Identification and Inference with Min-over-max Estimators for the Measurement of Labor Market Fairness

TL;DR

This paper develops a method for statistical inference on the Demographic Parity fairness metric, using smooth approximations of min and max functions to enable valid asymptotic analysis even at non-differentiable points.

Contribution

It introduces a novel approach to approximate complex min-max functions for inference, ensuring validity across the entire domain, including non-differentiable points.

Findings

Provides a smooth approximation method for DP inference

Derives asymptotic distribution for the approximations

Enables confidence interval computation and hypothesis testing for fairness thresholds

Abstract

These notes shows how to do inference on the Demographic Parity (DP) metric. Although the metric is a complex statistic involving min and max computations, we propose a smooth approximation of those functions and derive its asymptotic distribution. The limit of these approximations and their gradients converge to those of the true max and min functions, wherever they exist. More importantly, when the true max and min functions are not differentiable, the approximations still are, and they provide valid asymptotic inference everywhere in the domain. We conclude with some directions on how to compute confidence intervals for DP, how to test if it is under 0.8 (the U.S. Equal Employment Opportunity Commission fairness threshold), and how to do inference in an A/B test.