3.5 The Algorithm
In this section, we first give some guidance on the selection of
the bandwidth, the estimation of e.d.r. directions and the
determination of the number of e.d.r. directions. Then we give an
algorithm for the calculation of the e.d.r. directions.
We first standardize the original data. Write
. Let
and
,
and
. We use the cross-validation method to select
the bandwidth
. For each
, let
and
be the arguments of
The bandwidth is then chosen as
With the bandwidth
, we now proceed to the calculation of
the e.d.r. directions below by reference to the minimization
(3.16). By (3.17) and Theorem 3.2, we can
estimate the e.d.r. directions using the backfitting method. To
save space, we give here the details for model (3.4) to
illustrate the general idea. For any
, let
with
as
the initial value and
and
,
. Minimize
where
is a
vector,
a scalar and
a
vector. This is a typical constrained
quadratic programming problem. See, for example, Rao (1973, p. 232). Let
With
given,
which minimizes
is given by
 |
|
|
|
 |
|
|
(3.37) |
If
,
and
are
given, then the
which minimizes
is given by
 |
|
|
(3.38) |
where
and
denotes
the Moore-Penrose inverse of a matrix
. Therefore, we can
then minimize
iteratively as follows.
- 0.
- initial
and
;
- 1.
- initialize
such that
,
and
and repeat the following steps
- a.
- Calculate
as in (3.37);
- b.
- Calculate
as in (3.38) and let
- 2.
- replace
by
, let
if
, 1 otherwise, return to step 1.
Repeat steps 1 and 2 until convergence is obtained.
The above algorithm is quite efficient.
Note that to search for the minimum point we do not
have to use the derivative of the unknown functions as required
by the Newton-Raphson method, which is hard to estimate. See for example
Weisberg and Welsh (1994). Indeed, the subsequent numerical
examples suggest the above algorithm has a wider domain of
convergence than the Newton-Raphson based algorithm.