Local polynomial fitting of an equipercentile equating function: uniform approximations

Equipercentile equating is a common technique that identifies scores on two tests or two health-related quality of life instruments that have the same percentile rank. It allows the scores on the two scales to be used interchangeably. More precisely, if we denote by $F$ and $G$ the cumulative distribution functions of two scores $X$ and $Y$, respectively, then two given scores $x$ and $y$ are said to be equivalent if and only if $F(x)=G(y)$. This last equation is known as the {\em equipercentile equation}. For instance, for a given $x$, its equipercentile equivalent $y(x)$ is given by $y(x)=G^{-1}\left(F(x)\right)$, where $G^{-1}$ is the inverse of $G$. In this work, we propose various nonparametric estimators of the equipercentile equating function $G^{-1}\left(F(x)\right)$. The proposed estimators are based on the local polynomial fitting approach. We also investigate their uniform strong consistency together with uniform approximations by Brownian bridges.