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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
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