|
The model suggested by Boncelet and Dickinson (1984) and Cipra and Romera (1991) uses the innovational representation (18.2) and reads
![]() |
(18.11) |
To get a robust estimator for model (18.9) we use an influence curve (IC) of Hampel-Krasker form:
where is a Lagrange multiplyer guaranteeing that the coorelation condition
is fullfilled; due to symmetry of
,
of form (18.12) is centered automatically;
furthermore
bounds the influence of
on
.
The reader not familiar to the notion of influence curve may recur to section 18.5 and look up some of the references given there.
Instead of replacing the ML-equation by an M-Equation to be solved for
,
we use a one-step with same asymptotic properties:
![]() |
![]() |
![]() |
(18.13) |
![]() |
![]() |
(18.14) |
We note the following properties:
As already seen in (18.10), is decomposed into a sum of two
independent variables
and
. They may be interpreted as
estimating
and
, thus they represent in some sense the sensitive point to
AO and IO respectively.
Instead of simultaneous clipping of both summands,
just clipping the ``IO-part'', i.e.
,
or ``AO-part'', i.e.
, separately will therefore lead to a
robustified version specialized to IO's or AO's.
For the AO-specialization we get
![]() |
(18.15) |
As in the rLS case, we propose the assurance criterium for
the choice of : We adjust the procedure to a given relative efficiency loss in the ideal model.
This loss is quantified in this case as the relative degradation of the ``asymptotic''
variance of our estimator which is in our situation just
, which in
the ideal model gives again the MSE.
Of course the lower the clipping , the higher the relative efficiency loss, so
that we may solve
For better comparability the examples for the rIC will use the same setups as those for rLS. So we just write down the modifications necessary to get from the rLS- to the rIC-example.
As the first example is one-dimensional,
calibrIC
uses a
simultaneous Newton procedure to determine
, so neither
a number of grid points nor a MC-sample size is needed
and parameter N is without meaning, as well as
fact and expl. Nevertheless you are to transmit
them to
calibrIC
and, beside the rLS setting we
write
fact=1.2 expl=2 A0=0 b0=-1 typ= 0 ; rIC-simNext we calibrate the influence curve
ergIC=calibrIC(T,Sig,H,F,Q,R,typ,A0,b0,e,N,eps,itmax, expl,fact,aus) A=ergIC.A b=ergIC.bCalling
res= rICfil(y,mu,Sig,H,F,Q,R,typ,A,b) frx = res.filtX
![]() |
N=300 eps=0.01 itmax=15 aus=4 fact=1.2 expl=2 A0=0 b0=-1 typ= 0 ; rIC-simNote that as we are in
ergIC=calibrIC(T,Sig,H,F,Qid,R,typ,A0,b0,e,N,eps,itmax, expl,fact,aus) A=ergIC.A b=ergIC.b res = kfilter2(y,mu,Sig, H,F,Qid,R) fx = res.filtX res= rICfil(y,mu,Sig,H,F,Qid,R,typ,A,b) frx = res.filtX
![]() |
Again, as in the third rLS-example, it is shown in the next example that we really loose some efficiency in the ideal model, using the rIC filter instead of the Kalman filter; the following modifications are to be done with respect to Example 6:
e=0.05 N=300 eps=0.01 itmax=15 aus=4 fact=1.2 expl=2 A0=0 b0=-1 typ= 1 ; rIC-sep-AO ergIC=calibrIC(T,Sig,H,F,Q,R,typ,A0,b0,e,N,eps,itmax, expl,fact,aus) A=ergIC.A b=ergIC.b res = kfilter2(y,mu,Sig, H,F,Q,R) fx = res.filtX res= rICfil(y,mu,Sig,H,F,Q,R,typ,A,b) frx = res.filtX fry=(H*frx')'
![]() |
As sort of an outlook, we only want to mention here the possibility of using different
norms to assess the quality of our procedures. The most important norms besides
the euclidean are in our context those derived from the Fisher
information of the ideal model (information-standardization) and the
asymptotic Covariance of itself (self-standardization).
Generally speaking these two have some nice properties compared
to the euclidean norm; so among others, optimal influence curves in
this norm stay invariant under smooth transformation in the
parameter space, c.f. Rieder (1994).
In the context of normal scores, they even lead to a drastic simplification
of the calibration problem even in higher dimensions, c.f. Ruckdeschel (1999).
Nevertheless the use of this norm has to be justified by the application,
and in the
XploRe
quantlib
kalman
, they have not yet been included.