LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2012

ST 5504/ST 5500 – ESTIMATION THEORY

 

 

 

Date : 25-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART-A

Answer ALL questions:                                                                                           (10×2=20 marks)

• Define consistent estimator.
• State the characteristics of a good estimator.
• Define sufficient statistic.
• State factorization theorem.
• Mention any two properties of MLE.
• Explain the concept of method of moment estimation.
• Define prior and posterior probability distribution.
• State Gauss-Markoff linear model.
• Define BLUE.
• Write down the normal equation of a simple linear regression model.

 

PART-B

 Answer any FIVE questions:                                                                                         (5×8=40 marks)

       11)  Show that the sample variance is consistent estimator for the population variance     

               of a normal distribution.

12)  Define Completeness with an example. Also, give an example of a family which

 is not complete.

 13) Let x₁, x₂, x₃… x_n, be a random sample from N(µ,σ²) population. Find sufficient 

       estimator for µ and σ² .

 14) Explain the method of minimum chi-square estimation.

 15) Find MLE for the parameter λ of a Poisson distribution on the basis of a sample  

        of size n and hence, obtain the MLE of P[ X ≤ 1].

 16) State and prove Factorization Theorem on sufficient statistics in one parameter

       discrete case.

 17) Obtain the method of moments estimator for Uniform distribution U(a, b).

(P.T.O)

 18) Obtain the Bayes estimator using a random sample of size ‘n’ when

                                   f( x ; q ) =  , x = 0, 1, 2, . . .

       and the p.d.f. of q is a two parameter gamma distribution.

 

PART-C

Answer any TWO questions:                                                                                   (2×20=40 marks)

19. (a) State and prove Chapman- Robins Inequality and also mention its importance.

      (b) Obtain the minimum variance bound estimator for µ in normal population

                  N(µ,σ²) where σ² is known.

20. (a) State and prove Rao-Blackwell Theorem.

     (b) Let x₁, x₂, x₃… x_n, be random sample from U(0,θ) population obtain            

         UMVUE for θ.

21. (a) Explain the concept of Maximum Likelihood Estimator

      (b) In random sampling from normal population N(µ,σ²) find the MLE for

          (i) µ when σ² is known                   (ii) σ² when µ is known.

22. (a) Explain the concept of Method of Least Squares.

      (b) State and prove the necessary and sufficient condition for a parametric function

            to be linearly estimable.

 

 

 

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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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