This exercise has to do with the MarcenkoPastur law of Theorem 16.2. You are asked to run

This exercise has to do with the Marˇcenko–Pastur law of Theorem 16.2. You are asked to run a simulation study to numerically verify the result of the theorem. Consider n = 10 and n = 100, and p = n/2 in each case.

(a) Generate independent random vectors, X1, . . . , Xn, from the N(0, Ip)

distribution, the p-dimensional multivariate normal distribution with mean vector zero and covariance matrix Ip, the p-dimensional identity matrix. Then, compute S

∗

n

= n

−1



n i=1 XiX



i (note that, here, Xi is a p × 1 vector, 1 ≤ i ≤ n).

(b) Compute the eigenvalues of S

∗

n, say, λn,1, . . . , λn,p (note that S

∗

n is a p × p nonnegative definite matrix). Save the result.

(c) Repeat (a),

(b) N = 1, 000 times. Then, for K = 10, compute the proportions, pn,k = 1 p

|{1 ≤ j ≤ p : λn,j ≤ a + (k/K)(b − a)}|, 0 ≤ k ≤ K.

(d) Compute the probabilities, pk = P[X ≤ a+(k/K)(b−a)], 0 ≤ k ≤ K, where X follows the M-P law (16.14) with τ = 1 and γ = 1/2. Note that p0 = 0 and pK = 1.

(e) Make a table that compares pn,k with pk, 0 ≤ k ≤ K. Do this for both n = 10 and n = 100. Does pn,k appear to approach pk, 0 ≤ k ≤ K, as n increases?