−a∣x∣−aI{a≤∣x∣≤π}).
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")