LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 24

FIFTH SEMESTER – April 2009

ST 5500 – ESTIMATION THEORY

 

 

 

Date & Time: 16/04/2009 / 9:00 – 12:00      Dept. No.                                                Max. : 100 Marks

 

 

PART A                                Answer all the questions                             [10×2=20]

 

1. Define the efficiency of an unbiased estimator. Give an example of a most

efficient estimator.

2. State the invariance property of consistent estimators..
3. State Factorization Theorem on sufficient statistic.
4. Show that exponential distribution with parameter λ is complete.
5. List out any two small sample properties of ML estimator.
6. Find ML estimator of θ in random sampling of size n from a population whose

pdf is  f(x, θ)   =          e ^{– (x – θ)},  for x > θ

=   0                      otherwise.

7. Define Loss Function. Is it a random variable? Justify.
8. When do we say that a statistic is Bayesian sufficient? Give an example.

9          Write down the normal equations of a simple linear regression model.

10. Mention the uses of Gauss-Markoff Model.

PART B                                            Answer any FIVE questions                       [5×8=40]

11. Show that the sample variance is a consistence estimator of the population

variance.

12. If X follows Binomial distribution with parameters n and p. Examine the

asymptotic unbiasedness of   T =    .

13. States and Prove Rao Blackwell Theorem.
14. Let (X₁, X₂, X₃, …X_n) is a random sample from Poisson population with parameter

λ. Use Lehman Scheffe Theorem to obtain a UMVUE of λ

15. Obtain the moment estimators of the parameters of a two-parameter gamma

distribution.

 

16. Illustrate the invariance property of ML estimator through an example
17. Explain the method of modified Chi-square estimation.
18. State and prove Gauss Markoff model on BLUE

 

PART C                                            Answer any TWO questions                       [2×20=40]

19. (a) State and prove Cramer Rao inequality in one parameter regular case. When

does the equality hold good?

(b) Establish a sufficient condition for a biased estimator to become a consistent

estimator.

20. (a) State and prove Lehman Scheffe theorem on UMVUE

(b) Obtain a joint sufficient statistic of the parameters of the bi-variate normal

population.

21. (a) Derive the moment estimators of the parameters of two parameter uniform

distribution.

(b)            Derive the ML estimators of the parameters of normal distribution by solving

simultaneous equations.

22. (a) Establish a necessary and sufficient condition for a linear parametric function to

be estimable.

(b) Let (X₁, X₂, X₃, …X_n) is a random sample of size n from Bernoulli population.

Obtain the Bayesian estimator of the parameter by taking a suitable prior

distribution..

 

 

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