LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3801/S916 – MULTIVARIATE ANALYSIS

05.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

1. Let X ~ N . Obtain the conditional distribution of X₁ given X₂ = x₂.
2. Write the characteristic function of bivariate normal distribution.
3. Explain how the collinearity problem can be solved in the multiple regression

Y = Xb + Î.

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

m =  and   compute  .

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?

 

SECTION-B

Answer any FIVE  questions.                                                                          (5×8=40 marks)

 

11. Define multiple correlation coefficient between X₁ and X₂, …., X_p. Show that the multiple correlation coefficient between X₁ and X₂, …., X_p has the expression

.

12. Let Y ~ N_p ( 0, å) . Show that Y¹ has – distribution.
13. Test at level 0.05 whether m = in a bivariate normal population with s₁₁ = s₂₂ = 5 and s₁₂ = -2 ,   by using the sample mean vector  based on a sample of size 10.
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.
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.                                                                          (2×20=40 marks)

 

19. a) Derive the MLE of å when the sample is from N_p (m, å).
20. b) Define Hottelling’s T² – statistic.
21. c) Using the likelihood ratio test procedure, show that the rejection region for testing m =

m_o against m ¹ m_o is given by

T² = n( S^-1 (³  T.                                                             (10+3+7)

20. a) Prove that under some assumptions (to be stated), Variance- covariance matrix can be

written as å = LL¹ +  in the factor analysis model.  Also discuss the effect of an

orthogonal transformation.

1. b) Let X₁,X₂,…., X_p have covariance matrix å with eigen value vector pairs (l₁, e₁),…,

(l_p, e_p),  l₁ l₂ ….. ≥ l_p 0,   then prove that

s₁₁ + s₂₂ + …..+ s_pp = ,

Where ,  Y_i  represents the i – th principal component.

1. 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.

1. b) State an establish the additive property of Wishart distribution. (10+10)
2. Write shot notes on:-
3. Roy’s Union – Intersection principle
4. Step – Wise regression
5. Mohalanobis Squared distance. (5+10+5)

 

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