On mean decomposition for summarizing conditional distributions

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 simulated data 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. KEYWORDS asymptotics, compound expectation, conditional quantile function, regression model, summary measure 1 INTRODUCTION Statistical summary measures aim to comprise relevant information from the distribution of a variable of interest. Despite their attractive simplicity, central tendency measures are not always appropriate when we intend to describe the entire distribution. The sample mean, forexample, provides practical but often insufficient information. Conversely, a set of sample quantiles can provide a more detailed picture but lack information on how the distribution behaves between elements of the set.On the basis of the same mathematical relation between these two extremes exploited in Wang and Zhou (2010), we propose to summarize the variable's distribution by specifying a grid of proportions that divide its mean into a sum of components. From these components, one can easily derive the variable's expected value on different segments of its support. Our approach results in a set-valued summary measure, which we refer to as compound expectation, that describes the variable's entire distribution in terms of expected values and whose elements relate to specific fractions of the variable's support.Following this same formulation, the compound expectation can be easily extended to a regression framework in which we summarize the variable's conditional distribution given covariates. Similarly to the univariate case, we obtain regression coefficients that measure average differences on the outcome variable over distinct fractions along its entire distribution. Other regression models that characterize entire conditional distributions have been previously proposed, for example, by Koenker and Bassett (1978), Newey and Powell (1987), or Breckling