LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 42

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2008

    ST 3808 / 3801 – MULTIVARIATE ANALYSIS

 

 

 

Date : 26/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer all the questions.                                                                         (10 X 2 = 20)

 

1. If X and Y are two independent standard normal variables then obtain the distribution of two times of the mean of these two variables.

• If X = ( X₁, X₂ )’~N₂ then write the c.f. of the marginal distribution

of X₂.

3. Mention any two properties of multivariate normal distribution.

4 . What is meant by residual plot?.

5. Explain the use of partial and multiple correlation coefficients.
6. Define Hotelling’s T² – statistics.
7. Define Fisher’s Z-transformation
8. Write a short on discriminant analysis.
9. Explain how canonical correlation is used in multivariate data analysis
10. Explain classification problem into two classes.

 

 

PART B

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

 

11.Find the multiple correlation coefficient between X₁ and X ₂ , X₃, …     , X _p.

Prove that the conditional variance of X₁ given the rest of the variables can not be

greater than unconditional variance of X₁.

12. Derive the c.f. of multivariate normal distribution.
13. Let Y ~ N_p ( 0 , S ). Show that Y’S ^-1 Y   has     distribution.
14. Obtain a linear function to allocate an object of unknown origin to one of the two

normal populations.

15. Giving a suitable example describe how objects are grouped by complete linkage

method.

16. Let X ~ N_p (m, S). If X⁽¹⁾ and X⁽²⁾ are two subvectors of X, obtain the conditional

distribution of X⁽¹⁾ given X⁽²⁾.

17. Prove that the extraction of principal components from a dispersion matrix is the

study of characteristic roots and vectors of the same matrix  .

18. Explain step-wise regression.

 

 

 

 

 

PART C

Answer any two questions.                                                                 (2 X 20 = 40)

 

 

               19 .  Derive the MLE of  m and S when the sample is from N ( m ,S ).

 

20. a) Derive the procedure to test the equality of mean vectors of two p-variate

normal populations when the dispersion matrices are equal.

 

1. 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 10.

(15 + 5)

 

21. a) Define i) Common factor ii) Communality  iii) Total variation
22. b) Explain the principal component ( principal factor )method of estimating L

in the factor analysis method.

1. c) Discuss the effect of an orthogonal transformation in factor analysis method.

( 6 + 6 + 8 )

22. What are canonical correlations and canonical variables? .Describe the extraction

of canonical correlations and their variables from the dispersion matrix. Also

show that there will be p canonical correlations if the dispersion matrix is of size p.

( 2+10+8 )

 

 

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