LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5504 – ESTIMATION THEORY

 

 

 

Date : 01/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL the questions:                                                                                                      (10 x 2 = 20)

1. Define Unbiasedness.
2. If T is an unbiased estimator of θ, show that T² is a biased estimator for θ².
3. Define Efficiency.
4. Let X₁, X₂, …, X_n be a random sample from a population with pdf

f( x, θ) = θ x ^{θ – 1}, 0 < x < 1, θ > 0.

Show that  is sufficient for θ.

5. Define BLUE.
6. What is meant by prior and posterior distribution?
7. Define sufficiency.
8. Write down the normal equation associated with a simple regression model.
9. Define Completeness.
10. Define MVB.

PART- B

Answer any FIVE questions:                                                                                                    (5 x 8 = 40)

11. State and prove the sufficient condition for an estimator to be consistent.
12. Let X₁, X₂, …, X_n be a random sample from a Bernoulli distribution:

 

Show that  is a complete sufficient statistics for θ.

13. Mention the properties of MLE.
14. A random sample X₁, X₂, X₃, X₄, X₅ of size 5 is drawn from a normal population with unknown mean µ. Consider the following estimators to estimate µ:

(i)    (ii)   (iii)  where λ is

such that t₃ is an unbiased estimator of m.

1. Find λ.
2. Are t₁ and t₂ unbiased?
3. Which of the three is the best estimator?

 

15. State and prove Cramer – Rao Inequality.
16. Samples of sizes n₁ and n₂ are drawn from two populations with mean T₁ and T₂ and common variance σ². Find the BLUE of l₁T₁ + l₂T₂.
17. Prove that if T₁ and T₂ are UMVUE of g(θ) then T₁ = T₂ almost surely.
18. Obtain the UMVUE of the parameter l for the poisson distribution based on a random sample of size n.

PART – C

Answer any TWO questions:                                                                                                   (2 x 20 = 40)

19. a) State and prove Rao – Blackwell theorem.
20. b) Show that if the MLE exists uniquely then it is a function of the sufficient statistic.
21. a) State and prove the necessary and sufficient for a parametric function to be linearly

estimable.

1. b) Prove that the MLE of α of a population having density function: 0 < x < α

for a sample of size one  is 2x, x being the sample value. Show also that the estimate is

biased.

21. a) State and prove factorization theorem.
22. b) Let (X₁, X₂, …, X_n) be a random sample from N(µ, σ²) . Obtain the Cramer – Rao

lower bound for the unbiased estimator of m.

22. a) Explain the method of moments.
23. b) Let X₁, X₂, …, X_n be a random sample from Bernoulli distribution b(1, θ). Obtain the

Bayes estimator for θ by taking a suitable prior.

 

 

 

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