Saurastra University 2006 M.Sc Computer Science Statistics : - XIII (Multivariate Analysis) - Question Paper

M. Sc. (Part - II) exam
April / May – 2006
Statistics : Paper - XIII
(Multivariate Analysis)
Time : three Hours] [Total Marks : 100
Instructions : (1) Attempt any 5 ques..
(2) All ques. carry equal marks.
1 (a) describe multinomial distributions and find their
variance covariance matrices. Justify the name singular
multinomial distribution.
(b) In usual notations prove :
(1) The marginal distribution of subset Xr+1, Xr+2, ...,
Xk of X1, X2, ... Xk having singular multinomial
distribution is non singular multinomial.
(2) The conditional distribution of X1, X2, ..., Xr provided
Xr+1, Xr+2, ..., Xk is singular multinomial.
2 Let the probability density function (pdf) of a random vector
x x xp = n 1, ... s be f ( x) = k. exp. - x - x - RST one -
2
d mi' 1b mg S where
S is positive definite.
obtain : (1) k. (2) E(x) and (3) E(xx' ).
OR
2 Let x be a P - variate normal vector subsequent Np dm, Si
distribution.
(a) Show that characteristic function of x is provided as
j m x (I ) exp iT T T = ' - ' F
H G 1
2
S
(b) Show that y = cx where c is a metrix of order m´ p
has Nm c C C m, e S' j distribution.
ST-2452] two [ Contd...
(c) obtain the distribution of y
y
y
x x x
x x
=F
H G
IK J
=
- +
+
FH G
IK J
1
2
1 two 3
2 3
3
2
if
m = 0 and S = I3.
3 (a) describe Hotelling T2 - statistic show that it is in variant
under Non - singular transformation.
(b) Let x1, x2. .., xn be a random sample from Np dm, Si
find the distribution of
T n n x x two one = b - 1g d - mi Ñ- d - mi '
where
X x V x X x X
n n
   
 
  

 
1  1
,
e i e i'
4 (a) Let X X X1, 2..., n be a random sample of size n from
Np dm, Si, S - unknown. Derive a test for testing
H0 0
:m = m against H1 0
:m ¹ m .
(b) discuss how will you test the symmetry of origin giving
clearly the test statistic and test criteria.
5 (a) What is discriment issue ? provide an illustration where
such issue arises.
(b) Consider the classification procedure :
Classify X0 to P1 if L > k and to P2 if L £ k. where
L = d'S-1 x - m
0 two e j and k = + c 1
2
D2 , c is a constant.
find probabilities P1 and P2 of both kinds of
misclassification. If P1 P2 = show that C = 0 and
P1 P2
1
2
= = - F
H G I
K J f D where f donate distribution function
for N 0,1 b g distribution.
6 (a) describe canonical correlations, canonical variates and
canonical vectors.
(b) Prove that canonical correlations are invariant under
Non - singular transformation.
7 Let x1 x2 xn , , ..., be a random sample from Np 0, S b g and let
the random variable X be portioned as
X
x
x
p q
q
=
F
H G
I
K J
-
( )
( )
1
2
and let A be described as
A x x
A A
A A
= = F
H G
IK J
å a aa
' 11 12
21 22
(a) find the distribution of A11 two A11 A12 A12 A
1
. 21 = - -
(b) describe sample multiple correction coefficient
R one q 1,......, p b g and find its distribution when population
multiple correlation coefficient in zero Hence suggest a
test for testeng
H0 P1 q one p
: b , ... g 0 where P1 q 1, ... p b g is the population
multiple coorrelation coefficient.
8 (a) elaborate principal components ? State their uses.
(b) Point out similarities and dissimilarities ranging from
principal component analysis and factor analysis.
(c) obtain the 1st principal component when the despersion
matrix of the random vector X is =
L
N M
O
Q P
å 1
1
1
3
1
3
.
9 (a) discuss step wire regression method.
(b) How will you build simultaneous confidence intervals.
10 Write detailed notes on subsequent :
(1) 1 way multivariate Analysis of variance.
(2) Negative multivariate multinomial distribution.
ST-2452] three [ 100 ]

Earning:   Approval pending.