Time Allowed: 3 Hours Maximum Marks: 300
Caniliilates shouldaJtempt Questions 1 and 5 which arecompulsory, andanythreeoftheremaining
questions selecting oJ least one question/rom e(f£h Section.
Assume suitahle data ifconsiderednecessary andinilicnte the same clearly.
Notations andsymbols used are as usual

SECTION A
(Probability and Statistical Inference) Answer any five parts 0 f the following 12x5=60

In every box of a certain cereal is a picture of a football player A full set is represented by m players. Find the probability that n boxes of cereals must be purchased to obtain a full set of pictures

A random variable X has the following distribution

x p(x)
0 0
1 k
2 2k
3 2k
4 3k
5 k2
6 2k2
7 7k2+k2

Find k and determine the distribution function 0f X
Explain, with one illustration of each, convergence in distribution and convergence in probability for a sequence of random variables. State the relation between these two
Explain the difference between classical estimation and sequential estimation. Describe SPRT

Given a random sample of observation from the rectangular distribution with p.d.f
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Briefly describe the likelihood ratio test for equality of variances of k univariate normal populations

2
For the geometric distribution

p(X pqk, k 3.....

find the p.g.f Hence find the mean and variance
Examine if WLLN and CLT hold for the sequence of mutually independent random variables where
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Let be a two-dimensional non-negative continuous random variable having the joint density
src='./qimages/1107-2c.jpg'>

3.
Define completeness of a statistic Show that if a sufficient statistic is complete, then it is minimal sufficient

Show that a necessary and sufficient condition for a distribution to admit an unbiased estimator with variance attaining the Cramer-Rao lower bound is that it belongs to the one­parameter exponential family

Let X have the p.m.f given by

P(X=X)=(1.theeta)2 theetax=O, 2,...
and theeta E(belongs to) Obtain the m.v.u.e. of(1.theetae)2

4
Define a monotone likelihood ratio family of distributions Prove that the hypergeometric distribution is a member of the MLR family

Let X(i be a random sample from the rectangular theeta) population Find a UMP test of the hypothesis Ho:theeta theeta0(given) against all alternatives

Define a family of UMA confidence sets of an unknown parameter (theeta) at confidence level 1-(alfa). State a result giVing a one-one correspondence between the family of UMA confidence sets and the fanuly of acceptance regions of relevant UMP tests at level (alfa). 20




SEcnON B

(Linear Inference, Multivariate Analysis Sampling Theory and Design Experiments)
Answer any five parts 0f the followmg
12x5=60

What are outliers in the context of regression analysis? How can you use Cook's distance to detect outliers?

Explam briefly the method of factor analysis and its uses

With reference to a completely balanced two-way random effects model, find unbiased estimators of the different variance components

Let Y1, Y2,..... be i.i.d. random vectors, each having distribution
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assuming E to be positive definite.

When do you use a split plot design? Give the blank ANOVA table for a split-plot experiment.

Describe Horvitz-Thompson estimator of a finite population total and give an estimate of its standard error

6
Let ynx1 be a random vector distributed as Derive necessary and sufficient conditions for the quadratic form Y'AY to be distributed as x2

Briefly explain the development and use of linear discriminant functions in the context of two multivariate populations.

In a Gauss-Markoff set-up, define an estimable parametric function Show that If then every parametric function is estimable where we consider the set-up Ynx1,Xnxp(beta)px1,(sigma)2ln)

7
Consider PPS sampling without replacement, based on positive size measures. Define the Des Raj estimator of the population total and prove its unbiasedness.

Assume that the population size is an integral multiple of the sample size. Compare the efficiencies of the sample mean based on stratified random sample with one unit per stratum and systematic sample when there is a linear trend in the population.

Define the regression estimator of a finite population mean and work out its approximate mean square error, assuming that the second sample is a sub-sample of the first sample

8
Explain the concept of confounding in a factorial experiment.
In a Latin Square design, analysis of the design two observations from the same row are missing. Derive the analysis of the design.

When is a BIB design said to be resolvable? Show that for a resolvable BIBD. v r (in usual notations)