代做linear留学生、R程序语言调试、代写R代写、代做matrix equal

日期：2019-03-27 10:16

Homework 7

distribution with mean and variance

. Show that are independent, and derive the distribution of.

Question 3

Consider the linear regression problem 8 = 9: + <, where 9 is an ×  matrix with full

column rank, : is a  × 1 vector of unknown parameters, and <~(0, (), i.e., < is a multivariate normal distribution with a × 1 mean vector of 0’s, and a covariance matrix equal

to the identity matrix. Given the prior distribution expC0.5(

GF, where is a  ×

strictly positive definite matrix, and ,, G are strictly positive constants:

1. Show that the marginal posterior distribution of

is an inverse gamma distribution

IG JG + RSS)M distribution, where RSS = )8 9:N+

E

(9?9E + ())8 9:N+ and

:N = (9E9 +)9E8.

2. Show that the marginal posterior distribution of : is a multivariate noncentral O distribution

with mean :N, scale matrix (G + RSS)(9E9 +)/( + 2,), and + 2, degrees of

freedom. Note: A multivariate noncentral O distribution with a  × 1 mean vector Q, scale

matrix , and R degrees of freedom has PDF∝ [1 + R(Q)E Q)](UV>)/.

Question2

Consider the linear regression model 8 = 9: + <. Let E = )8 8N be the

vector of leave-one-out prediction error in ridge regression. Show that

h = diag(m)m8,

where m = ( 9(9E9 + X()9E

and mis the nth diagonal element of m.

Question 1(i.e.,Let , … ,is a random sample from the univariate normal