Exercise 7.2 Consider the random variable

(b) Use the Edgeworth expansion to try to approximate the distribution function of X.

(c) Estimate the quantiles of order 90%, 95%, and 99% using the Cornish-
Fisher expansion. Also plot the graph of the quantile function determined by the Cornish-Fisher expansion.
Is the approximation better than the one obtained by the Edgeworth expansion?

(d) Generate 100000 approximations of X by truncating the infinite sum to a sum over k ≤ 100, i.e., generate

(e) Using the simulated values of ˜X , estimate its distribution function and compare it to the one obtained in (b). Is the Edgeworth expansion precise enough? Also estimate the quantiles of order 90%, 95%, and 99%.
Compare them with the theoretical values. Finally, compare the empirical quantiles curve with the Cornish-Fisher expansion.

X = 4 IM k=1 Z (2k - 1)'