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) ___ ________ ______ ________ _________ __________ _________ _________ _________ __________ ____.
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__________ _____ _______ __________ ________ _____ ________.
_________ _____ _________ _______ _________ ____ _____ _____ _________.
_______ _____ ________ __________ _______ ___.
_________ __________ ______ _____ __________.
___ ____ __________ ________ __________ ___ _______ _________ _______.
______ _______ ________ _______ _______.
______ ______ _____ ________ ___ ________ __________ ___ _______ ____ _________.
___ _________ _____ _________ ________ _______ __________ _______ ______ _______ ____ ________.
__________ ________ __________ ____ __________ ______ _______ __________.
______ _________ ___ __________ ___ ____ _________ _________ _____ ________ ______.
____ ____ _____ _____ _______ ______ __________.
_____ _____ ________ _________ _____ _________ ___ ____ ___ _________ ______ _____.
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_________ __________ _______ ___ ___ ___.
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If with
and
. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.
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
Let , where
and
. Check the independence
of (i) X2 and X1 (ii) (X2, X4) and (X1, X3) (iii) (X1, X2) and (X3, X4).
Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.
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.$
Let , where
and
.
Then find and
.
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
)
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