Shape-Enforcing Operators for Point and Interval Estimators

TL;DR

This paper introduces a set of shape-enforcing operators that improve point and interval estimates of functions with shape restrictions, ensuring closer approximation to true functions and better interval coverage.

Contribution

It proposes a unified method to enforce shape restrictions ex post on estimates, applicable to six common shape constraints, enhancing estimation accuracy and inference reliability.

Findings

Shape-enforcing operators improve estimate accuracy.

Operators increase interval coverage and reduce length.

Applications demonstrate practical benefits in growth charts and production functions.

Abstract

A common problem in econometrics, statistics, and machine learning is to estimate and make inference on functions that satisfy shape restrictions. For example, distribution functions are nondecreasing and range between zero and one, height growth charts are nondecreasing in age, and production functions are nondecreasing and quasi-concave in input quantities. We propose a method to enforce these restrictions ex post on point and interval estimates of the target function by applying functional operators. If an operator satisfies certain properties that we make precise, the shape-enforced point estimates are closer to the target function than the original point estimates and the shape-enforced interval estimates have greater coverage and shorter length than the original interval estimates. We show that these properties hold for six different operators that cover commonly used shape restrictions in practice: range, convexity, monotonicity, monotone convexity, quasi-convexity, and monotone quasi-convexity. We illustrate the results with two empirical applications to the estimation of a height growth chart for infants in India and a production function for chemical firms in China.