Show that the ISB in a bin containing the origin of the double exponen-tial density, f(x) = exp(-|x|)/2, is O(h^{3}); 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 innite because of the jump at zero (integral of the square of the Dirac delta function at 0), but R(f') is well-dened 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.