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

FIFTH SEMESTER – APRIL 2008
ST 5500 – ESTIMATION THEORY
Date : 280408 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
PARTA
Answer ALL the questions: (10×2=20)
 Define ‘bias’ of an estimator.
 When do you say an estimator is consistent?
 Define a sufficient statistic.
 What do you mean by bounded completeness?
 Describe method of moments in estimation.
 State invariance property of maximum likelihood estimator.
 Define Loss function and give an example.
 Explain ‘prior distribution’ and ‘posterior distribution’.
 Explain least square estimation.
 Mention any two properties of least squares estimator.
PARTB
Answer any FIVE questions: (5×8=40)
 If Tn is asymptotically unbiased with variance approaching 0 as , then show that Tn is consistent.
 Show that is an unbiased estimate of , based on a random sample drawn from .
 Let be a random sample of size n from population. Examine if is complete.
 State and prove RAoBlackwell theorem.
 Estimate by the method of moments in the case of Pearson’s Type III distribution with p.d.f .
 State and establish Bhattacharya inequality.
 Describe the method of modified minimum Chi square.
 Write a note on Baye’s estimation.
PARTC
Answer any TWO questions: (10×2=20)
 a) and is a random sample of size 3 from a population with mean value and variance . are the estimators used to estimate mean value , where and .
 Are T_{1} and T_{2} unbiased estimators?
 Find the value of such that T_{3} a consistent estimator?
 With this value of is T_{3} a consistent estimator?
 Which is the best estimator?
 b) If are random observations on a Bernoulli variate X taking the value 1 with probability p and the value 0 with probability (1p), show that is a consistent estimator of p(1p).
 a) State and Prove cramerRao inequality.
 b) Given the probability density function
Show that the CramerRaolower bound of variance of an unbiased estimator of is 2/n, where n is the size of the random sample from this distribution. [12+8]
 a) State and prove Lehmann – Scheffe theorem
 b) Obtain MLE of in based on an independent sample of size n. Examine whether this estimate is sufficient for . [12+8]
 a) Show that a necessary and sufficient condition for the linear parametric function to be linearly estimable is that
ank (A) = rank
where and
 b) Describe Gauss – Markov model [12+8]
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