Density estimation

Density estimation constructs an estimate of an unknown probability density function from sampled observations.

A kernel for density estimation is an integrable function $K$ with unit integral, usually nonnegative and centered, whose rescaling $K_h(u)=h^{-1}K(u/h)$ averages observations over bandwidth $h$.

A kernel is of order $\ell$ when its moments of orders $1,\dots ,\ell -1$ vanish and its absolute moment of order $\ell$ is finite.

Given independent and identically distributed random variables $X_1,\dots ,X_n$, the kernel density estimate is

For a symmetric bandwidth-$h$ averaging kernel and a density with bounded first derivative, the interior pointwise bias is at most a constant times $h\Vert f^{\prime }{\Vert }_{\infty }$. Near a support boundary, an uncorrected symmetric kernel can instead have nonvanishing bias.