In this article, we develop a multivariate theory for analyzing multivariate datasets that have fewer observations than dimensions. More specifically, we consider the problem of testing the hypothesis that the mean vector
μ of a
p-dimensional random vector
x is a zero vector where
N, the number of independent observations on
x, is less than the dimension
p. It is assumed that
x is normally distributed with mean vector
μ and unknown nonsingular covariance matrix ∑. We propose the test statistic
F+ =
n−2 (
p −
n + 1)
N ¯
x′
S+¯
x, where
n =
N − 1 <
p, ¯
x and
S are the sample mean vector and the sample covariance matrix respectively, and
S+ is the Moore-Penrose inverse of
S. It is shown that a suitably normalized version of the
F+ statistic is asymptotically normally distributed under the hypothesis. The asymptotic non-null distribution in one sample case is given. The case when the covariance matrix ∑ is singular of rank
r but the sample size
N is larger than
r is also considered. The corresponding results for the case of two-samples and
k samples, known as MANOVA, are given.
View full abstract