Methods have been proposed for the determination of the number of
the e.d.r. directions. See, for example, Li (1992), Schott (1994),
and Cook (1998). Their approaches are based on the assumption of
symmetry of the distribution of the explanatory variable . We
now extend the cross-validation method (Cheng and Tong (1992);
Yao and Tong (1994)) to solve the above problem, having selected the
explanatory variables using the referenced cross-validation
method. A similar extension may be effected by using the approach
of Auestad and Tjøstheim (1990), which is asymptotically
equivalent to the cross-validation method
Suppose that
are the e.d.r. directions, i.e.
with
. If
, we can nominally extend the number of
directions to
, say
, such that they are perpendicular to one another.
Now, the problem becomes the selection of the explanatory
variables among
. However, because
are unknown,
we have to replace
's by their estimator
's. As we have proved that the convergence rate of
's is faster than that of the nonparametric function estimators,
the replacement is justified.
Let
be the minimizers of
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In theory, we ought to select the explanatory variables among all
possible combinations of
. However, in practice because
have been ordered according to
their contributions (see the algorithm in the next section), we
need only calculate
and compare
their values.
After determining the number of directions, which is usually less
than , we can then search for the e.d.r. directions on a
lower dimensional space, thereby reducing the effect of high
dimensionality and improving the accuracy of the estimation.
Denote the corresponding estimate of
by
. Let
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To illustrate, we apply the procedure to models (3.20) and (3.21) to see the improvement of the rMAVE method. The mean of the estimation absolute errors against the bandwidth is shown in Figure 3.6. The improvement is significant.
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As a special case, we can estimate the direction in a single-index
model by the rMAVE method (with ). The root-
consistency can be achieved even if we use the bandwidth
. A similar result was obtained by Härdle, Hall, and Ichimura (1993)
by minimizing the sum of squares of the residuals
simultaneously with respect to the direction and the bandwidth.