查看更多>>摘要：Abstract The statistical decision theory pioneered by (Wald, Statistical decision functions, Wiley, 1950) has used state-dependent mean loss (risk) to measure the performance of statistical decision functions across potential samples. We think it evident that evaluation of performance should respect stochastic dominance, but we do not see a compelling reason to focus exclusively on mean loss. We think it instructive to also measure performance by other functionals that respect stochastic dominance, such as quantiles of the distribution of loss. This paper develops general principles and illustrative applications for statistical decision theory respecting stochastic dominance. We modify the Wald definition of admissibility to an analogous concept of stochastic dominance (SD) admissibility, which uses stochastic dominance rather than mean sampling performance to compare alternative decision rules. We study SD admissibility in two relatively simple classes of decision problems that arise in treatment choice. We reevaluate the relationship between the MLE, James–Stein, and James–Stein positive part estimators from the perspective of SD admissibility. We consider alternative criteria for choice among SD-admissible rules. We juxtapose traditional criteria based on risk, regret, or Bayes risk with analogous ones based on quantiles of state-dependent sampling distributions or the Bayes distribution of loss.
查看更多>>摘要：Abstract We consider inference on optimal treatment assignments. Our methods allow inference on the treatment assignment rule that would be optimal given knowledge of the population treatment effect in a general setting. The procedure uses multiple hypothesis testing methods to determine a subset of the population for which assignment to treatment can be determined to be optimal after conditioning on all available information, with a prespecified level of confidence. A Monte Carlo study confirms that the inference procedure has good small sample behavior. We apply the method to study Project STAR and the optimal assignment of a small class intervention based on school and teacher characteristics.
查看更多>>摘要：Abstract Statistical treatment rules map data into treatment choices. Optimal treatment rules maximize social welfare. Although some finite sample results exist, it is generally difficult to prove that a particular treatment rule is optimal. This paper develops asymptotic and numerical results on minimax-regret treatment rules when there are many treatments. I first extend a result of Hirano and Porter (Econometrica 77:1683–1701, 2009) to show that an empirical success rule is asymptotically optimal under the minimax-regret criterion. The key difference is that I use a permutation invariance argument from Lehmann (Ann Math Stat 37:1–6, 1966) to solve the limit experiment instead of applying results from hypothesis testing. I then compare the finite sample performance of several treatment rules. I find that the empirical success rule performs poorly in unbalanced designs, and that when prior information about treatments is symmetric, balanced designs are preferred to unbalanced designs. Finally, I discuss how to compute optimal finite sample rules by applying methods from computational game theory.
查看更多>>摘要：Abstract This study considers the treatment choice problem when the outcome variable is binary. We focus on statistical treatment rules that plug in fitted values from a nonparametric kernel regression, and show that the maximum regret can be calculated by maximizing over two parameters. Using this result, we propose a novel bandwidth selection method based on the minimax regret criterion. Finally, we perform a numerical exercise to compare the optimal bandwidth choices for binary and normally distributed outcomes.
查看更多>>摘要：Abstract The Maximin and Choquet expected utility theories guide decision-making under ambiguity. We apply them to hypothesis testing in incomplete models. We consider a statistical risk function that uses a prior probability to incorporate parameter uncertainty and a belief function to reflect the decision-maker’s willingness to be robust against the model’s incompleteness. We develop a numerical method to implement a test that minimizes the risk function. We also use a sequence of such tests to approximate a minimax optimal test when a nuisance parameter is present under the null hypothesis.
查看更多>>摘要：Abstract We consider a decision maker who faces a binary treatment choice when their welfare is only partially identified from data. We contribute to the literature by anchoring our finite-sample analysis on mean square regret, a decision criterion advocated by Kitagawa et al. in (2022) "Treatment Choice with Nonlinear Regret" . We find that optimal rules are always fractional, irrespective of the width of the identified set and precision of its estimate. The optimal treatment fraction is a simple logistic transformation of the commonly used t-statistic multiplied by a factor calculated by a simple constrained optimization. This treatment fraction gets closer to 0.5 as the width of the identified set becomes wider, implying the decision maker becomes more cautious against the adversarial Nature.
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