**Sankhya:
The Indian Journal of Statistics**

2005, Volume 67, Pt. 2, 305--334

**Estimation and Comparison of
Fractile Graphs Using Kernel Smoothing
Techniques**

By

Bodhisattva Sen, University of Michigan, USA

SUMMARY. The concept of Fractile Graphical Analysis (FGA) was introduced by Prasanta Chandra Mahalanobis (see Mahalanobis, 1960). It is one of the earliest nonparametric regression techniques to compare two regression functions for two bivariate populations $(X,Y)$. This method is particularly useful for comparing two regression functions where the covariate ($X$) for the two populations are not necessarily on comparable scales. 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. In this paper, we consider a smooth estimate of the 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 also investigate some procedures based on the idea of resampling to test the equality of the fractile regression functions for two different populations. These methods are applied to some real data sets obtained from the Reserve Bank of India, and this leads to some interesting and useful observations. In course of our investigation, we review many of Mahalanobis' original ideas relating to FGA {\it vis a vis} some of the key ideas used in nonparametric kernel regression.

*AMS (1991) subject classification. *Primary 62G08; Secondary 62G09,
62G10.

*Key words and phrases. *Fractile graphical analysis, fractile graphs,
smooth estimates, weight functions, consistency and asymptotic normality,
resampling methods.