Exact mean integrated squared error of a kernel density estimator ($\mathrm{MISE}=R(K_h\ast f-f)+n^{-1}[R(K_h)-R(K_h\ast f)]$)

For independent observations with common density f, the kernel density estimator has mean $K_h\ast f$. Its pointwise variance is $n^{-1}[(K_h^2\ast f)-(K_h\ast f{)}^2]$. Integrate the variance and add integrated squared bias to get the displayed exact mean integrated squared error, assuming the integrals are finite. Since $R(K_h)=R(K)/h$, this also separates the smoothing bias from the stochastic variance.