M.Sc. (Semester III) (CBCS) Examination Oct/Nov-2017
Statistics
MULTIVARIATE ANALYSIS
Day Date: Saturday, 18-11-2017 Max. Marks: 70
Time: 02.30 PM to 05.00 PM
Instructions: Attempt five questions
Q.1 and Q.2 are compulsory.
Attempt any three questions from Q. 3 to 7.
Figures to the right indicate full marks.
Q.1 Select the correct alternatives of the following questions: 05
Hotelling's is a multivariate generalization of
F-statistic t-statistic
Chi-square statistic statistic
A principal component analysis was run and the following eigen values
were obtained: 2.731,2. 218, 0.442. How many components would you
retain so that 50% of the variation present in the old variables will be
explained?
1 2
3 1 or 2
All of the following techniques are useful for dimension reduction
except
Factor analysis Principal component analysis
Discriminant analysis Clustering
Let and denotes the sample mean based on a random
sample of size n. Then
Which of the following is not true about K-means clustering?
Need to specify the number of clusters in advance
It is relatively scalable and efficient in processing large data sets.
Works well when the clusters are of non-convex shape.
It is sensitive to noise and outlier data
Fill in the blanks: 05
Based on a random sample of size n from say an
unbiased estimator of Σ is
Sampling distribution of MLE of in is
In orthogonal factor model with usual notations,
A p-variate normal distribution is said to be non-singular if
A Wishart matrix is said to have Pseudo Wishart distribution of n
degrees of freedom and associate covariance matrix Σ if
Write whether the following statements True or False: 04
Marginal normality always implies multivariate normality.
Multiple correlation coefficient is proportion of variation in a variable
explained by each of the remaining variables in a group.
K-means is categorized as a non-hierarchical clustering method.
Hotelling's can be used to test hypothesis related to mean of a
multivariate normal distribution.
Q.2 Attempt the following:
Show that two p-variate normal vectors and are independent iff
Describe average linkage method of clustering. 03
Define Mahalanobis State its interpretation. Give its relationship with
error of misclassification.
03
State null distribution of sample correlation coefficient. 04
Q.3 Derive null distribution of sample correlation coefficient. 07
In usual notations, show that and are independently distributed when
sampling from
Describe minimum ECM rule. Derive the same to discriminate between
and population.
07
Q.5 Describe Roy's Union-Intersection principle. Show that Roy's Union-
Intersection principle leads to Hotelling's T2 statistic.
07
Derive density function of Wishart distribution in canonical case. 07
Q.6 Based on random sample of size n from derive an expression for MLE of Σ.
07
Describe canonical variable and canonical correlations. State and prove
any two properties of canonical variables.
07
Q.7 Derive expressions for principle components. Show that total variation
explained by principal components is same as total variation in original
variables.
07
Describe a test for testing whether additional variables are needed for
discrimination purpose.