On Fractile Transformation of Covariates in Regression

Abstract

The need for comparing two regression functions arises frequently in statistical applications. Comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different 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 article, we describe a method of standardizing the covariates and estimating the transformed regression function, which then become comparable. We develop smooth estimates of the fractile regression function and study its statistical properties analytically as well as numerically. We also provide a few real examples that illustrate the difficulty in comparing the usual regression functions and motivate the need for the fractile transformation. Our analysis of the real examples leads to new and useful statistical conclusions that are missed by comparison of the usual regression functions.

KEY WORDS:

• Asymptotic consistency
• Fractile regression
• Groups of transformations
• Invariance and equivariance
• Kernel smoothing
• Nonparametric regression

Acknowledgments

The authors are grateful to three referees and an associate editor for their insightful comments that helped improve the article. They would also like to thank Partha P. Majumder (Human Genetics Unit, Indian Statistical Institute, Kolkata, India) for providing them with the blood pressure data for the Bhutia and the Toto tribes and the Reserve Bank of India for supplying the data on sales and profits of companies in India.