# Sur l'estimation non paramétrique de la fonction d'égalisation équipercentile. Application à la qualité de vie.

Abstract : Let $X$ and $Y$ be two random variables with cumulative distribution functions $F$ and $G$ respectively. Two given realizations $x$ and $y$ are said to be equivalent if and only if $F(x)=G(y)$. This last equation is known as equipercentile equation''. For instance, for a given $x$, its equipercentile equivalent $y(x)$ is given by $y(x)=G^{-1}(F(x))$, where $G^{-1}$ is the inverse of $G$. In this work, we propose various nonparametric estimators of the equipercentile equating function $G^{-1}(F(x))$. The proposed estimators are based on local polynomial fitting approach. Their asymptotic properties are investigated. The obtained results are : First, we show the almost sure uniform convergence. Then, we establish the approximation by an appropriate Brownian bridge. We evaluate the performance of local polynomial estimators of the equipercentile equating function by considering a measure of loss known as mean square error. Finally, we provide some simulations in R to illustrate our results and compare estimators by applying a set of real data from a longitudinal study of multi-center cohort ANRS C08.
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https://tel.archives-ouvertes.fr/tel-00425330
Contributor : Kaouthar El Fassi <>
Submitted on : Tuesday, October 20, 2009 - 6:46:20 PM
Last modification on : Wednesday, December 9, 2020 - 3:16:51 PM
Long-term archiving on: : Tuesday, June 15, 2010 - 10:52:04 PM

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• HAL Id : tel-00425330, version 1

### Citation

Kaouthar El Fassi. Sur l'estimation non paramétrique de la fonction d'égalisation équipercentile. Application à la qualité de vie.. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2009. Français. ⟨tel-00425330⟩

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