TABLE OF CONTENTS Given a wedge domain Q, the centroid c is defined as the geometric centroid of Q, under any radially symmetric density function. Of course, c is defined only up to a positive scalar constant, and hence the radial structure of the density is not important.
The simplest way to calculate c is by Monte Carlo. Let x be a sample from an n-dimensional uncorrelated Gaussian, and for the single-sector case, let y be the vector whose components are the components of x sorted into decreasing order. Then y ? Q, and since the sorting operation consists only of interchanges of components which are equivalent to planar reflections, the density of y is a radially symmetric Gaussian restricted to Q. The estimate of c is then the sample average of many independent draws of y.
The multiple-sector case is handled simply by sorting only within each sector. Note that this automatically determines the relative weights between the sectors.
The case with comparison to zero is also easily handled. The initial Gaussian vector is sign-corrected so that its first ? components are non-negative and its last n components are non-positive; then a sort is performed within each section. Clearly, each of these operations preserves measure.
For more complicated inequality information structures, the geometry is not always so simple; it is not always possible to reduce a general...
