LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2004

ST 5500/STA 505/S 515 – ESTIMATION THEORY

03.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

Answer ALL the questions                                                                          (10 ´ 2 = 20 marks)

 

1. Define ‘bias’ of an estimator in estimating a parametric function.
2. Explain ‘consistent estimator’
3. Describe ‘efficiency’ of an estimator.
4. Define ‘Uniformly Minimum Variance Unbiased Estimator’.
5. Define Cramer – Rao lower bound (CRLB).
6. Explain bounded completeness.
7. Define complete sufficient statistic.
8. Let X₁, X₂ denote a random sample of size 2 from B(1, q), 0< q <1. Show that X₁ + 3X₂ is sufficient for q.
9. State Chapman – Robbins inequality.
10. Give an example where MLE is not unique.

 

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 s².
12. State and derive Cramer – Rao inequality.
13. Let T₁ and T₂ be two unbiased estimators of a parametric function with finite variances. Obtain the best unbiased linear combination of T₁ and T₂.
14. State and establish Rao – Blackwell theorem.
15. Give an example of an UMVUE which does not take values in the range of the parametric function.
16. State and prove Bhattacharya inequality.
17. State and prove invariance property of MLE.
18. Describe the method of moments and illustrate with an example.

 

SECTION – C

Answer any TWO questions                                                                       (2 ´ 20 = 40 marks)

 

19. a) If T_n is consistent for y (q) and g is continuous, show that g (T_n) is consistent for

g (y (q)).

1. b) Show that UMVUE is essentially unique. (10+10)

 

20. a) Give an example to show that bounded completeness does not imply completeness.
21. State and establish factorization theorem in the discrete case.      (10+10)

 

21. a) Explain the method of maximum likelihood.
22. b) Let X₁, X₂, …, X_n denote a random sample of size n from N (m, s²). Obtain MLE of

q = (m, s²).                                                                                                              (5+15)

 

22. a) Describe the method of minimum chi-square and method of modified minimum chi-

square.

1. b) Obtain the estimate of p based on a random sample of size n from B (1, p),0 < p <1 by the method of i) Minimum chi-square and ii) Modified Minimum Chi-Square. (10+10)

 

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