ST 2808 – ESTIMATION THEORY

Date : 17/04/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

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

Give an example of a parametric function for which unbiased estimator does not exist.
Define a loss function for simultaneous estimation problem and give an example.
If δ is a UMVUE, then show that 5δ is also a UMVUE.
Find the Fisher information in the Bernoulli distribution with the parameter θ.
Define completeness and bounded completeness.
Given a random sample of size 2 from N(0, σ²), σ>0, suggest two ancillary statistics.
Give two examples for location equivariant estimator.
Let X follow E( θ,1), θ = 0.1,0.2. Find the MLE of θ .
Define a consistent estimator and give an example.
Explain prior distribution and Conjugate family.

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

‌11. Let X follow DU{1,2,…N}, N = 2,3,4,… Find the class of unbiased estimators of N .

State and prove Cramer-Rao inequality for the multiparameter case.
Discuss the importance of Bhattacharyya inequality with a suitable example.
Let X₁,X₂,…,X_n be a random sample from N(θ, θ²), θ >0. Find a minimal sufficient statistic and

examine whether it is complete.

Using Basu’s theorem show that the sample mean and the sample variance are independent in the

case of N( θ, 1), θ ε R.

16.Given a random sample from E(0, τ), τ > 0, find MREE of τ and τ² with respect to standardized

squared error loss.

17.Give an example in which MREE of a location parameter exists with respect to squared error loss but

UMVUE does not exist.

Let X₁,X₂,…,X_n be a random sample from B(1, θ), 0<θ<1. If the prior distribution is U(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 Bhattacharya inequality

b) Let X follow DU{1,2,…,N}, N = 3,4,…Find the UMVUE of N using Calculus approach.

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

b) Given a random sample from N(μ,σ²), μ ε R, σ > 0, find UMRUE of (μ, μ/σ) with

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

21 a) Discuss the problem of equivariant estimation of the scale parameter.

b) Given a random sample of size n from U(ξ, ξ+1), ξ ε R,find the MREE of ξ with respect to

standardized squared error loss.

22 a) Give an example for an MLE which is consistent but not CAN.

b) Stating the regularity conditions, show that the likelihood equation estimator is CAN.