Estimation in Financial Models - RiskLab

tion
F (x) =
Z x
,1
f(y)dy:
The latter statement can be made precise for instance through the Borel-
Cantelli Lemma and its various renements (see [68]) .
In order to construct a density estimator f n (x) starting from the empirical
distribution function F n (x) in (4.1), we have to smooth F n . One possibility
is by using so-called kernel density estimators
nX x , Xk
f n (x) = 1
nh n
k=1
where the kernel V :IR ,! IR + satises
Z 1
,1
V
V (x)dx =1;
and where the so-called bandwidth sequence (h n )issuch that
h n
h n ,! 0; nh n ,!1:
This class of estimators is referred to as the Parzen-Rosenblatt estimators.
Under certain restrictions on f(x) the sequence f n (x) converges in some
probabilistic sense to f(x). See [38], x4.4, and [69] for a detailed discussion.
Ibragimov and Has'minskii [38], x7, present some interesting approaches to
and examples of nonparametric estimators of a signal S(t), belonging to a
certain functional space. Two models for signals are considered. A rst one
has the form
dX(t) =S(t)dt + "db(t); (4.2)
with 0 t 1, (b(t)) a Wiener process, " small (typically " # 0) and X(t) is
an observed signal on [0; 1]. A second model is of the form
;
dX(t) =S(t)dt + db(t); (4.3)
with 0 t n, S(t) a one-periodic function and X(t) is observed over n
time periods.
As a basic example consider estimation of the functional F on S(t)
F (S) =
Z 1
0
f(t)S(t)dt;
with S; f 2 L 2 (0; 1). The functional F has to be estimated based on observations
of (4.2), respectively (4.3). The estimator
^F 1 =
Z 1
0
f(t)dX(t)
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