LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2012

ST 5504/ST 5500 – ESTIMATION THEORY

 

 

 

Date : 25-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART-A

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

• Define consistent estimator.
• State the characteristics of a good estimator.
• Define sufficient statistic.
• State factorization theorem.
• Mention any two properties of MLE.
• Explain the concept of method of moment estimation.
• Define prior and posterior probability distribution.
• State Gauss-Markoff linear model.
• Define BLUE.
• Write down the normal equation of a simple linear regression model.

 

PART-B

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

       11)  Show that the sample variance is consistent estimator for the population variance     

               of a normal distribution.

12)  Define Completeness with an example. Also, give an example of a family which

 is not complete.

 13) Let x₁, x₂, x₃… x_n, be a random sample from N(µ,σ²) population. Find sufficient 

       estimator for µ and σ² .

 14) Explain the method of minimum chi-square estimation.

 15) Find MLE for the parameter λ of a Poisson distribution on the basis of a sample  

        of size n and hence, obtain the MLE of P[ X ≤ 1].

 16) State and prove Factorization Theorem on sufficient statistics in one parameter

       discrete case.

 17) Obtain the method of moments estimator for Uniform distribution U(a, b).

(P.T.O)

 18) Obtain the Bayes estimator using a random sample of size ‘n’ when

                                   f( x ; q ) =  , x = 0, 1, 2, . . .

       and the p.d.f. of q is a two parameter gamma distribution.

 

PART-C

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

19. (a) State and prove Chapman- Robins Inequality and also mention its importance.

      (b) Obtain the minimum variance bound estimator for µ in normal population

                  N(µ,σ²) where σ² is known.

20. (a) State and prove Rao-Blackwell Theorem.

     (b) Let x₁, x₂, x₃… x_n, be random sample from U(0,θ) population obtain            

         UMVUE for θ.

21. (a) Explain the concept of Maximum Likelihood Estimator

      (b) In random sampling from normal population N(µ,σ²) find the MLE for

          (i) µ when σ² is known                   (ii) σ² when µ is known.

22. (a) Explain the concept of Method of Least Squares.

      (b) State and prove the necessary and sufficient condition for a parametric function

            to be linearly estimable.

 

 

 

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