Gamma kernel Density estimator

Question

The standard kernel Density estimator of $f$ is given by $$\hat{f}(x)=\frac{1}{nh}\sum_{i=1}^n K\left(\frac{x-X_i}{h}\right)$$ and the gamma kerenel $$ K_{x/b+1,b}(t)=\frac{t^{x/b}e^{-t/b}}{b^{x/b+1}\Gamma(x/b+1)}$$ where $b$ is the smoothing parameter ($b=h$ in this case) satisfying $b \to 0$ and $nb \to \infty$ as $n \to \infty$

I read a peaper, where the author obtain directly

$$\hat{f}(x)=\frac{1}{n}\sum_{i=1}^n K_{x/b+1,b}(X_i)$$ but I do not know how this happen directly.

Could someone help me?

Any participation will be appreciated.

Answer 1

This is $$\hat{f_1}(x)=\frac{1}{n}\sum_{i=1}^n K_{x/b+1,b}(X_i)$$ and there is $$\hat{f_2}(x)=\frac{1}{n}\sum_{i=1}^n K_{\rho_b(x),b}(X_i)$$ which are definitions .

Answer 2

I think you've misunderstood the paper. The author isn't saying that the definition $$\hat f_{\!1}(x) = \frac{1}{n} \sum_{i=1}^n K_{x/b+1,b}(X_i)$$ corresponds to a specific choice of kernel and bandwidth in the previous equation $$\hat f(x) = \frac{1}{nh} \sum_{i=1}^n K\left(\frac{x-X_i}{h}\right)$$ (which is called Eq. 2.1 in the paper). Rather, the author is defining a gamma kernel density estimator with that form. Indeed, you can see that this is necessary because with (2.1), the kernel must be defined for $x < X_i$. There are no choices of $K$ from the gamma family that would make that work since its support is on $[0, \infty)$.