LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 41

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.

2. 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.

3. Mention any two properties of multivariate normal distribution.
4. Explain the use of partial and multiple correlation coefficients.
5. Define Hotelling’s T² – statistics. How is it related to Mahlanobis’ D²?
6. Outline the use of discriminant analysis.
7. What are canonical correlation coefficients and canonical variables?
8. Write down any four similarity measures used in cluster analysis.
9. Write the c.f. of X where

X~N₂ { ,   }.

10.Write  a short  note on data mining.

 

PART B

Answer any FIVE questions.                                                                    (5 X 8 = 40)

 

11. Obtain the maximum likelihood estimator S of p-variate normal distribution with

mean vector known.

12. Let X₁, X₂,…, X _n be independent N( 0 , 1 ) random variables. Show that X’ A X

is chi-square if A is idempotent, where  X= ( X₁,X₂,…,X _n )’.    

13. 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?.

14. 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:

15. Outline single linkage and complete linkage clustering procedures with an

example.

16. Giving suitable examples explain how factor scores are used in data analysis.
17. Consider a multivariate normal distribution of X with

m =      ,       S =                    .

 

Find i )  the conditional distribution of ( X₁, X₃ ) / ( X₂, X₄ )

33. ii) s₄₂  
34. 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)

 

19. a) Derive the distribution function of the generalized T² – statistic.
20. b) Test at level 0.05 ,whether µ = ( 0 0 )’ in a bivariate normal population with

σ₁₁ = σ₂₂= 10 and  σ₁₂= -4 , by using the sample mean vector= (7  -3)^‘   based

on a sample size 20.                                                                                 (15 + 5)

20. a) What are principal components?. Outline the procedure to extract principal

components   from a given covariance matrix.

1. b) Define partial correlation between X_i and X_j .Also prove that

______   ______

r_12.3=  ( r₁₂-r₁₃r₂₃)/ {Ö(1-r²₂₃) Ö(1-r²₁₃)}.                           (  12+8)

21.a) Consider the two data sets

X₁=     and   X₂ =

for which         .

1) Calculate the linear discriminant function.

2) Classify the observation x₀‘= ( 2  7 ) as population π₁ or  population π₂ using

the decision rule with equal priors and equal costs.

1. 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.

1. 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)

 

 

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