Asymptotics for Statistical Treatment Rules

One major goal of treatment evaluation in the social and medical sciences is to provide guidance on how to assign individuals to treatments. For example, a number of studies have examined the problem of “profiling” individuals to identify those likely to benefit from a social program.1 Manski (2000, 2002, 2004) and Dehejia (2005) suggest placing the problem within a decisiontheoretic framework, and specifying a loss function that quantifies the consequences of choosing different treatments under different states of nature. Schlag (2006) and Stoye (2006) derive exact minmax-regret rules for randomized experiments with a discrete covariate and a bounded continuous outcome. Despite these important results, it is difficult to obtain exact optimality results in many empirically relevant settings, in the same way that it is difficult to obtain exactly optimal estimators or hypothesis tests. In this paper, we develop large sample results to compare treatment rules, and show how to construct approximately optimal procedures from efficient estimates of treatment effect parameters. The data could come from a randomized experiment or an observational data source, and we allow for unrestricted outcome and covariate distributions (including continuously distributed covariates). The key requirements are a local asymptotic normality condition and that a welfare contrast parameter be point-identified. When social welfare contrasts are point-identified, there will typically exist many treatment rules that are consistent, in the sense that they assign the “better” treatment with probability approaching one. Our goal here is to make finer comparisons among rules, and to base these comparisons on risk rather than conventional statistical criteria that are not tightly connected to the underlying decision problem. We first study regular parametric models, using a local parametrization so that the problem of determining whether to assign the treatment does not become trivial as the sample size increases. Using Le Cam’s limits of experiments framework (see Le Cam (1986)) we show that the treatment