05.11.2003
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
THIRD SEMESTER – NOVEMBER 2003
ST-3801/S916 - MULTIVARIATE ANALYSIS
Max:100 marks
SECTION-A
Answer ALL the questions.
(10x2=20 marks)
 0   1 0.8  
  . Obtain the conditional distribution of X1 given X2 = x2.
1. Let X ~ N  , 
0
0
.
8
1





2. Write the characteristic function of bivariate normal distribution.
3. Explain how the collinearity problem can be solved in the multiple regression
Y = X + .
4. If X and Y are two independent standard normal variables, obtain the distribution of two
times of the mean of these two variables.
5. Let X be trinormal with
 2 0 3


   0 5 0  compute 13.2 .
 3 0 5


6. Define Fisher's Z - transformation.
7. Explain classification problem into two classes.
8. Write down any four similarity measures used in cluster analysis.
9. Distinguish between principal component and factor analysis.
10. What is meant by residual plot?
6
 
 =   1 and
3
 
SECTION-B
Answer any FIVE questions.
(5x8=40 marks)
11. Define multiple correlation coefficient between X1 and X2, …., Xp. Show that the
multiple correlation coefficient between X1 and X2, …., Xp has the expression
12  221  21
 11
.
12. Let Y ~ Np ( 0, ) . Show that Y1  1 Y has  2 - distribution.
0
13. Test at level 0.05 whether  =   in a bivariate normal population with 11 = 22 = 5
0
 7 
and 12 = -2 , by using the sample mean vector X    based on a sample of size 10.
  3
14. How will you test the equality of covariance matrices of two multivariate normal
distributions on the basis of independent samples drawn from two populations?
15. Derive the characteristic function of Wishart distribution.
1
16. In Principal component analysis derive the first principal component.
17. Obtain the rule to assign an observation of unknown origin to one of two p-variate normal
populations having the same dispersion matrix.
18. Outline single linkage and complete linkage clustering procedures with an example.
SECTION-C
Answer any TWO questions.
(2x20=40 marks)
19. a) Derive the MLE of  when the sample is from Np (, ).
b) Define Hottelling's T2 - statistic.
c) Using the likelihood ratio test procedure, show that the rejection region for testing  =
o against   o is given by
T2 = n( X   o )1 S-1 ( X   o )  T 02 .
(10+3+7)
20. a) Prove that under some assumptions (to be stated), Variance- covariance matrix can be
written as  = LL1 +  in the factor analysis model. Also discuss the effect of an
orthogonal transformation.
b) Let X1,X2,…., Xp have covariance matrix  with eigen value vector pairs (1, e1),…,
(p, ep), 1  2  ….. ≥ p  0, then prove that
11 + 22 + …..+ pp =
p
V (Y ) ,
i 1
i
Where , Yi represents the i - th principal component.
c) Explain the principal component (principal factor) method of estimating L in the factor
analysis model.
(10+5+5)
21. a) Explain the method of extracting canonical correlations and canonical variables. Also
explain how the theory of canonical correlation is helpful in the analysis of multivariate
data.
b) State an establish the additive property of Wishart distribution.
(10+10)
22. Write shot notes on:a) Roy's Union - Intersection principle
b) Step - Wise regression
c) Mohalanobis Squared distance.
(5+10+5)

2