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Affinely invariant matching methods with discriminant mixtures of proportional ellipsoidally symmetric distributions. (English) Zbl 1246.62159
Summary: 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 D.B. Rubinand N. Thomas [ibid. 20, No. 2, 1079-1093 (1992; Zbl 0761.62065)]. 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., D.B. Rubin and N. Thomas, Biometrics 52, No. 1, 249–264 (1996; Zbl 0881.62121)]. Extensions to conditionally affinely invariant matching with conditionally DMPES distributions are also discussed.

##### MSC:
 62H99 Multivariate analysis 62H10 Multivariate distribution of statistics 62D05 Sampling theory, sample surveys
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