Multivariate outlier detection

Abstract

The concepts of parametric outlier testing extend with some difficulty but only relatively minor modification to multivariate data. Suppose that X ₁, X ₂, . . . , X _n are n vectors ofp components, the null hypothesis being that they are a random sample from the multivariate normal distribution with mean vector _ξ and covariance matrix ∑

$$ H_0 :{\text{X}}_i \sim {\text{MN}}(\xi ,\Sigma ){\text{ }}i = 1\,{\text{ }}to{\text{ }}\,n $$

((8.1))

Under the alternative hypothesis of, say, k outliers, there is some unknown permutation j(i) of the integers 1, 2, . . . , n such that

$$ X_{j(i)} \sim MN(\xi ,\Sigma {\text{) }}i = k + 1\,{\text{ }}to{\text{ }}\,n $$

((8.2))

while X _j(1)’ . . . , X _j(k) follow some other distribution or distributions. It is convenient, and seems not to result in much loss of generality, to suppose that the outliers also follow multivariate normal distributions, but with at least one parameter different from (ξ, ∑).