^{1}

^{2}

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.

We assume that on the interval

is given, 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,

Definition. The function

a)

b)

c)

Here

The points

Later on for convenience we let

Let _{N}(x) be cubic spline interpolating the values y_{k} = F_{n}(x_{k}) in the points x_{k} = kh, _{(n)} with “boundary conditions”

Here F_{n}(x) is the empirical function of the distribution of the sample

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

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) [

More detailed review on spline estimation is given in works of Wegman, Wright [

Using the results of the work Lii [

Let F_{n}(x) be empirical function of the distribution constructed by simple sample _{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

then the derivative

Here

and

C_{i}_{,j}(x) are defined by the following relations:

where

Let F_{n}(x) be empirical function of the distribution constructed by simple sample _{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

Then the derivative

where

and C_{i}_{,j} are defined by formula (2).

We introduce the following denotations:

We give the auxiliary lemmas.

There exists a probability space (Ω, F, P).

On which it can be defined version _{n}(t) such that for all x > 0

where a = 3.26, b = 4.86, с = 2.70.

Let _{n}(t),

and

Here

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

Let

1) If

2) If

where 0 < x < 1,

[y] is the integer part of the number y.

3) Suppose

Proof. By virtue of

Suppose

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

From Lemma 1 it follows that with probability 1 for

If we denote the modulus of continuity

Lemma 2

where

with probability

This, combining (3)-(6) and using Theorem 3 we get the sufficiency condition of Theorem 4.

Necessity. Let with probability 1

Hence, from continuity of

Therefore, the sequence random variables

are uniformly integrable. Therefore according to Theorem 5 from Shiryaev [

it follows that for

By virtue of (7) it is easy to see that the sequence of functions

uniformly converges to some continuous function g_{0}(x), i.e. for

We show now continuity of F(x) on the interval [0, 1].

We assume the inverse that there exists a point x_{0},

it follows continuity of F(x) on the interval [0, 1].

By (8) for all

From another side, according to Theorem 11 from Stechkin and Subbotin (1976)

By virtue of (9)-(11)

Theorem 4 is proved.

Mukhammadjon S. Muminov,Khaliq S. Soatov, (2016) Strong Consistency of the Spline-Estimation of Probabilities Density in Uniform Metric. Open Journal of Statistics,06,373-379. doi: 10.4236/ojs.2016.62032