# Loyola College B.Sc. Statistics April 2007 Estimation Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 AC 16

DEGREE EXAMINATION –STATISTICS

FIFTH SEMESTER – APRIL 2007

ST 5500ESTIMATION THEORY

Date & Time: 27/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

# Answer all the questions.                                                                  10 X 2 = 20

1. State the problem of point estimation.
2. Define asymptotically unbiased estimator.
3. Define a consistent estimator and give an example.
4. Explain minimum variance bound estimator.
5. What is Fisher information?
6. Write a note on bounded completeness.
7. Examine if { N (0, σ2), σ2 > 0 } is complete.
8. Let X1, X2 denote a random sample of size 2 from P(q),q > 0. Show that X1 + 2X2 is not sufficient for q.
9. State Chapman – Robbins inequality.
10. Explain linear estimation.

# Answer any five  questions.                                                              5 X 8 = 40

1. Let X1, X2, … ,Xn denote a random sample of size n from B(1,q), 0<q<1. Show that  is an unbiased estimator of q2, where T = .
2. If Tn  is consistent for Y(q) and g is continuous, show that g(Tn ) is consistent for g{Y(q)}.
3. State and establish Cramer – Rao inequality.
4. Show that the family of binomial distributions { B (n, p), 0 < p < 1, n fixed } is complete.
5. State and establish Rao – Blackwell theorem.
6. Let X1, X2, … , Xn denote a random sample of size n from U (0, q), q > 0.

Obtain the UMVUE of q.

1. Give an example for each of the following
1. MLE, which is not unbiased.
2. MLE, which is not sufficient.
1. Describe the method of minimum chi-square and the method of modified minimum

chi-square.

##### Part C
###### Answer any two questions.                                                                     2 X 20 = 40

1. a). Show that the sample variance is a biased estimator of the population variance.

Suggest an unbiased estimator of s2.

b). If Tn is asymptotically unbiased with variance approaching zero as n

approaches infinity then show that Tn is consistent.                              (10 + 10)

1. a). Let X1, X2, … , Xn denote a random sample of size n from U (q-1, q +1).

Show that the mid – range U = is an unbiased estimator of q.

b). Obtain the estimator of p based on a random sample of size n from

B(1, p), 0 < p < 1by the method of

i). Minimum chi-square

ii). Modified minimum chi-square.                                                    (12 + 8)

1. a). Give an example to show that bounded completeness does not imply completeness.

b). Stat and prove invariance property of MLE.                                           (10 +10)

1. a). State and establish Bhattacharyya inequality.

b). Write short notes on Bayes estimation.                                                    (12 + 8)

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