18.5 Excursion: Generating Multivariate Robust Influence Curves for Normal Scores
- {A, b, V, ctrl} =
ICerz
(e, FI, A0, b0, N, eps, itmax, expl, fact0, aus)
- generates--if possible--a (simultaneously clipped) IC to a given
efficiency loss for normal scores
- {A, b, V, ctrl} =
ICerzsep
(e, S1, S2, A0, b0, N, eps, itmax, expl, fact0, aus)
- generates--if possible--a (separately clipped) IC to a given efficiency
loss for normal scores, with the same output list as
ICerz
.
|
Coming from the local i.i.d. asymptotic setup,
we have applied estimators to the regression model (18.9) that
have proven to be optimal there; so at this point we want to give a short
abridge of the theory behind it and of how these
optimal IC's may be obtained, numerically.
18.5.1 Definition of IC
In the local i.i.d. asymptotic setup,
we consider a parametric family
and want to estimate the true
based on observations
. To do so we only
allow for asymptotically linear estimators
for this parameter
, i.e.,
 |
(18.17) |
for some
-Variable
.
As is shown in Rieder (1994), to get local Fisher consistency of
--i.e.
has to converge to
in
-probability for all
--we must necessarily have
![$\displaystyle \mathop{\rm {{}E{}}}\nolimits _{\theta}[\psi_{\theta}]=0, \qquad ...
...p{\rm {{}E{}}}\nolimits _{\theta}[\psi_{\theta}\Lambda_{\theta}']={\rm\bf I}_n.$](xaghtmlimg2039.gif) |
(18.18) |
Optimality results also to be looked up in Rieder (1994) show that for the i.i.d. setup,
- the IC minimizing the trace of the covariance in the ideal model subject to
a bias bound in a neighbourhood around the ideal situation is just of Hampel-Kraker form;
[in short this is
subject to
].
- for
not too small, there exists exactly one Hampel-Krasker IC
and as
is continuous,
also is unique.
- to each
not too big there is exactly one pair
fullfilling
(18.16) and (18.18).
18.5.2 General Algorithm
In general, for
given,
is determined by the implicite equation
![$\displaystyle A^{-1} \stackrel{!}{=} \mathop{\rm {{}E{}}}\nolimits \left[\Lambda \Lambda' \min\left\{1,\frac{b}{\Vert A\Lambda\Vert}\right\}\right].$](xaghtmlimg2042.gif) |
(18.19) |
As clipping is done continuously, and by the integration
, we achieve an arbitrary
smoothness of (18.19) in
and
.
As we know that for
,
, for
not too small we can use
as a starting value for the fixed point iteration
![$\displaystyle A_{i+1}^{-1} \stackrel{!}{=} \mathop{\rm {{}E{}}}\nolimits \left[[\Lambda \Lambda' \min\left\{1,\frac{b}{\Vert A_i\Lambda\Vert}\right\}\right].$](xaghtmlimg2046.gif) |
(18.20) |
Proofs for local convergence may be found in Ruckdeschel (1999).
For smaller
, we first solve (18.19) for a larger
and then take
as a starting value for (18.20); as a criterium whether (18.20) converges or
not we take the development of the size of
which is controlled by the parameter
expl, the stepsize from
to
is controlled by fact.
Once we have determined for given
, we control whether
is smaller or
larger than
. To determine the pair
for given
, we
use a bisection algorithm, as
is strictly dicreasing in
.
In dimension
, the problem is only
-dimensional, so it pays off, using
a simultaneous Newton procedure to determine
.
This is done by the auxiliary routines
abinfonewton
and
absepnewton
for the simultaneously and the separately clipped case, respectively.
18.5.3 Polar Representation and Explicite Calculations
In the case of
,
, we have some nice
properties, which make calculations in (18.20) easier.
We write
as
 |
(18.21) |
with
,
,
. Then
,
and
,
independent.
Now we have to solve
with
18.5.3.2 Explicit Calculations for the Absolute Value
For given
,
is constant, and
calculation of
,
using the proposition that clipped moments of
can be calculated by means of (higher)
clipped moments of the standard normal--c.f. Ruckdeschel (1999).
18.5.4 Integrating along the Directions
For the remaining integration along the directions
we do
: nothing has to be done;
are calculated simultaneously by a Newton procedure.
:
-valued integration along the unit-circle; done by a Romberg procedure.
:
-valued integration along the surface of the
-dim unit ball;
done by MC-integration.
For simultaneous clipping we additionally have (c.f. Ruckdeschel (1999)) the interesting proposition that
clipping only effects the spectrum of
but not the invariant spaces.
This is used to transform integration from
,
to
,
, thus reducing the problem
from
to
dimensions.
18.5.5 Short Description of the Auxiliary Routines Used
The further quantlets in the quantlib
kalman
just being
auxiliary routines for the ones described
up to now, we confine ourselves to shortly giving a survey of
these routines in form of a table;
note that in the table,
, and
,
stand for random variables,
and
.
quantlet |
input |
output |
function |
itera |
(A0,FI,b,N, |
(A,V,ctrl) |
Fixed-Point-Iteration (18.20) |
|
eps,itmax, |
|
for sim. clipping; also decides |
|
expl) |
|
if there was convergence or not |
iteras |
(A0,S1,S2,b, |
(A,V,ctrl) |
Fixed-Point-Iteration (18.20) |
|
N,eps,itmax, |
|
for sep. clipping; also decides |
|
expl) |
|
if there was convergence or not |
numint2 |
(aIh, b, N) |
(r,s) |
: (reduced problem) |
|
|
|
Romberg-integration |
numint2m |
(aIh, b, N) |
(r,s) |
: (full problem) |
|
|
|
Romberg-integration |
stointp |
(aIh, b, N) |
(r,s) |
: (reduced problem) |
|
|
|
MC-integration |
stointpm |
(aIh, b, N) |
(r,s) |
: (full problem) |
|
|
|
MC-integration |
ewinn |
(c,n) |
y |
calculates  |
ew2inn |
(c,n) |
y |
calculates  |
betrnormF |
(t,n) |
y |
calculates
. |
betrnormE |
(t,n) |
y |
calculates
. |
betrnormV |
(t,n) |
y |
calculates
. |
nmomnorm |
(t,n) |
y |
calculates
. |