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


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.

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