Strong Consistency of the Spline-Estimation of Probabilities Density in Uniform Metric

In the present paper as estimation of an unknown probability density of the spline-estimation is constructed, necessity and sufficiency conditions of strong consistency of the spline-estimation are given.


Introduction
We assume that on the interval [ ] , , , , 1 Let 1 2 , , , N X X X  be independent identical distributed random variables with unknown density distribution f(x) concentrated and continuous on the interval [0, 1], and S N (x) be cubic spline interpolating the values a and N b are given real numbers.Concrete choice of these numbers depends on the considered problem.
As estimation of an unknown probability density we take the statistics In the present work as estimation of the unknown density f(x) we take the statistics ′ defined as in Theorem 1 and in Theorem 2 as well.
It is clear that, in Theorems 1 and 2 spline estimations are constructed with different boundary conditions.Theorem 3 is devoted to asymptotic unbiasedness of the spline estimation.Also for completeness of the results the dispersion and the covariance of the spline-estimation are given.
In the main Theorem 4 necessity and sufficiency conditions for strong consistency of the spline-estimation are given.
Similar result for the Persen-Rozenblatt estimation is obtained in the book of Nadaraya (1983) [2].More detailed review on spline estimation is given in works of Wegman, Wright [3], Muminov [4].

Auxiliary Results
Using the results of the work Lii [5] the following theorems are easily proved.

Theorem 1
Let F n (x) be empirical function of the distribution constructed by simple sample 1 2 , , , N X X X  and S N (x) be cubic spline interpolating the values F n (x k ) in the nodes of the mesh (1).If we choose the boundary conditions for S N (x) in the form , , , , , , 0, are defined by the following relations: ) ) ) for the other i and j.

Theorem 2
Let F n (x) be empirical function of the distribution constructed by simple sample 1 2 , , , n X X X  and S N (x) be cubic spline interpolating the values F n (x k ). in the mesh (1).If we choose the boundary conditions for S N (x) in the form , , and C i,j are defined by formula (2).We introduce the following denotations: is the simple sample from the general population is the sequence of wiener processes; We give the auxiliary lemmas.

Lemma 1 [6]
There exists a probability space (Ω, F, P).On which it can be defined version ( )

Lemma 2 [7]
Let ω be modulus of continuity of the brownian bridge B n (t), ( ) ( ) . Then with probability 1 ω does not exceed the quantity ( ) .
Here v ε is the random variable which is not less than 1 almost everywhere and 4 2 Mv ε < .

Main Results and Proofs
The following theorem characterizes the asymptotic behavior of the bias, the covariance and the dispersion of the spline estimation.

Theorem 3
Let ( ) 3 2, , , [y] is the integer part of the number y.

Theorem 4
Suppose ln 0 n nh → as n → ∞ .Then in order with probability 1 ( ) ( ) it is necessary and sufficient that the function g(x) is the density of the distribution F(x) concentrated and continuous on the interval [0,1] with respect to Lebesgue measure.Proof.Sufficiency.It is clear that where ( ) ( ) ( ) ( ) First we estimate the term N ε in the right hand part of (3).We have From Lemma 1 it follows that with probability 1 for n → ∞ ( ) ( ) If we denote the modulus of continuity where ( ) By virtue of (7) it is easy to see that the sequence of functions where N is a natural number.Let P k be the set of polynomials of degree ≤ k and С k [a, b] be the set of continuous on the [a, b] functions having continuous derivative of order k, 1, 2, k = .In the book of Stechkin and Subbotin[1] the following is given.Definition.interpolation cubic spline with respect to the mesh (1) for the function F(x), if: a) The points { } i x are called by the nodes of the spline.Later on for convenience we let [ ] [ ] , 0,1 a b = and the obtained results will remain valid for any finite interval [a, b].
function is defined by the equality of the spline function is defined by the equality

1 (
This, combining (3)-(6) and using Theorem 3 we get the sufficiency condition of Theorem 4. Necessity.Let with probability of g(x) on the interval [0, 1].Therefore, the sequence random variables some continuous function g 0 (x), i.e. for n → continuity of F(x) on the interval [0, 1].We assume the inverse that there exists a point x 0 ,