LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2011

ST 2811 / 2808 – ESTIMATION THEORY

 

 

 

Date : 2/4/2011                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

 

Answer all the questions                                                                                           (2×10=20)

1. Define Minimal Sufficient Statistic
2. Define Efficient Estimator
3. Define Ancillary Statistic
4. State the different approaches to identify UMVUE
5. Define Likelihood Equivalence
6. Define D –optimality
7. Define Location-Scale Family
8. Define Minimum Risk Equivariant Estimator(MREE)
9. Define CAN estimator
10. Define Maximum Likelihood Estimator

 

SECTION – B

 

Answer any five questions                                                                                        (5×8 = 40)

1. Obtain UMVUE of θ(1- θ) using a random sample of size n drawn from a Bernoullie population with parameter θ
2. State and Establish Rao-Blackwell theorem
3. State and Establish Neyman-Fisher Factorization theorem
4. i) Let L be squared error then MREE of θ is unique                                        (4)
5. ii) Let X₁,X₂,…,X_n be a random sample from N(θ,1), Show that (4)
6. Let δ be a LEE and L be invariant then show that    i)The Bias of δ is free from θ

and ii) Risk of δ is free from θ                                                                                    (4+4)

6. i) State and Establish Basu’s theorem (6+2)
7. ii) Define UMRUE
8. Determine MREE of θ in the following cases i) N(θ,1) , θ Î R ii)E(θ,1) , θ ÎR

 

 

 

8. Let X₁,X₂,…,X_n be a random sample from population having pdf

 

obtain MLE of P(X>2)

 

SECTION – C

Answer any two questions                                                                                        (2×20 = 40)

1. i) Establish: If UMVUE exists for a parametric function Ψ(θ), It has to be essentially unique (10)
2. ii) State and Establish Cramer-Rao Inequality for multi-parameter case and hence deduce the inequality for single parameter (10)
3. Establish: δ^*Î U_g is D-optimal if and only if each component of δ^* is UMVUE
4. i) Let X₁,X₂,…,X_n be a random sample from N(µ,σ²). Obtain Cramer-Rao lower bound for estimating (16)
5. i) µ ii) σ²                 iii) µ+σ                                 iv) σ/ µ
6. ii) Establish: Let T be a sufficient statistic such that T(x) = T(y) then           (4)
7. i) Establish: Let δ^* belong to the class of LEEs. Then δ^* is a MREE with respect to squared error if and

only if E(δ^*u)=0                                                                                           (10)

1. ii) Let X₁,X₂,…,X_n be a random sample drawn from a normal population with mean θ and variance σ²

Find the MLE of θ and σ² when both θ and σ² are unknown                                (10)

 

 

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