LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 36

SECOND SEMESTER – April 2009

ST 2811 / 2808 – ESTIMATION THEORY

 

 

 

Date & Time: 20/04/2009 / 1:00 – 4:00       Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A                                  Answer all the questions                                   (10 x 2 = 20)

 

01.Give an example of a parametric function for which unbiased estimator is unique.

02.State any two loss functions for simultaneous estimation problem.

03.Show that UMVUE of a parametric function is unique.

04.Define Fisher information for the multiparameter situation.

05.Define bounded completeness and give an example.

06.Given a random sample of size 2 from E(0, σ), σ>0, suggest two ancillary statistics.

07.Define a scale equivariant estimator and give an example.

08.Let X follow N( θ,1), θ = 0, 0.1. Find the MLE of  θ .

09.If  δ is consistent for θ, show that there exists infinitely many consistent estimators of θ.

10.Describe Conjugate family and give an example.

 

SECTION – B     ‌‌                              Answer any  five questions                                (5 x 8 = 40)

 

‌11.Let X follow DU{1,2,…N}, N = 2,3. Find the class of unbiased estimators of  zero.

Hence find the class of unbiased estimators of N and N².

12.State Cramer-Rao inequality for the multiparameter case. Hence find the Cramer- Rao

lower bound for estimating  σ/μ  based on a random sample from N(μ,σ²), μ ε R, σ > 0.

13. Discuss the importance of Fisher information in finding a sufficient statistic.
14. Let X₁,X₂,…,X_n be a random sample from U(0, θ), θ >0. Find a minimal sufficient

statistic and examine whether it is complete.

15.State and establish Basu’s theorem.

16.Given a random sample from N(0, τ²), τ > 0, find MREE of τ ²  with respect

to  standardized squared error loss. Is it unbiased ?

17.Find  MREE of the location parameter with respect to absolute error loss based on a

random sample from E(ξ, 1), ξ ε R.

18. Let X₁,X₂,…,X_n be a random sample from P(θ), θ > 0. If the prior distribution is E(0,1),

find the Bayes estimator of θ with respect to the squared error loss.

 

SECTION – C                              Answer any two questions                                    (2 x 20 = 40)

 

19 a) State and establish any two properties of Fisher information.

1. b) Let X have the pdf

P( X = x) = (1- θ)² θ^x , x = 0,1,…  ; 0< θ < 1

=  θ,  x = -1.

Using Calculus approach examine whether UMVUE of the following parametric functions

exist:  i) θ     ii) (1 – θ)².

20 a) Show that an estimator δ is D – optimal if and only if each component of δ is a UMVUE.

1. b) Given a random sample from E(μ,σ), μ ε R, σ > 0, find UMRUE of (μ, μ + σ) with

respect to any loss function, convex in the second argument.

21 a) Show that the bias and the risk associated with a location equivariant estimator do not depend

on the parameter.

1. b) Show that a location equivariant estimator δ is an MREE if and only if E₀(δu) = 0 for each

invariant function u.
22 a) Given a random sample from N(μ,σ²), μ ε R, σ > 0, find the maximum likelihood

estimator of (μ,σ²). Examine whether it is consistent.

1. b) Stating the regularity conditions, show that the likelihood equation admits a solution which

is consistent.

 

 

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