A generalized inequality is a partial ordering on \(real^n\) that has many, but not all, of the properties of conventional inequalities and the standard ordering on \(\real\).
Formally, a generalized inequality is defined with respect to a proper cone \(K\) in the following manner:
\[\x \gle \y \Longleftrightarrow \y - \x \in K.\]The cone in question is usually clear from context, but if not it can be specified with the notation \(\gle_K\).
For example, with vectors, unless otherwise specified the cone is assumed to be \(\real_+^d\), the nonnegative orthant. Thus, \(\x \gle \a\) means \(\a - \x \in \real_+^d\), or \(a_j - x_j \ge 0\) for all \(j\).
An example where this idea is relevant to statistics is the CDF of a multivariate distribution, where \(F_\x(\a)\) means \(\Pr\{\x \gle \a \}\).
The strict inequality is defined similarly, with \(\x \gl \a\) defined to mean \(\a - \x \in \textrm{ interior} (\real_+^d)\), or \(x_j < a_j\) for all \(j\).
For matrices, the cone is usually taken to be the set of positive semidefinite matrices, such that \(\A \gle \B\) means that the matrix \(\B - \A\) is positive semidefinite (and \(\A \gl \B\) meaning that \(\B - \A\) is positive definite).