A note on prediction via estimation of the conditional mode function

doi:10.1016/0378-3758(86)90099-6

Journal of Statistical Planning and Inference

Volume 15, 1986–1987, Pages 227-236

https://doi.org/10.1016/0378-3758(86)90099-6 Get rights and content

Abstract

Let ${(X_{i}, Y_{i})}_{i∈N} ⊂ E × R, E⊂ R^{d}$ be a strictly stationary process. The conditional density of Y given X is estimated by the kernel method. It is shown that the (empirically determined) mode of the kernel estimate is uniformly (in a compact) convergent to the conditional mode function when the process is Φ-mixing. This result is applied to a strictly stationary time series {Z_k}_k∈N which is markovian of order q. It is seen that the so-called model predictor of Z_{N + 1} from the observed data is converging to the predictor that is based on the full knowledge of the conditional density of Z_{N + 1} given {Z₁,…,Z_N}.

References (9)

G. Collomb et al.
Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations
Stochastic Process. Appl.
(1986)
P. Billingsley
Convergence of Probability Measures
(1968)
G. Collomb
Estimation non-parametrique de la regression: Revue bibliographique
Internat. Statist. Rev.
(1981)
G. Collomb
Proprietés de convergence presque complète du predicteur a noyan
Z. Wahrsch. Verw. Geb.
(1984)

There are more references available in the full text version of this article.

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Citation Excerpt :
It is well known that the conditional mode is more informative than the classical regression in some particular cases. We cite for instance, when the conditional density is un-symmetric, has multi-modal or if the white noise has heavy-tail distribution (see Collomb et al., 1987). While all these authors estimate the conditional mode by maximizing the kernel estimator of the conditional density function, in this paper we estimate this model using the robust estimator of the quantile regression.
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Citation Excerpt :
Since the pioneer work of Parzen (1962), many studies have been dealing with the density estimation from which they have deduced the mode estimation for real or vectorial variables (see, e.g., Chernoff (1964), Van Ryzin (1969) and Samanta (1973)). Later, in view of efficient prediction, there has been an important interest for the conditional mode estimation with vectorial explanatory variables (see Collomb et al. (1987), Berlinet et al. (1998) and Louani and Ould-Said (1999)) and, more recently, convergence of the conditional mode estimator with a functional covariable has been obtained (Ferraty et al. (2005, 2006), Dabo-Niang and Laksaci (2007) and Ezzahrioui and Ould-Saïd (2008)). But, to the best of our knowledge, the conditional mode has never been estimated for continuous time processes, and all the results with functional explicative variables suppose that the variable of interest is real valued.
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^∗: Presently at Universität Bonn, 5300 Bonn, Fed. Rep. Germany. Work supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 ‘Stochastische Mathematische Modelle’.

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A note on prediction via estimation of the conditional mode function

Abstract

Stochastic Process. Appl.

Convergence of Probability Measures

Estimation non-parametrique de la regression: Revue bibliographique

Internat. Statist. Rev.

Proprietés de convergence presque complète du predicteur a noyan

Z. Wahrsch. Verw. Geb.