Bias for kernel density estimator (periodic case)

Question

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?

Answer 1

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:

However, when using data on the circle, we cannot use distance in Euclidean space, so all differences θ − θ_i should be replaced by considering the angle between two vectors:

$d_i\theta)= \| \theta -\theta_i \| = \min(|\theta-\theta_i|, 2π -|\theta-\theta_i|).$

-- 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).