Kernel Density Estimation and Bandwidth Selection

A kernel density estimate smooths a sample into a density by placing a Gaussian bump of width h (the bandwidth) on every point:

\hat f(x) = \frac{1}{n h}\sum_i \phi\!\left(\frac{x - x_i}{h}\right)

Implement for 1-D data:

kde(sample, x, h): the density at each point of x
silverman(sample): Silverman's rule of thumb, 0.9 * min(std, IQR / 1.34) * n ** (-1/5) (population std; IQR from np.percentile)
loo_log_likelihood(sample, h): the leave-one-out log-likelihood, $\sum_i \log \hat f_{-i}(x_i)$ , where each point's density is estimated from the other n − 1 points
best_bandwidth(sample, grid): the h in grid with the highest leave-one-out log-likelihood (ties: the smallest)