A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models

TL;DR

This paper explores how moment inequality models in empirical economics facilitate minimax efficiency comparisons, emphasizing the importance of a specific distance measure that links confidence intervals and tests.

Contribution

It demonstrates that the structure of moment inequality models in low-dimensional settings enables stronger minimax efficiency results through a duality between tests and confidence intervals.

Findings

Duality between minimax tests and confidence intervals established

Structure of moment inequalities enhances efficiency comparisons

Low-dimensional inference benefits from specific distance definitions

Abstract

This note uses a simple example to show how moment inequality models used in the empirical economics literature lead to general minimax relative efficiency comparisons. The main point is that such models involve inference on a low dimensional parameter, which leads naturally to a definition of "distance" that, in full generality, would be arbitrary in minimax testing problems. This definition of distance is justified by the fact that it leads to a duality between minimaxity of confidence intervals and tests, which does not hold for other definitions of distance. Thus, the use of moment inequalities for inference in a low dimensional parametric model places additional structure on the testing problem, which leads to stronger conclusions regarding minimax relative efficiency than would otherwise be possible.