Statistical Theory For Economist

1. Let 1 2 , ,…n X X X be a random sample of size n from a Poisson distribution with parameter l. Show
that the sample mean X is the minimum variance unbiased estimator of l. (8 marks)
2. Let 1 2 , ,…n X X X be a random sample of size n from a normal distribution with parameters
(μ,s 2 ) . (12 marks)
a. Derive the Cramer-Rao lower bound matrix for an unbiased estimator of the vector of
parameters . (μ,s 2 ) .
b. Using the Cramer-Rao lower bound prove that the sample mean X is the minimum
variance unbiased estimator of μ .
c. Is the maximum likelihood estimator of ( )2 2 2
1
ˆ 1 n
i i X X
n
s s = = = å – unbiased?
Compute the variance of sˆ 2 and compare it to the Cramer-Rao lower bound derived in
(a).
3. State and prove the Lindberg-Levy form of the Central Limit Theorem. Use the moment generating
function approach for the proof. (8 marks)
4. Let 1 2 , ,…n X X X be a random sample of size n from a random variable with mean and variance
given by (μ,s 2 ) . (10 marks)
a. Show that the sample mean X is a consistent estimator of mean μ .
b. Show that the sample variance of ( )2 2 2
1
ˆ 1 n
i i X X
n
s s = = = å – converges in probability
to s 2 . Clearly state any theorems or results you may have used in this proof.
5. Let { , }, 1,2,…, i i X Y i = n be independently and identically distributed with common mean μ > 0
and common variance s 2 . Let 1 1
1 ; 1 n n
i i i i X XY Y
n = n = = å = å . Assume that { } i X and
{ } i Y are independent. Assume also that Yi >0 for all i. Obtain the probability limit and the
asymptotic distribution of X + logY . (8 marks)
6. Let { , }, 1,2,…, i i X Y i = n be independently and identically distributed sequence of random
variables with means ( , ) X Y μ μ and variances ( 2 , 2 ) X Y s s respectively. Further i i X andY re
independently distributed. Obtain the asymptotic distribution of: (i) X 2 ; (ii)1/ X; and (iii)
exp(X ) . Clearly state any theorems or results you may have used in this proof. (12 marks)
7. A particular drug was given to a group of 100 patients and found that 60 patients recovered.
Construct a 90% confidence interval on the mean rate of recovery by patients who used the
particular drug. (6 marks)
8. If 50 students in an econometrics class took on the average 35 minutes to solve an examination
problem with a variance of 10 minutes, construct a 95% confidence interval for the true standard
deviation of the time it takes students to solve the given problem. Answer using exact distribution
under normality of the distribution of time taken to solve the problem and also using asymptotic
methods.(6 marks)
9. Let 1 2 , ,…n X X X be a random sample of size n from a normal distribution with unknown μ and
known variance s 2 . Let i x be the observed value of i X . Find the best critical region for testing
0 0 1 1 H :μ =μ vs H :μ =μ . (10 marks)
10. Let X be a random variable that has density function ( ) x , for 0, 0
X f x =b e-b x > b > and 0
otherwise. Drive the Neyman-Pearson optimal region assuming a = 0.10 for testing
0 1 H :b =1 vs H :b =11 on the basis of one observation on X. (10 marks)
11. Let X Uniform(0,h). Derive and draw the power function of the test based on the critical region
[0.75,¥) that uses one observation for testing 0 1 H :h =1 vs H :h >1. (10 marks)