# Non-parametric estimation of probability distributions

## Kernels

### Smoothing kernel estimation

#### Smoothed kernels

We have $$K(x-x_i)$$

We can smooth this to:

$$K_h(x-x_i)=\dfrac{1}{h}K(\dfrac{x-x_i}{h})$$

Where $$h>0$$ is the smoothing bandwidth.

$$f(x)=\dfrac{1}{n}\sum_{i=1}^nK_h(x-x_i)$$