Kernel density estimator is given by
$$\hat{f}(x,h)=\frac{1}{nh}\sum_{i=1}^{n}K(\frac{x-X_{i}}{h})$$
where $X_1,…X_n$ i.i.d with some unknown density $f$, $h$ – bandwith,
$K$ – kernel function (
$\int_{-\infty}^{\infty}K(x)dx=1$,
$\int_{-\infty}^{\infty}K(x)xdx=0$,
$\int_{-\infty}^{\infty}K(x)x^2dx<\infty$).
The bias can be computed using Taylor expansion:
$$\int_{-\infty}^{\infty}\frac{1}{h}K(\frac{x-y}{h})f(y)dy-f(x)=\int_{-\infty}^{\infty}K(y)\left(f(x-hy)-f(x)\right)dy$$ $$=\int_{-\infty}^{\infty}K(y)\left(f'(x)hy+\frac{1}{2}f''(x)(hy)^{2}+o(h^{2})\right)dy=\frac{1}{2}f''(x)h^{2}+o(h^{2})$$
How to deal with periodic kernel and $f$ ($\int_{0}^{1}K(x)dx=1$,$\int_{0}^{1}K(x)xdx=0$,$\int_{0}^{1}K(x)x^2dx<\infty$)?
How can I use taylor expansion?($\int_{0}^{1}\frac{1}{h}K(\frac{y-x}{h})f(y)dy=\int_{-\frac{x}{h}}^{1-\frac{x}{h}}K(y)f(x-yh)dy\neq\int_{0}^{1}K(y)f(x-yh)dy$ -I can't use kernel properties)
Could you recommend a good book about kernel smoothing for circular data?
Best Answer
A quick google brings up this, which indicates that when working with circular data you'll need a different definition of 'bias' for a start:
-- Charles C Taylor. Automatic bandwidth selection for circular density estimation. Computational Statistics & Data Analysis Volume 52, Issue 7, 15 March 2008, Pages 3493-3500. doi: 10.1016/j.csda.2007.11.003
He references these books:
S. Rao Jammalamadaka and A. SenGupta, Topics in Circular Statistics, World Scientific, Singapore (2001).
K.V. Mardia and P.E. Jupp, Directional Statistics, John Wiley, Chichester (1999).