Characterizations of Noncentral Chi-Squared-Generating Covariance Structures for a Normally Distributed Random Vector

• Published in: Sankhya. Series. A Vol. 78; no. 2; pp. 231 - 247
• Main Authors: Young, Phil D., Young, Dean M.
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01-08-2016; Indian Statistical Institute
• Subjects: Mathematics and Statistics; Matrix; Random variables; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Symmetry; 15A24; Moore-Penrose pseudo-inverse; Positive-definite covariance matrix; Non-central chi-squared random variable; Nonnegative-definite covariance matrix; 62H10; Generalized inverse; 15A09
• ISSN: 0976-836X; 0976-8378
• DOI: 10.1007/s13171-016-0081-3

Let
y
∼
N
n
μ
,
V
, where
y
is a
n
×1 random vector and
V
is a
n
×
n
covariance matrix. We explicitly characterize the general form of the covariance structure
V
for which the family of quadratic forms
y
′
A
i
y
i
=
1
k
for
i
∈
1
,
...
,
k
, 2≤
k
≤
n
, is distributed as multiples of mutually independent non-central chi-squared random variables. We consider the case when the
A
i
’s and
V
are both nonnegative definite, including several cases where the
A
i
’s have special properties, and the case where the
A
i
’s are symmetric and
V
is positive definite. Our results generalize the work of Pavur (Sankhyā
51
, 382–389,
1989
), Baldessari (Comm. Statist. - Theory Meth.
16
, 785–803,
1987
), and Chaganty and Vaish (Linear Algebra Appl.
264
, 421–437,
1997
).