LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

## SECOND SEMESTER – APRIL 2004

**ST 2801 – ESTIMATION THEORY**

05.04.2004 Max:100 marks

1.00 – 4.00

**SECTION – A**

Answer **ALL** the questions (10 ´ 2 = 20 marks)

- What is the problem of point estimation?
- Show that UMVUE of a given parametric function is unique almost surely.
- Define Q
_{A}– optimality criterion. - Let X ~ N ( 0, s
^{2}), s > 0. Find a minimal sufficient statistic. - Classify the following as location none of the two:
- a) BVN (0,0, q, q, 1/2) b) BVN (q, 0, 1, 1, 0.6).
- State Rao-Blackwell theorem.
- Define Exponential family.
- Let X ~ B (n, p), n = 2, 3 and p = . Obtain MLE of (n, p) based on X.
- Define scale equivariant estimator.
- Explain Minimax estimation.

**SECTION – B**

** **

Answer any **FIVE** questions (5 ´ 8 = 40 marks)

- Find the Jackknified estimator of m
^{2}in the case of f(x) = , x ≥ m; m Î

- State and establish Basu’s theorem.

- Let X
_{1}, X_{2}, …, X_{n}be a random sample from N (m, s^{2}), m Î R, s > 0. Find UMRUE of

(m, m/s) with respect to any loss function which is convex in its second argument.

- Let X
_{1}, X_{2},…, X_{n}be iid U (q – , q + ), q Î Find minimal sufficient statistic and examine whether it is boundedly complete.

- Given a random sample of size n from N (m, s
^{2}), mÎR, s > 0, find Cramer – Rao lower bound for estimating ( m/s^{2}). Compare it with the variance of UMVUE.

- State and establish the invariance property of CAN estimator.

- Given a random sample from a location family with the location parameter x, show that is MREE of with respect to any invariant loss function, where d
_{O}is an LEE, = and^{*}minimizes E_{o}{ P (½with respect to .

- Let X ~ N (q, 1), qÎ Find Bayes estimator of q with respect to squared error loss if the prior of q is N (0, 1).

**SECTION – C**

** **

Answer any **TWO** questions (2 ´ 20 = 40 marks)

- a) Give an example for each of the following:
- i) U
_{g}is empty ii) U_{g}is singleton.

- b) Let X be DU {1,2,…, N}, N = 1,2,3,4,… . Find Q
_{A}– optimal estimator of (N, N^{2}).

(12+8)

- a) Show that a vector unbiased estimator is D – optimal if and only if each of its

components is a UMVUE.

- b) State and establish Lehmann – Scheffe theorem. (12+8)

- a) Let X
_{1}, X_{2}, …, X_{n}be iid N (0, ), Find MREE of^{r}with respect to

standardized squared error loss.

- b) Let (X
_{i}, Y_{i}), i = 1,2, …, n be a random sample from ACBVE distribution with pdf.

f(x,y) = {(2a+b) (a + b) / 2} exp {-a(x+y) – b max. (x,y)}, x, y > 0.

Find i) MLE of ( a, b) and (ii) examine whether the MLE is consistent. (8+8+4)

- Write short notes on:-
- Jackknifing method.
- Fisher information
- Location – scale family (10+5+5)