**Sankhya:
The Indian Journal of Statistics**

2005, Volume 67, Pt. 2, 378--398

**Conditional Quantiles for
Dependent Functional Data with Application to the Climatic {\em El Ni\~no} Phenomenon**

By

Fr\'ed\'eric Ferraty, Universit\'{e}
Paul Sabatier, Toulouse, France

Abbes Rabhi, Universit\'e Djillali Liabes, Sidi Bel Abbes, Algeria

Philippe Vieu, Universit\'e Paul Sabatier, Toulouse, France

SUMMARY. This paper deals with a scalar response conditioned by a functional random variable. The main goal is to estimate nonparametrically the quantiles of such a conditional distribution when the sample is considered as an $\alpha$-mixing sequence. Firstly, a kernel type estimator for the conditional cumulative distribution function ({\em cond-cdf}) is introduced. Afterwards, we derive an estimate of the quantiles by inverting this estimated {\em cond-cdf}, and asymptotic properties are stated. This approach can be applied in time series analysis. For that, the whole observed time series has to be split into a set of functional data, and the functional conditional quantile approach can be used both to forecast and to build confidence prediction bands. The {\em El Ni\~no} time series illustrates this.

*AMS (1991) subject classification. *Primary 62G05, 62G99; Secondary
62M10.

*Key words and phrases. *Conditional quantile, conditional cumulative
distribution, %derivatives of conditional cumulative distribution, {\em El
Ni\~no} series, functional random variable, %kernel estimator, nonparametric
estimation, %prediction confidence band, %time series, semi-metric,
$\alpha$-mixing.