Linköpings Universitet
IDA/Statistik
BW
732A37 Multivariate Statistical Methods, 6 hp
Exam in Multivariate Statistical Methods, 2015-01-09
Time allowed:
Allowed aids:
Examinator:
Grades:
14-18
Calculator, The book: Johnson, Wichern: Applied Multivariate
Statistical Analysis. Notes in the book and Copy of the book are
allowed.
Bertil Wegmann, 070 - 1128321
A=22-24 points, B=19-21p, C=16-18p, D=13-15p, E=10-12p
Provide a detailed report that shows motivation of the results.
GOOD LUCK!
_________________________________________________________________________________________
1
Let the random vector X  ( X1, X 2 )' have covariance matrix
 9 3
  
.
 3 4
Show that  is a positive definite matrix. Derive the correlation matrix, R,
from  . Then, by using the correlation matrix R, determine the principal
components and find the proportion of the total variance of X explained by
the first component.
3p
2
Observations on two responses (X1 = height (in metres), X2 = percentage of
soluble sugars of the sugarcane stalk), were collected for sugarcanes from a
sugar mill. The observation vectors x1 , x2  are:
Treatment A:
Treatment B:
(5.2, 15.2), (4.5, 15.6), (5.9, 15.1), (5.5, 14.8), (5.2, 15.7)
(4.3, 14.7), (4.6, 14.3), (4.5, 14.9), (5.0, 14.1)
Assume for each treatment that the data on  X1, X 2  is a random sample from
a bivariate normal population.
a) Obtain the 90 % confidence region for mean height and mean
percentage in treatment B and sketch the resulting ellipse. 3p
b) Construct the one-way MANOVA table for comparing population mean
vectors between treatments.
4p
c) Evaluate Wilks’ lambda,  * , and use Table 6.3 to test for differences
in the population mean vectors between the treatments. State your
conclusions for the test by considering the three different significance
levels  = 0.01, 0.05, and 0.10.
2p
3
Let X be distributed as 𝑁3 (𝝁, 𝚺) where 𝛍′ = [3, − 2, 0] and
5 0 −3
Σ=( 0 9 0 )
−3 0 2
a) Are X 2 and 2 X 1  X 3 independent? Explain.
b) Find the distribution of
 X 1  3X 3



2X1  X 2

1p
1.5p
c) Find the conditional distribution of X 3 , given that X 1  2 and X 2  3 .
1.5p
4
In a study of poverty, crime, and deterrence, the sample correlation matrix
below is shown for certain summary crime statistics:
|
R12
R11
𝐑 = ( − − − | − − −)
R21
|
R22
1.0
0.6
0.6
1.0
= −−− −−−
−0.1
−0.2
−0.1
( −0.3
| −0.1
| −0.2
| −−−
|
1.0
| −0.3
−0.3
−0.1
−−−
−0.3
1.0 )
The variables are
X 1(1)  nonprimary homicides
X 2(1)  primary homicides
X 1( 2 )  severity of punishment
X 2( 2 )  certainty of punishment
Find the first sample canonical variate pair, (U 1 ,V1 ) , and their sample
canonical correlation.
4p
5
In one part of a study, a factor analysis of accounting profit measures and
market estimates of economic profits was performed by using a sample
correlation matrix. This resulted in the following rotated principal
component estimates of factor loadings for an m=3 factor model:
Estimated factor loadings
Variable
Historical return on assets
Historical return on equity
Historical return on sales
Replacement return on assets
Replacement return on equity
Replacement return on sales
Market Q ratio
Market relative excess value
F1
F2
F3
0.43
0.13
0.30
0.41
0.20
0.33
0.93
0.91
0.61
0.89
0.24
0.71
0.90
0.41
0.16
0.08
0.50
0.23
0.89
0.48
0.28
0.79
0.29
0.36
a) Using the estimated factor loadings, determine the specific variances
and communalities for the first six variables and interpret your
results.
3p
b) Using the estimated factor loadings, determine the proportion of total
sample variance due to the first factor.
1p