Compute the roughness of several parametric densities, Applied Statistics

An approximation to the error of a Riemannian sum:

156_Compute the roughness of several parametric densities.png

where Vg(a; b) is the total variation of g on [a, b] de ned by the sup 1145_Compute the roughness of several parametric densities1.png 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.

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