Statistical inference for statistical decisions

TL;DR

This paper explores inference-based statistical decision functions within Wald's framework, emphasizing their practical use in binary choices, optimization, and decision-making under uncertainty, evaluated through finite-sample regret.

Contribution

It introduces inference-based SDFs as practical decision procedures, extending Wald's theory to include hypothesis testing, point estimation, and optimization strategies.

Findings

Inference-based SDFs can effectively guide treatment choices.

Finite-sample maximum regret evaluates decision performance.

Illustrative examples demonstrate practical applications.

Abstract

The Wald development of statistical decision theory addresses decision making with sample data. Wald's concept of a statistical decision function (SDF) embraces all mappings of the form [data -> decision]. An SDF need not perform statistical inference; that is, it need not use data to draw conclusions about the true state of nature. Inference-based SDFs have the sequential form [data -> inference -> decision]. This paper motivates inference-based SDFs as practical procedures for decision making that may accomplish some of what Wald envisioned. The paper first addresses binary choice problems, where all SDFs may be viewed as hypothesis tests. It next considers as-if optimization, which uses a point estimate of the true state as if the estimate were accurate. It then extends this idea to as-if maximin and minimax-regret decisions, which use point estimates of some features of the true state as if they were accurate. The paper primarily uses finite-sample maximum regret to evaluate the performance of inference-based SDFs. To illustrate abstract ideas, it presents specific findings concerning treatment choice and point prediction with sample data.