LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 32

SECOND SEMESTER – APRIL 2007

ST 2808/2806/2801 – ESTIMATION THEORY

 

 

 

Date & Time: 17/04/2007 / 1:00 – 4:00 Dept. No.                                              Max. : 100 Marks

 

 

SECTION – A

Answer all the questions                                                                                   (10 x 2 = 20)

 

1. Explain the problem of Point estimation.
2. Give two examples of loss function for simultaneous estimation.
3. If δ is a UMVUE, then show that δ + 2 is also a UMVUE.
4. Define Fisher information in the multi-parameter case.
5. Define minimal sufficient statistic.
6. Give an example of a family of distributions which is not complete.
7. Give two examples of scale equivariant estimator.
8. Let X follow B(1, θ), θ = 0.1,0.2. Find MLE of θ .
9. Given a random sample from DU{1,2,…, N}, N ε I₊, find a consistent estimator of N.
10. Explain Bayes estimation.

 

SECTION – B

‌‌‌Answer any  five questions                                                                                (5 x 8 = 40)

 

11. If δ₀ is an unbiased estimator of g, show that the class of unbiased estimators of g is

{ δ₀ + u‌‌ │‌‌u ε U₀}.

12. Given a random sample from N(μ, σ²), μ ε R , σ > 0, find Cramer-Rao lower bound for

estimating  σ/ μ.

13. State and establish Bhattacharya inequality.
14. Let X₁,X₂,…,X_n be a random sample from U(θ – 1, θ + 1), θ ε R. Show that

(X₍₁₎, X_(n)) is minimal sufficient but not complete.

15. State and establish Basu’s theorem.
16. Given a random sample from E(ξ,1), ξ ε R, find MREE of ξ with respect to i) squared error loss and
17. ii) absolute error loss.
18. State and prove the theorem providing MREE of a scale parameter.
19. Given a random sample from U(0, θ), θ ε R, show that MLE is not CAN. Suggest a CAN estimator.

 

SECTION – C

Answer any two questions                                                                               (2 x 20 = 40)

 

19 a) State and establish Cramer-Rao inequality for the multiparameter case.

1. b) Let X follow DU{1,2,…,N}, N = 3,4,… Find the UMVUE of g(N). Hence find the UMVUE of N.

20 a) Show that an estimator is Q_A – optimal if and only if it is D – optimal.

1. b) Given a random sample from E(ξ, τ), ξ ε R, τ > 0, find UMRUE of (ξ , ξ + τ) with

respect to any loss function, convex in the second argument.

 

21 a) Discuss the problem of equivariant estimation of the percentiles of a location – scale model.

1. b) Given a random sample of size n from N(μ, τ²), μ ε R, τ > 0, find MREE of (μ+3τ) with respect to

standardized squared error loss.

22 a) State and establish invariance property of CAN estimator.

1. b) Let (X_i,Y_i) , i= 1,2,…,n be a random sample from a bivariate distribution with pdf

 

 

Find MLE of

 

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