Dimensionality reduction in nonparametric conditional density estimation with applications to nonlinear time series.
PhD thesis, The London School of Economics and Political Science.
Nonparametric methods of estimation of conditional density functions when the dimension of the explanatory variable is large are known to su§er from slow convergence rates due
to the ëcurse of dimensionalityí. When estimating the conditional density of a random variable Y given random d-vector X, a signiÖcant reduction in dimensionality can be achieved, for example, by approximating the conditional density by that of a Y given �
TX, where the
unit-vector � is chosen to optimise the approximation under the Kullback-Leibler criterion.
As a Örst step, this thesis pursues this ësingle-indexí 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 Örst-order asymptotic properties as it would have if the optimal � was
known. We then proceed and generalise this result to a ëmulti-indexíapproximation using
a Projection Pursuit (PP) type approximation. We propose a multiplicative PP approximation of the conditional density that has the form f (yjx) = f0 (y)
where the projection directions �m 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 Örst 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 inÖnity.
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.
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