State whether the following statements are True or False. Give reasons in support of your answer: 5×2=10
(a) The eigenvalues of a positive definite matrix are less than zero.
(b) is an orthogonal and idempotent matrix.
(c) A factor model postulates that a random vector is linearly dependent upon a few observable common factors.
(d) A given px1 vector is a plausible value for the mean of a multivariate normal distribution can be tested using Student's t test.
(e) In single linkage method, we find the minimum distance in the distance matrix and merge the minimum distanced clusters
(a) False. A positive definite matrix has all its eigenvalues strictly greater than zero, not less than zero, because for any non-zero vector (x), (x'Ax > 0) implies positivity of eigenvalues. (b) False. A matrix cannot in general be both orthogonal ________ _________ _______ _____ ________ ___ _____ __________ ______ _______ _________ ____.
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Define clustering. Differentiate between single linkage and complete linkage method of clustering.
(b) Consider the following partitioned sample variance-covariance matrix S, obtained from a sample of size 10.$
If , then test for the independence of the sub-vectors
and
.
(Use
)
---
Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.
Let , where
and
. Check the independence
of (i) X2 and X1 (ii) (X2, X4) and (X1, X3) (iii) (X1, X2) and (X3, X4).
Let , where
and
.
Then find and
.
If with
and
. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.
Obtain the square root matrix corresponding to a matrix . Also
verify that .
Let two independent samples of size and
from trivariate normal populations have following mean vectors and variance-covariance matrices:
and
and
, respectively.
---$
$
samples are significantly different from each other at 5% level of significance or not.$