In the paper, the deviation of the spline estimator for the unknown probability density is approximated with the Gauss process. It is also found zeros for the infimum of variance of the derivation from the approximating process.
The present work is a continuation of the work [
Let
where
Remind that
where
of Wiener processes.
Denote by
and by
where
In the second section of the work, Theorem 2 and 3 are proven:
and
And it is also stated (Theorem 5) that
It holds the following
Theorem 1. Let
The proof of this statement is easy, therefore we omit it.
Theorems 2 and Theorem 3 will be proved by the mthods given in [
Theorem 2. Let
Then under our assumption a) and b) concerning
Proof. By the main Theorem from [
and for any
Set
Theorem 2 follows now from Theorem 1, relations (2) from [
random variables
Theorem 3. If conditions of Theorem 2 hold and
where
Proof. From the interpolation condition
we have
One can easily note that
in the points of interpolation
where
The relation (5) implies that for arbitrary
It remains to choose
Relations
Theorem 4. First order mean square derivations of the Gauss process
Let now
Theorem 5. 1) The variance of mean square derivations of the Gauss process
2) If the variance vanishes also in intervals
Proof. At the beginning of the proof of the theorem, we proceed as in [
we get for
Substituting into (6)
and taking into account that
or
We find analogously
and also
Generalizing the obtained results, we have
Denote
implies
On the other hand,
where
Obviously,
i.e. at
The first part of Theorem 5 is proved.
Let pass to the proof of the second part. Both in the case of
is valid for
The explicit form of
Note, in this case
One can easily see that
The first part of Theorem 5 is proved.
At last, Theorems 2 and 3 imply that limit distributions of the random variables
coincide. However, the Gauss process
polation points for the spline, and
to investigate the distribution of the maximum of