A quantile-copula approach to conditional density estimation

Abstract

a b s t r a c tA new kernel-type estimator of the conditional density is proposed. It is based on an efficient quantile transformation of the data. The proposed estimator, which is based on the copula representation, turns out to have a remarkable product form. Its large-sample properties are considered and comparisons in terms of bias and variance are made with competitors based on nonparametric regression. A comparative simulation study is also provided.

Application of probabilistic wind power forecasting in electricity markets

“…The QCE was introduced in Faugeras and further enhanced with a time‐adaptive version in Bessa et al The basic idea is to represent the joint density function f P , X ( p t + k , x t + k | t ) in equation by a copula function which models the dependency structure among the explanatory variables and the target variable. The modified equation is as follows: where c is a copula density function, and u and v are quantile transforms of the data − u = F P ( p ) and v = F X ( x ), respectively.…”

A note on the adaptive estimation of a bi-dimensional density in the case of knowledge of the copula density

“…Such ''plug-in estimator'' has been investigated in, e.g., Liebscher (2005), Bouezmarni and Rombouts (2008) and Faugeras (2009) via non-adaptive kernel methods. Their mean integrated squared error (MISE) properties, uniform strong consistency and asymptotic normality was proved.…”

Development and testing of improved statistical wind power forecasting methods.

“…The quantile-copula estimator was introduced by Faugeras [37]. According to the authors, its main advantages over the existing methods are that: the methods based on the NW estimator are numerically unstable when the denominator is close to zero; for a problem with several explanatory variables, this method has only one kernel product, instead of two; at a conceptual level, density estimation should only be based on density estimation methods and not on regression approaches (like the NW estimator).…”

“…Almost at the same time, Faugeras [37] and Bouezmarni and Rombouts [38] proposed the idea of using a copula for modeling the dependency structure between Y and X. Regarding copulas, the Sklar theorem [39] says the following for the bivariate case:…”

“…The idea proposed by Bouezmarni and Rombouts [38] was a semiparametric approach, where a parametric model is considered for the copula, and the marginal distributions are represented by a nonparametric model (empirical distribution function). However, we followed the idea described by Faugeras [37], where the copula density is estimated with KDE.…”