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)
- Let X ~ N . Obtain the conditional distribution of X1 given X2 = x2.
- Write the characteristic function of bivariate normal distribution.
- Explain how the collinearity problem can be solved in the multiple regression
Y = Xb + Î.
- If X and Y are two independent standard normal variables, obtain the distribution of two times of the mean of these two variables.
- Let X be trinormal with
m = and compute .
- Define Fisher’s Z – transformation.
- Explain classification problem into two classes.
- Write down any four similarity measures used in cluster analysis.
- Distinguish between principal component and factor analysis.
- What is meant by residual plot?
SECTION-B
Answer any FIVE questions. (5×8=40 marks)
- Define multiple correlation coefficient between X1 and X2, …., Xp. Show that the multiple correlation coefficient between X1 and X2, …., Xp has the expression
.
- Let Y ~ Np ( 0, å) . Show that Y1 has – distribution.
- 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.
- How will you test the equality of covariance matrices of two multivariate normal distributions on the basis of independent samples drawn from two populations?
- Derive the characteristic function of Wishart distribution.
- In Principal component analysis derive the first principal component.
- Obtain the rule to assign an observation of unknown origin to one of two p-variate normal populations having the same dispersion matrix.
- Outline single linkage and complete linkage clustering procedures with an example.
SECTION-C
Answer any TWO questions. (2×20=40 marks)
- a) Derive the MLE of å when the sample is from Np (m, å).
- b) Define Hottelling’s T2 – statistic.
- c) Using the likelihood ratio test procedure, show that the rejection region for testing m =
mo against m ¹ mo is given by
T2 = n( S-1 (³ T. (10+3+7)
- 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.
- b) Let X1,X2,…., Xp have covariance matrix å with eigen value vector pairs (l1, e1),…,
(lp, ep), l1 l2 ….. ≥ lp 0, then prove that
s11 + s22 + …..+ spp = ,
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)
- 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)
- Write shot notes on:-
- Roy’s Union – Intersection principle
- Step – Wise regression
- Mohalanobis Squared distance. (5+10+5)