LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST-5500/STA 505/S 515 – ESTIMATION THEORY

03.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

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

 

1. State the problem of point estimation.
2. Define ‘bias’ of an estimator in estimating a parametric function.
3. Define a ‘Uniformly Minimum Variance Unbiased Estimator’ (UMVUE).
4. Explain Cramer-Rao lower bound.
5. Define completeness and bounded completeness.
6. Examine if is complete.
7. Let X₁, X₂ denote a random sample of size 2 from B(1, q), 0<q<1. Show that X₁+3X₂ is sufficient of q.
8. Give an example where MLE is not unique.
9. Define BLUE
10. State Gauss – Markoff theorem.

 

SECTION-B

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

 

11. Show that the sample variance is a biased estimator of the population variance. Suggest an UBE of .
12. If T_n is asymptotically unbiased with variance approaching zero as n , show that T_n is consistent.
13. Show that UMVUE is essentially unique.
14. Show that the family of Binomial distributions is complete.
15. State and establish Lehmann – Scheffe theorem.
16. State and prove Chapman – Robbin’s inequality.
17. Give an example where MLE is not consistent.
18. Describe the linear model in the Gauss – Marboff set-up.

 

SECTION-C

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

 

19. a) Let X₁, X₂,….., X_n (n > 1) be a random sample of size n from P (q), q > 0. Show that the class of unbiased estimator of q is uncountable.
20. b) Let X₁, X₂,….., X_n denote a random sample of size n from a distribution with pdf

 

 

 

f(x) ;q) =

0              ,   other wise.

Show that X₍₁₎ is a consistent estimator of q.                                                          (10+10)

 

20. a) Obtain CRLB for estimating q, in the case of

f  based on random sample of size n.

1. b) State and establish factorization theorem in the discrete case. (8+12)
2. a) Explain the method of maximum likelihood.
3. b) Let X₁, X₂, …., X_n denote a random sample of size n from N (. Obtain MLE of q = (.                                                                                           (5+15)
4. a) Let Y = Ab + e be the linear model where E (e) = 0. Show that a necessary and sufficient condition for the linear function  of the parameters to be linearly estimable is that rank (A) = rank .
5. b) Explain Bayesian estimation procedure with an example. (10+10)

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