Nonparametric methods of estimation of conditional density functions when the dimension of the explanatory variable is large are known to suffer from slow convergence rates due to the 'curse of dimensionality'. When estimating the conditional density of a random variable Y given random d-vector X, a significant reduction in dimensionality can be achieved, for example, by approximating the conditional density by that of a Y given θ TX, where the unit-vector θ is chosen to optimise the approximation under the Kullback-Leibler criterion. As a first step, this thesis pursues this 'single-index' approximation by standard kernel methods. Under strong-mixing conditions, we derive a general asymptotic representation for the orientation estimator, and as a result, the approximated conditional density is shown to enjoy the same first-order asymptotic properties as it would have if the optimal θ was known. We then proceed and generalise this result to a 'multi-index' approximation using a Projection Pursuit (PP) type approximation. We propose a multiplicative PP approximation of the conditional density that has the form f(y|x) = f₀(y)πM m=1 hm (y,θTmx), where the projection directions θm and the multiplicative elements, hm, m = 1,...,M, are chosen to minimise a weighted version of the Kullback-Leibler relative entropy between the true and the estimated conditional densities. We first establish the validity of the approximation by proving some probabilistic properties, and in particular we show that the PP approximation converges weakly to the true conditional density as M approaches infinity. An iterative procedure for estimation is outlined, and in order to terminate the iterative estimation procedure, a variant of the bootstrap information criterion is suggested. Finally, the theory established for the single-index model serve as a building block in deriving the asymptotic properties of the PP estimator under strong-mixing conditions. All methods are illustrated in simulations with nonlinear time-series models, and some applications to prediction of daily exchange-rate data are demonstrated.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:571000 |
| Date | January 2012 |
| Creators | Rosemarin, Roy |
| Publisher | London School of Economics and Political Science (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.lse.ac.uk/387/ |
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