ST 5504                       / ST 5500         ESTIMATION THOERY

 

Section A

Answer all the questions                                                                            10×2=20  

 

 

1          Define an Unbiased Estimator.

2. Define consistency of an estimator. Mention any one of its properties.
3. What is the importance of factorization theorem?
4. Define a Sufficient Statistic.
5. Briefly explain the method of moment estimation.
6. Explain prior distributions and posterior distributions with reference to Bayesian Estimation.
7. List out any two properties of Maximum Likelihood estimators.
8. Define Squared Error Loss function.
9. Define Best Linear unbiased Estimation. Give an example.
10. Write down the normal equations of a simple linear regression model.

 

Section B

Answer any five questions                                                                    5×8=40 

 

11. Let X₁,X₂, ..X_n be a random sample taken from a normal population with unknown mean and unknown variance. Examine the unbiasedness and consistency of T(x) = (X_i – )²
12. State and prove a sufficient condition for an estimator to be consistent.

(P.T.O)

13 Let X₁,X₂, ..X_n be a random sample taken from a population whose probability density function is

f(x, ) =  exp{–  },  >0,  x>0

Use Factorization Theorem to obtain a sufficient statistic for .

 

14. Show that Poisson distribution is complete.

 

15. Explain the method of minimum chi-square estimation.

 

16. Obtain the Maximum Likelihood estimators of the parameters of a normal distribution.

 

17. X₁,X₂, X₃,and X₄ are four independent Poisson random variables with mean . Define T₁(x) =  {X₁ +3X₃}

                          T₂(x) ={X₁ +2X₂+3X₃}

                          T₃(x) ={X₁+X₂ +X₃+X₄}

Examine their unbiasedness and compute their variances. Which one is the best estimator among the three? Find the efficiency of T₁(x) and T₂(x) with respect to T₃(x).

 

18. Explain in detail Gauss-Markov model.

 

 

 

(P.T.O)

 

 

Section C

Answer any two questions                                                               2×20=40 

 

19        (a)       State and prove Chapman-Robbins inequality. Bring out its importance.

(b) Let X₁,X₂, ..X_n be a random sample taken from a normal population with unknown mean and unit variance. Obtain Cramer-Rao Lower bound for an unbiased estimator of the mean.                                                 (12 + 8)

 

20. (a) Show that UMVUE for a parametric function is unique.

(b) State and Prove Rao-Blackwell Theorem.                            (10 + 10)

 

21. (a) Obtain the moment estimators for Uniform distribution U(a, b).

(b) Show that Maximum Likelihood Estimator need not be unique with an example. Also, show that when MLE is unique, it is a function of the sufficient statistic.                                                                                       (10 + 10)

 

22. (a) Let X₁,X₂, ..X_n be a random sample from Bernoulli distribution b(1, q). Obtain the Bayes estimator for q by taking a suitable prior.

(b) State and prove a necessary and sufficient condition for a parametric function to be linearly estimable.                                                     (10 + 10)

 

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