An approximation to the error of a Riemannian sum:

where V_g(a; b) is the total variation of g on [a, b] de ned by the sup over all partitions on [a, b], including (a; b) = (-∞, ∞). Con- clude that if f'(.
)² has nite total variation, then the remainder term in the bias (3.14) is O(h³).

Compute the roughness of several parametric densities: Cauchy, Student's t, Beta, lognormal. For each compute the optimal bin width. Express the optimal bin width in terms of the variance, if it exists.