Exercise 8.2 Consider the random variable

where the Zijs are independent and identically distributed standard Gaussian variables.

(a) Find the first six cumulants of X. Note that the famous zeta6 function is defined by

(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 i, j ≤ 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%.
Finally, compare the empirical quantiles curve with the Cornish-Fisher expansion.

Transcribed Image Text:

T = 36 IM8 i=1 j=1 12j2