LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2004

ST 5500/STA 505/S 515 – ESTIMATION THEORY

25.10.2004                                                                                                           Max:100 marks

9.00 – 12.00 Noon

 

SECTION – A

 

Answer ALL the 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 ‘Consistent estimator’.
4. Define ‘efficiency of an estimator.
5. Explain ‘Uniformly Minimum Variance Unbiased Estimator’.
6. What is Cramer – Rao lower bound?
7. Define ‘bounded completeness’.
8. Examine if {N (0, s²), s² > 0} is complete.
9. 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.
10. Explain BLUE.

SECTION – B

 

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

 

11. Show that the sample variance is a biased estimator of the population variance s². Suggest an UBE of s².
12. If T_n is a consistent estimator of j (q), show that there exists infinitely many consistent estimators of j (q).
13. State and derive Cramer – Rao inequality.
14. Show that UMVUE is essentially unique.
15. Give an example to show that bounded completeness does not imply completeness.
16. Show that the sample mean is a complete sufficient statistic in the case of P (q), q > 0.
17. State and establish Lehmann – Scheffe theorem.
18. State and prove ‘Invariance property’ of MLE.

 

SECTION – C

 

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

 

 

19. a) Let f (x;q) =

 

• ,  otherwise

 

Based on a random sample of size n, suggest an UBE of

1. q when s is known and
2. s when q is known          (5+5)

 

1. b) Obtain CRLB for estimating q, in the case of f(x; q) = x Î R, q Î R,

based on a random sample of size n.                                                                          (10)

20. a) State and establish factorization theorem in the discrete case.

 

1. b) Obtain a sufficient statistic for q = (m, s²) based on a random sample of size n from

N (m, s²), m Î R, s² > 0.                                                                                     (12 + 8)

 

21. a) Explain the method of maximum likelihood.

 

1. 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 the method of modified minimum

chi-square.

 

1. b) Describe the linear model in the Gauss – Markov set – up.

 

1. c) Let y = Ab + e be the linear model where E (e) = 0. Show that a necessary and

sufficient condition for the linear function  b of the parameters to be linearly

estimable is that rank (A) = rank .                                                           (10 + 6 + 4)

 

 

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