LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
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FIFTH SEMESTER – APRIL 2008
ST 5500 – ESTIMATION THEORY
Date : 28-04-08 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
PART-A
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.
PART-B
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 RAo-Blackwell 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.
PART-C
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 T1 and T2 unbiased estimators?
- Find the value of such that T3 a consistent estimator?
- With this value of is T3 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 (1-p), show that is a consistent estimator of p(1-p).
- a) State and Prove cramer-Rao inequality.
- b) Given the probability density function
Show that the Cramer-Rao-lower 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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