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)
Attempt any five sub-parts
Let X be an absolutely continuous random variable with p.d.f and distribution function Show thaI
has distribution

letx1,x2,..be i.i.d random variables common p.d.f
<img images/1096-1b.jpg'>

Show that the weak law of large numbers holds for the sequence

(c)Let X random variable with p.m.f
p(x Pex 1
p(x P(x 11130
Obtain the distribution 0f Y X2

Let T1 and T2 be two unbiased estimators of e having the same variance Show that their correlation coefficient p cannot be smaller than 2e, where lS the efficiency of each estimator

If a sufficient statistic T elements for the family <img src='./qimages/1096-1e.jpg'> show that the Neyman-Pearson MP testlS a function ofT

Outline the Wilcoxon-Mann-Whitney test for the two-sample problem. Also, giVe the relevant large-sample test-statistic

If X lS any non-negative integer valued variate and lS any pOSitive number, show that
<img src='./qimages/1096-2a.jpg'>

(b)let xn be a sequence of distribution functions defined by

<img src='./qimages/1096-2b.jpg'>

Does Xn converge in distribution to If so, identify the distribution function of X Also, examine whether

For any k as
The characteristic function of a random variable X is pe it)n. Find the distribution of X

<img src='./qimages/1096-3a.jpg'>
src='./qimages/1096-3b.jpg'>
Find the MP test of Size a for testing Ho:beta against H1:beta=2 Also 0btain its power
Let the powder function of level test 0f
Ho 8 l@o against 8 be continuous is 0 Show that a UMP a-Similar testlS UMPU provided that its Size lS a for testing Ho against

4 Let Xl, x2, ...Xn be a random sample of Size n from a distribution 0 80< 1 Obtain the UMVU estimator of

(b)State and prove a result which shows that, compared to any other sequential test procedure, the SPRT has the least ASN

For each 00 E letA(00) be the acceptance reglOn of the level a UMPtest for Ho theta= 00 against all alternatives. Show that defined by
<img src='./qimages/1096-4c.jpg'>
lS, for varying Ix a family of UMA confidence sets at the level!
SECTION B
(Linear Inference, Multivariate Analysis, Sampling Theory and Design of Experiments)
Attempt any five sub-parts

if X (With P components) lS distributed according to Np(mew, then prove that DX lS distributed according to D:S where D lS a q x p matrix of rank P

Show that a linear function of observations belongs to the error space If and only If its coefficient vector lS orthogonal to the colunUls of the matrix X m the usual linear model
Y=X beta+e psla
A Simple random sample of Size n with mean y is drawn from a finite population, and a Simple random sub-sample of Size fillS drawn from it with mean Show that

Cov(y,

Bring out, giving an example, the relationship between a sampling scheme and a sampling deSign

Find a confounding arrangement of a 25 factorial experiment m 4 blocks such that no main effect or first order interaction lS confounded

WhatlS the purpose of MANoVA? Explain the one-way MANGVA model


For the model


y1= beta1 +beta2 +sigma1
y2 =beta1 beta3 +sigma2
y3= beta1+ beta2+sigma3
show that A3A lS estimable If and only If J., A3
Explain the purpose of discriminant analySiS WhatlS Fisher's discriminant function for two populations?

Give the allocation rule based on it

Outlining the purpose of factor analySiS, describe the orthogonal factor model along With its covariance structure

7 Describe a method of selecting a sample With probability proportional to Size and with replacement

Obtain the variance and its estimator for the Hansen-Hurwitz estimator



Show that the usual regressiOn estimator iS unbiased under a suitable model to be specified by you. Also, obtain the model variance of this estimator

Using Warner's randomized response technique; obtain the estimate of the population proportion relating to the sensitive characteristic

Give the analySiS of an LSD If the yield of one of the plots lS mlSsmg

Derive the analySis of variance table for 2-way c1asslfication With is observations per cell m the case of a random effects model

Construct a confounded factorial experiment is to 8 blocks of 8 treatments each by
confounding the effects ABEF, ABCD and ACE Also, Identify the other effects which are confounded