Show that the ISB in a bin containing the origin of the double exponen-tial density, f(x) = exp(-|x|)/2, is O(h³); hence, the discontinuity in the derivative of f does not have any asymptotic eect on consistency. Compare when 0 and h/2 are bin edges. Formally, if f is the ordinary negative exponential density, R(f') is in nite because of the jump at zero (integral of the square of the Dirac delta function at 0), but R(f') is well-de ned for the double exponential.

Consider the exact MISE of a histogram when f = U(0; 1).

(a) If the mesh t_k = kh is chosen where h = 1/m, show that MISE(m, n) = (m - 1)/n.

(b) If t_k = (k + 1/2 )h with h = 1/m, show that

(c) Conclude that

Verify the inequalities for the adaptive MISE.

Find the optimal adaptive meshes for a skewed Beta density using a numerical optimization program.