LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc.

DEGREE EXAMINATION –STATISTICS
FIFTH SEMESTER – APRIL 2007
ST 5500 – ESTIMATION THEORY
Date & Time: 27/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
Part A
Answer all the questions. 10 X 2 = 20
 State the problem of point estimation.
 Define asymptotically unbiased estimator.
 Define a consistent estimator and give an example.
 Explain minimum variance bound estimator.
 What is Fisher information?
 Write a note on bounded completeness.
 Examine if { N (0, σ^{2}), σ^{2} > 0 } is complete.
 Let X_{1}, X_{2} denote a random sample of size 2 from P(q),q > 0. Show that X_{1} + 2X_{2} is not sufficient for q.
 State Chapman – Robbins inequality.
 Explain linear estimation.
Part B
Answer any five questions. 5 X 8 = 40
 Let X_{1}, X_{2}, … ,X_{n} denote a random sample of size n from B(1,q), 0<q<1. Show that is an unbiased estimator of q^{2}, where T = .
 If T_{n } is consistent for Y(q) and g is continuous, show that g(T_{n }) is consistent for g{Y(q)}.
 State and establish Cramer – Rao inequality.
 Show that the family of binomial distributions { B (n, p), 0 < p < 1, n fixed } is complete.
 State and establish Rao – Blackwell theorem.
 Let X_{1}, X_{2}, … , X_{n} denote a random sample of size n from U (0, q), q > 0.
Obtain the UMVUE of q.
 Give an example for each of the following
 MLE, which is not unbiased.
 MLE, which is not sufficient.
 Describe the method of minimum chisquare and the method of modified minimum
chisquare.
Part C
Answer any two questions. 2 X 20 = 40
 a). Show that the sample variance is a biased estimator of the population variance.
Suggest an unbiased estimator of s^{2}.
b). If T_{n} is asymptotically unbiased with variance approaching zero as n
approaches infinity then show that T_{n} is consistent. (10 + 10)
 a). Let X_{1}, X_{2}, … , X_{n} denote a random sample of size n from U (q1, 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 chisquare
ii). Modified minimum chisquare. (12 + 8)
 a). Give an example to show that bounded completeness does not imply completeness.
b). Stat and prove invariance property of MLE. (10 +10)
 a). State and establish Bhattacharyya inequality.
b). Write short notes on Bayes estimation. (12 + 8)
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