On fractile regression

Abstract

The need for comparing two or more regression functions arises frequently in statistical applications. Comparison of the usual regression functions is not very meaningful in situations where the distribution and the range of the covariates have changed for the populations. For instance, in econometric studies, the prices of commodities and people’s incomes observed at different time points may not be on comparable scales due to inflation and other economic factors. In this paper we describe, motivated by an idea of Mahalanobis (1960), a method of standardizing the covariates and estimating the transformed regression function, which now become comparable. We develop smooth estimates of fractile regression function and study its statistical properties. We prove the consistency and asymptotic normality of the estimated fractile regression function defined through general weight functions. We illustrate our method through analysis of three data sets: blood pressure and related measurements of two tribes in India, profit and sales of private companies in India at two time points, and data on household income and expenditure of two East European transitional economies.