LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 10

FIFTH SEMESTER – November 2008

ST 5500 – ESTIMATION THEORY

 

 

 

Date : 03-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer ALL questions.                                                                              10 X 2 = 20

1. Define Consistent estimator. Give an example.
2. Give two examples for unbiased estimator.
3. Define UMVUE.
4. Describe the concept of Bounded completeness.
5. Describe the Method of Minimum Chi-square estimation.
6. State an MLE ofλ based on a random sample of size n form a Poisson Distribution

with parameter λ.

1. Describe the concept of Baye’s estimation.
2. Define Loss function.
3. Describe the Method of Least Squares.
4. Define BLUE.

SECTION – B

Answer Any FIVE questions.                                                                                       5 X 8 = 40

11. Derive an unbiased estimator of , based on a random sample of size n form B (1,).
12. Let { Tn = 1, 2,3, ….. } be a sequence of estimators such that

and  .Then show that T_n is

consistent for .

13. If is a random sample from P (λ),, then show that

is a sufficient statistic for .

14. Show that the family of Binomial distributions {B (1,).0 < θ < 1} is complete.
15. Describe estimation of parameters by “Method of Maximum Likelihood”
16. Describe any two properties of MLE, with examples.
17. Explain prior and posterior distributions.
18. Derive the least square estimator of β₁ under the model Y = β₀ + β₁X+

 

SECTION – C

Answer any TWO questions.                                                                          2 X 20 = 40

19. a. State and prove Chapman-Robbin’s inequality. [12]

b Using Factorization theorem derive a sufficient statistic for μ based on a random

sample of size n from N (μ, 1), MϵR                                                                      [8]

 

20. a. State and prove a necessary and sufficient condition for an unbiased estimator to be a

UMVUE.                                                                                                                      [15]

1. If T₁ and T₂ are UMVUES of y₁(q) and  y₂(q) respectively, then show that T₁+T₂ is the UMVUE of  y₁(q) and  y₂(q).                                                                                                                [5]

 

 

21 a.  Explain the concept of estimation by the method of modified minimum chi-square.  [8]

1. Let be a random sample from a distribution with density function

f (x, θ) =                                                            [12]

Find the maximum likelihood estimator of  and examine whether it is consistent.

 

22. Explain: i) Risk function.              ii) Method of Moments

 

iii) Completeness     iv). Gauss –Markov model                           [ 4 x 5 ]

 

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