An important problem in multivariate analysis is the test of sphericity of the data, that is, the null hypothesis Ho : ) =021, where of is unspecified. This hypothesis expresses the fact that the error is cross-sectionally uncorrelated (independent if the data is normally distributed ) and have the same variance (homoscedasticity). Clearly, data sampled from a multivariate normal No(u, [) would exhibit sphericity as the density function is constant on the ellipsoids ( x - H ) T E - 1 ( x - H ) = k for every positive value of k and x 6 RP. A general class of distributions with this property is the class of elliptical distributions. A random vector x with zero mean follows an elliptical distribution if (and only if) it has the stochastic representation * = [Au, (* ) where the matrix A 6 ROXP is nonrandom and rank(A) = p, & 2 0 is a random variable representing the radius of x, and u e RP is the random direction, which is independent of and uniformly distributed on the unit sphere 50-1 in RP, denoted by u ~ Unif(SP-1). Question 1 (a) Write a function runifsphere(n, p) that samples n observations from the distribution Unif(SP-1) using the fact that if 2 - No(0, 1) then z/||z| | ~ Unif(SP-1). Check your results by: (1) set p = 10, n = 100 and show that the (Euclidean) norm of each observation is equal to 1, (2) generate a scatter plot in the case p = 2, n = 500 to show that the samples lie on a circle. (b) A classic statistic for testing sphericity (called John's test) that is proposed in [A] and [B] is U = = tr Sn (1/p) tr S, (1/p) tr S, ((1/p) trS,] 1, where is it shown that when p is fixed and n -+ co, under the null hypothesis, it holds that "U 4 x, with p := jp(p+ 1) -1. Perform a simulation to show that "U is distributed like x, under the null hypothesis in the case n = 5000 observations, p = 5, and with data generated from No(0, /,). (c) Check the impact on the distribution of 2 0 when the data is sampled from a double exponential distribution (i.e., a particular case of an elliptic distribution). This can be generated using (* ) with & ~ Gamma(p, 1) and A = /p