1.   [25 marks] Assume that X1 , . . . ,Xn are independent, identically distributed random variables with probability density function

fX (x;✓) = (x + ✓)4 ,  x > 0,  ✓ > 0.

Also assume that x1 , . . . ,xn are observations of X1 , . . . ,Xn .

(a)  [5 marks] Show that E(Xi ) = ✓/2 and hence that ✓˜ = 2 is an unbiased estimator of ✓, where  =  P Xi .

(b)  [1 mark] Write down the log-likelihood function for ✓ given x = (x1 , . . . ,xn ).

(c)  [2 marks] Find the score for ✓ for a sample of size n.

(d)  [5 Marks] Show that the Fisher information for ✓, for a sample of size n, is

3n

I(✓) =

(e)  [3 marks] Give a lower bound for the variance of unbiased estimators of ✓ and show that this cannot be attained.

(f)  [3 marks] Given that Var(Xi ) = ✓2 , find the efficiency of ✓˜.

(g)  [6 marks] Use the Central Limit Theorem to find the asymptotic distribution of ✓˜ and hence an approximate 95% confidence interval for ✓ .

2.   [25 marks]

(a)  [3 marks] Consider the following two linear models both with Y = Xβ + ✏, but where under M0

✏ ⇠ N(0 , σ 2In )

and under M1

✏ ⇠ N(0 , ⌃),

where ⌃ is an invertible n ⇥ n covariance matrix.

Show that the ordinary least squares estimator of β ,

 = (XT X)−1XTY ,

is unbiased under both models and find its covariance matrix under both models. (b)  [2 marks] Consider from now on the following special cases of M0 and M1 :

Yi = 

where under M0

✏i ⇠ N(0, σ 2 ),  i = 1, . . . , 2m,

and under M1

✏i ⇠

Further assume that the ✏i ,i = 1, . . . , 2m, are independent.

Write-out the vectors and matrices Y , X , β and ⌃ in terms of

Y1 , . . . ,Y2m , β0 , β1 , σ 1(2) and σ 2(2), when m = 2.

(c)  [6 marks] Show that under M1 the maximum likelihood estimates of β0 , β1 , σ1(2) and σ2(2) are given by

βˆ0   =  y¯1 ,

βˆ1   =  y¯2 − y¯1 ,

m

2(2)  =   i (yi − βˆ0 − βˆ1 )2 ,

where y¯1  =  P yi and y¯2  =  P yi . There is no need to check that

the Hessian is negative definite.

(d)  [3 marks] Show that under model M0 , the maximum likelihood estimate of σ 2

can be written as

2  =  ( 1(2) + 2(2)).

(e)  [5 marks] Show that the generalised likelihood ratio test for testing M0 against M1 rejects M0 when

(1 + F) ✓ 1+ ◆ > k,

where

1(2)

F =

(f)  [3 marks] Explain why the generalised likelihood ratio test for testing M0 against M1 rejects M0 when F < 1/c or when F > c.

(g)  [3 marks] Explain why F follows an F distribution under M0 and give its degrees of freedom. You may state any standard results from distribution theory without    proof.

3.   [25 marks]

Assume that X1 ,X2 , . . . ,Xnare independent identically distributed N(✓ , σ 2 ) observations. Suppose that the joint prior distribution for ✓ and σ 2 is

⇡(✓ , σ 2 )

/  1

✓ 2 ( −1, 1), σ 2  2 (0, 1).

(a)  [3 marks] Derive, up to a constant of proportionality, the joint posterior density of ✓ and σ 2 .

(b)  [4 marks] Derive the conditional posterior distributions of ✓ given σ 2 and of σ 2 given ✓ . Name those distributions.

(c)  [8 marks] Derive the marginal posterior density of ✓ . Name that distribution and write down its mean and variance for n > 3.

(d)  [10 marks] Now assume that σ 2 is a known constant. Find the Bayes factor where one model corresponds to ✓ = 0 and the other model has ✓ following N(0, ⌧ 2 ) a-priori.