  In observational studies designed to estimate the effects of interventions or
exposures, such as cigarette smoking, it is desirable to try to control
background differences between the treated group (e.g., current smokers) and
the control group (e.g., never smokers) on covariates $X$ (e.g., age,
education). Matched sampling attempts to effect this control by selecting
subsets of the treated and control groups with similar distributions of such
covariates. This paper examines the consequences of matching using affinely
invariant methods when the covariate distributions are ``discriminant mixtures
of proportional ellipsoidally symmetric'' (DMPES) distributions, a class herein
defined, which generalizes the ellipsoidal symmetry class of Rubin and Thomas
[Ann. Statist. 20 (1992) 1079--1093]. The resulting generalized results help
indicate why earlier results hold quite well even when the simple assumption of
ellipsoidal symmetry is not met [e.g., Biometrics 52 (1996) 249--264].
Extensions to conditionally affinely invariant matching with conditionally
DMPES distributions are also discussed.
