  We consider an open (scattering) quantum system under the action of a
perturbation of its closed counterpart. It is demonstrated that the resulting
shift of resonance widths is a sensitive indicator of the non-orthogonality of
resonance wavefunctions, being zero only if those were orthogonal. Focusing
further on chaotic systems, we employ random matrix theory to introduce a new
type of parametric statistics in open systems, and derive the distribution of
the resonance width shifts in the regime of weak coupling to the continuum.
