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 a series of papers, we describe a method of standardizing the covariates and estimating the transformed regression function, which now become comparable, in both a univariate and multivariate regression setting. We develop nonparametric estimates of the transformed regression function and study their statistical properties analytically as well as numerically. We also study invariance/equivariance properties of transformations of covariates in a regression setup. We illustrate through many real applications the difficulty in comparing the usual regression functions and motivate the need for the proposed fractile transformation.

References

1. 1.Sen, B. (2005). Estimation and Comparison of Fractile Graphs using Kernel Smoothing Techniques. Sankhya, 67, 305-334. (Special issue on Quantile Regression and Related Methods)
   

2. 2. Nag, A.K., Sen, B. and Bhaumik, D. (2006). Interest Rate and Size of Credit - A Nonparametric Exploratory Analysis.
   

3. 3.Sen., B. and Chaudhuri, P. (2011). Mahalanobis’s Fractile Graphs: Some History and New Developments. International Journal of Statistical Sciences, 11, 17-35 (Invited paper for a special issue in honor of Prasanta Chandra Mahalanobis).
   

4. 4.Sen, B. and Chaudhuri, P. (2011). On fractile transformation of covariates in regression. J. Amer. Statist. Assoc. (to appear).