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The distribution of
k is absolutely continuous with density f , which is assummed
to be unknown. Moreover, we have the following assumptions on the
density of with positive constans , p, L and M :
• (A1) F(0) = R0
−1
f(u)du = , i.e, quantile of the distribution of
k is 0,
for all 0 < < 1.
• (A2) inf|x|< f (x) p, i.e, in the neighborhood of (any) x, the density of
is strictly positive.
• (A3) |f (0)| < 1
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• (A4) The function f (u) satisfies the Lipschitz condition:
|f (u2) − f (u2)| L |u1 − u2|, u1, u2 2 R
In this study, the design {tk} is assumed to be almost equidistant. In other words,
the following conditions are assumed on the design:
• (D1). 0 = t0 t1 . . . tn = 1;
• (D2). |tl − tm| D|l − m|/n for all 0 l,m n.
The unknown function (x) on the interval [0, 1] belongs to the Lipschitz function
class = (H, L) with smoothness . That is, for some positive H, L and 0 <
1:
(H, L) = { : |(0)| H, |(u) − (v)| L|u − v| , u, v 2 [0, 1]}.
Our aim is to estimate a function value for (x), for x 2 I. An estimator ˆ
n
=
ˆ
n
(Y
0 , Y
1 , . . . , Y
n ) is an arbitrary measurable function of the observations.
We denote the frequency of the observations per unit interval by parameter n. We
study the sequence of models:
Y
k,n = (tk,n) +
k,n, k = 0, 1, . . . , n.
For the sake of simplicity in the notation, we omit the subscript n, and superscript
thus, the notation of our model is
Yk = (tk) + k, k = 0, 1, . . . , n. (3.4)
Observe that the conditional expectation does not necessarily exist under the assumptions
(A1) - (A4), therefore, our model 3.3 is in general not a non-parametric
regression. The aim of this study is predicting one step ahead function value of
given historical observations and estimates of .
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3.5.1 A Recursive Estimator
In this section, we try to find a recursive estimator based on a stochastic approximation
procedure and derive its rate of convergence, as the frequency of the observations
per unit interval, n, tends to infinity, based on the results achieved by
Belitser and van de Geer [2000] ([5]).
Remeber the piecewise linear loss function:
(u) = u 1{u 0} + u( − 1)1{u < 0}
Now define the derivative of (u − v) with respect to v as follows:
d(u − v) =
d(u − v)
dv = − 1{u − v 0} − ( − 1)1{u − v < 0}
Moreover, for some fixed positive h, define the function:
S (u, v) =8>><>>:
−d(u − v) if |v| H + L + h
−v if |v| > H + L + h
C0 = 2/min{p, p/2(H + L) + h, 1/2}
and the sequence
n = n−2 /(2 +1) log n, where the constants , H and L appear in
the definition of the class .
For any 0 < < 1, the following recursive formula gives an estimator for the
function value k:
ˆ
k+1 = ˆk +
nS (Yk,ˆk), k = 0, 1, . . . , n − 1
with the initial value ˆ0 = 1.
In our case, the observations occur successively and at a time moment, t, the information
based on only the observations occuring by t is available. In the above
formula, since k stands for (tk), and tk stands for the time-moment, this estimation
procedure is a filtering algorithm and filtering algorithms are most appropriate
in situations such as ours. Observe that, the estimator of the function k is a measurable
function with respect to Y1, Y2, . . . , Yk, i.e, the information available upto
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the time-moment tk.
As one can see, we need to know a constant in the algorithm which must be bigger
than H + L (we took it as H + L + h). Although this seems to be restrictive feature
of the algorithm, we can assume this without loss of generality, because we
can prove the result with a sequence Hnconverging to infinity sufficiently slowly
instead of constant H + L + h, based on Belitser and de Geer[2000]([?, Bel].
This would make the proof of the result lengthier. In practise, one can use simply
S (u, v) = 1{u − v 0} + (1 − )1{u − v < 0}.
If we examine the above algorithm, we see that the expectation of ˆk+1 given the
observations upto the time-monment tk−1 is:
E(ˆk+1|Y1, Y2, . . . , Yk−1) = ˆ k+
n( −F( ˆ k−k))1{|ˆk| H+L+h}−
nˆk1{|ˆk| > H+L+h}
Because we assumed the quantile of the error term is 0, this expectation shows
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