Consider the treated and control populations of size
a set of matched variables recorded on all
units. Due to cost considerations, outcomes and additional covariates are recorded for matched subsamples of sizes
chosen such that the distributions of
matched units are more similar than for random subsamples. The standard matched sample estimator of the treatment’s effect on an outcome
based on matched units. The bias of this estimator is less compared to the difference based on random subsamples. Consider ellipsoidal distributions for which there exists a linear transformation of the variables that results in a spherically symmetric distribution for the transformed variables. Matching methods based on population or sample inner products, such as discriminant matching or Mahalanobis metric matching or methods using propensity scores based on logistic regression estimators which are called affinely invariant, are used with ellipsoidal distributions. Furthermore, canonical forms for conditionally ellipsoidal distributions using conditionally affinely invariant matching methods are considered.