LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – April 2009
ST 3808 – MULTIVARIATE ANALYSIS
Date & Time: 16/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
PART – A
Answer all the questions. (10 X 2 = 20)
 Give an example in the bivariate situation that the marginal distributions are normal but the bivariate distribution is not.
 Let X, Y and Z have trivariate normal distribution with null mean vector and covariance matrix
2 3 4
3 2 1
4 1 1 ,
find the distribution of Y+X.
 Mention any two properties of multivariate normal distribution.
 Explain the use of partial and multiple correlation coefficients.
 Define Hotelling’s T^{2} – statistics. How is it related to Mahlanobis’ D^{2}?
 Outline the use of discriminant analysis.
 What are canonical correlation coefficients and canonical variables?
 Write down any four similarity measures used in cluster analysis.
 Write the c.f. of X where
X~N_{2 }{ , }.
10.Write a short note on data mining.
PART B
Answer any FIVE questions. (5 X 8 = 40)
 Obtain the maximum likelihood estimator S of pvariate normal distribution with
mean vector known.
 Let X_{1}, X_{2},…, X _{n }be independent N( 0 , 1 ) random variables. Show that X’ A X
is chisquare if A is idempotent, where X= ( X_{1},X_{2},…,X _{n })’.
 13. How will you test the equality of covariance matrices of two multivariate normal
distributions on the basis of independent samples drawn from two populations?.
 Let (X_{i}, Y_{i})’ , i = 1, 2, 3 be independently distributed each according to bivariate
normal with mean vector and covariance matrix as given below. Find the joint
distribution of six variables. Also find the joint distribution of and .
Mean vector: (m, t)’, covariance matrix:
 Outline single linkage and complete linkage clustering procedures with an
example.
 Giving suitable examples explain how factor scores are used in data analysis.
 Consider a multivariate normal distribution of X with
m = , S = .
Find i ) the conditional distribution of ( X_{1}, X_{3 }) / ( X_{2}, X_{4 })
 ii) s_{42 }
 a) Define i ) Common factor ii) Communality iii) Total variation
b)Explain classification problem into two classes and testing problem.
PART C
Answer any two questions. (2 X 20 = 40)
 a) Derive the distribution function of the generalized T^{2} – statistic.
 b) Test at level 0.05 ,whether µ = ( 0 0 )’ in a bivariate normal population with
σ_{11} = σ_{22}= 10 and σ_{12}= 4 , by using the sample mean vector= (7 3)^{‘} based
on a sample size 20. (15 + 5)
 a) What are principal components?. Outline the procedure to extract principal
components from a given covariance matrix.
 b) Define partial correlation between X_{i} and X_{j} .Also prove that
______ ______
r_{12.3}= ( r_{12}r_{13}r_{23})/ {Ö(1r^{2}_{23}) Ö(1r^{2}_{13})}. ( 12+8)
21.a) Consider the two data sets
X_{1}= and X_{2 }=
for which .
1) Calculate the linear discriminant function.
2) Classify the observation x_{0}‘= ( 2 7 ) as population π_{1 }or population π_{2} using
the decision rule with equal priors and equal costs.
 b) Explain how the collinearity problem can be solved in the multiple regression.
( 14+6)
22.a) Explain the method of extracting canonical correlations and their variables
from a dispersion matrix.
 b) Prove that under some assumptions (to be stated), variance and covariance can
be written as S = LL’ + y in the factor analysis model. Also discuss the effect
of an orthogonal transformation. (8 + 12)