On mean decomposition for summarizing conditional distributions

TL;DR

This paper introduces a mean decomposition method for summarizing conditional distributions, providing a new way to analyze how covariates influence different parts of a variable's distribution in regression models.

Contribution

It develops a novel summary measure based on expected values over distribution subsets and derives its asymptotic properties within a regression framework.

Findings

The method effectively captures covariate effects across distribution segments.

Simulation shows the variance bound relates to grid and estimator precision.

Applications demonstrate usefulness in resource allocation and intervention assessment.

Abstract

We propose a summary measure defined as the expected value of a random variable over disjoint subsets of its support that are specified by a given grid of proportions, and consider its use in a regression modeling framework. The obtained regression coefficients provide information about the effect of a set of given covariates on the variable's expectation in each specified subset. We derive asymptotic properties for a general estimation approach that are based on those of the chosen quantile function estimator for the underlying probability distribution. A bound on the variance of this general estimator is also provided, which relates its precision to the given grid of proportions and that of the quantile function estimator, as shown in a simulation example. We illustrate the use of our method and its advantages in two real data applications, where we show its potential for solving resource-allocation and intervention-evaluation problems.