﻿Asymptotic Analysis for U-Statistics and Its Application to Von Mises Statistics

Open Journal of Statistics
Vol.1 No.3(2011), Article ID:8065,6 pages DOI:10.4236/ojs.2011.13016

Asymptotic Analysis for U-Statistics and Its Application to Von Mises Statistics

Timur Zubayraev

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

E-mail: tzubayraev@gmail.com

Received July 30, 2011; revised August 31, 2011; accepted September 14, 2011

Keywords: U-Statistics, Von Mises Statistics, Symmetric Statistics

Abstract

Let be i.i.d. random variables taking values in a measurable space (). Let: and: be measurable functions. Assume that is symmetric, i.e., for any. Consider U-statistic, assuming that, for all, ,. We will provide bounds for , where is a distribution function of and, are its limiting distribution function and Edgeworth correction respectively. Applications of these results are also provided for von Mises statistics case.

1. Introduction

Consider the measurable space (), with measure. Let denote the real Hilbert space of square integrable real functions. Let denote the Hilbert-Schmidt operator associated with the kernel and defined via

Let be its eigenvalues. Without loss of generality we shall assume that.

Let denote an orthonormal complete system of eigenfunctions of of the corresponding eigenvalues. Then

(1.1)

since is a Hilbert-Schmidt operator and the kernel is degenerate. The series in (1.1) converges in . Consider the subspace generated by and eigenfunctions corresponding to nonzero eigenvalues. Introducing, if necessary, an eigenvalues ,we can assume that is an orthonormal basis in . Thus, we have

in, ,(1.2)

with and, for all j. Therefore is an orthonormal system of random variables with zero means.

Hilbert space consists of , such that

.

Consider the random vector

,(1.3)

which takes values in. Since is a system of mean zero uncorrelated random variables with variances 1, the random vector has mean zero and and is Kronecker’s symbol. Using (1.1) and (1.2), we can write

, , (1.4)

where we define for and. The equalities (1.4) allow us to assume that the measurable space is. Let be a random vector taking values in with mean zero and covariance and that

,. (1.5)

Without loss of generality we shall assume that the kernels and are linear functions in each of their arguments ([2]).

Introduce the definitions:

, , , and assume that

,.

for the statistic we can write

.

The statistic is called degenerate since ensures that the quadratic part of the statistic is not asymptotically negligible and therefore statistic is not asymptotically normal. More precisely, the asymptotic distribution of is non-Gaussian and is given by the distribution of the random variable

,(1.6)

is a sequence of i.i.d. standard normal variables, denotes a sequence of square summable weights and denote eigenvalues of the Hilbert-Schmidt operator, say, associated with the kernel.

Consider the concentration functions of statistic

, ,(1.7)

where is an arbitrary statistic depending only on, is as well arbitrary but independent of. Note that the class of statistics is slightly more general than the class of statistics T. We shall denote constants. If a constant depends on, say s, we shall write.

Consider the distibution functions

, , , ,

denotes an Edgeworth correction.The Edgeworth correction is defined as a function of bounded variation satisfying and with the Fourier-Stieltjes transform given by

.

Let us notice that vanishes if or if

(1.8)

holds for all. Using the technique presented in this work we may obtain the result for approximation bound of order for U-statistic distribution function which has an order (see Theorem 3, 2) below) or (see Theorem 3, 1) below) with respect to dependence on first nine or thirteen eigenvalues of operator, respectively.

2. Auxiliary Results

Consider the vector with values in, where standard normal variables. Let us formulate lemma in which equalities for the moments of determinants of random matrices consisting of the scalar products such as are obtained. Analogue of this lemma is proved in [1] for matrices consisting of the scalar products such as where G-Gaussian vector.

Lemma 1. Let be random elements in a Hilbert space such that, where standard normal variables. Let be the eigenvalues of Hilbert-Schmidt operator., where,.

Then

,.

Nondegeneracy condition

We shall assume that random vector, a kernel, parameters and satisfy the nondegeneracy condition if

, ,

, , ,(2.1)

where, , , are independent copies of.

Here parameter is small and parameter is close to 1. Let denote the set of all vectors satisfying the nondegeneracy condition.

Notice that satisfies the nondegeneracy condition. Let vectors and have equal means and covariances, then

, ,

,

.

The following Lemma 2 means that increase of yields equivalence of nondegeneracy conditions fulfillments for sum and Gaussian vector.

Lemma 2. Let be a Gaussian random vector and

. Then for we have, where is random sum.

Further, it is necessary to bound the characteristic function of the statistic. That will be done in Lemmas 3, 4 and Theorem 1.

The following Lemma 3 has a similar proof to Lemma 6.5 from [2].

By we shall denote independent copies of a symmetric random variable with nonnegative characteristic function and such that

,.(2.2)

Lemma 3. Let and. Assume that vector takes values in. Write

, , where and are independent copies of Y. Then

, , where denotes the supremum over all nonrandom matrices such that.

U and V denote independent vectors in which are sums of n independent copies of.

In the following lemma the bound from above for the characteristic function is received. This results was proved in [1]. The received estimation contains the determinant of matrix in right-hand side of inequality. This fact allows to use eigenvalues of operator for the estimation of characteristic function.

Lemma 4. Let A be a nondegenerate matrix. Let denote a random vector with covariance. Assume that there exists a constant such that

, ,.(2.3)

Let and denote independent random vectors which are sums of n independent copies of. Then

for, where for.

Using our Lemmas 3 and 4 we may obtain a bound for characteristic function for statistic.

Theorem 1. Let. Assume that the sum . Then, for any statistic we have

.

The proof of this theorem is similar to proof of Theorem 6.2 in [2].

Write :

.(2.4)

In following lemma a multiplicative inequality for characteristic function of is given. This inequality yields the desired bound for an integral of the characteristic function of a U-statistic. Similar result was proved in Lemma 7.1 in [2]

Lemma 5. Let and. Assume that . Then there exist constants and such that the event

,(2.5)

satisfies

.(2.6)

For, define the integrals

, where denotes the Fourier-Stieltjes transform of the distribution function. The estimation for these integrals is received in following lemma, which has a proof similar to Lemma 3.3 in [2].

Lemma 6. Let. Assume that the random vector

and. Let

, , , where, are some positive constants.

Then

,.(2.7)

3. Approximation Accuracy Estimation

For and functions, introduce the statistic

(3.1)

where

for, for.

Write and put

,(3.2)

where

,(3.3)

,(3.4)

where supremum is taken over all linear statistics, that is, over all functions which can be represented as with some functions .

Consider the following Lemma 7, which has a similar proof as Lemma 4.2 in [2].

Lemma 7. Let, and . Assume that the random vector satisfies the nondegeneracy condition. Then, for, the distribution function of satisfies

,(3.5)

where.

The Edgeworth correction is defined as a function of bounded variation satisfying and with the Fourier-Stieltjes transform given by

.(3.6)

Lemma 8. Assume that the nondegeneracy condition is fulfilled.

1) Let and . Then

(3.7)

2) Assume that the condition (1.8) holds and that. Then

(3.8)

To prove this lemma we need to make the same steps as in Lemma 4.1 in [2] replacing Theorem 6.2 by Theorem 1.

Now we can formulate a following Theorem 2, where bounds for are obtained. This theorem were proved in [4]:

Theorem 2. 1) Let

,. Then

(3.9)

2) Assume that (1.8) holds and. Then

(3.10)

4. An Extension of Bounds to Von Mises Statistics. Applications

Assuming that the kernels and are degenerate, consider the von Mises statistic

.(4.1)

Introducing the function with, we can rewrite (4.1) as

(4.2)

In this section we shall extend the bounds to statistics of type (4.2), assuming that and .

Similarly to the case of, we can represent the kernel (respectively, and) as a bilinear (respectively, linear) function, defined on. However in this case we have to assume that has an additional coordinate since can be linearly independent of and of the eigenfunctions of. To fix notation, we shall assume that consists of vectors . Then all representations and results of Section 2 concerning and still hold, and for we have with some such that. Write.

Introduce the function of bounded variation (provided that) with the Fourier-Stieltjes transform

and such that. Bellow we shall show that (see Lemma 9.3 [2])

.(4.3)

Notice that whenever.

Write, and let denote the distribution function of. Define

,.

Theorem 3. 1) Assume that. Then we have

(4.4)

2) Assume that (1.8) is fulfilled and. Then we have

Proof. We shall use the following estimates. Write

,.(4.5)

Expanding with remainder, splitting the sum in parts and conditioning , we have

.(4.6)

Proceeding similarly to the proof of Lemma 8.2 from [2], we obtain

.(4.7)

Applying the Bergstrom-type identity

,

with and proceeding similarly to the proof of Lemma 8.3 from [2], we get

(4.8)

Arguments similar to the proof of Lemma 8.5 from [2] allow proving

,(4.9)

and, for,

, (4.10)

.(4.11)

The estimates (4.6)-(4.11) allow proceeding similarly to the proof of Theorem 2, using a lemma similar to Lemma 8. Proving such a lemma, we have to apply Lemma 8 to the distribution function. This is possible since that statistic is a statistic of type (3.1). The estimates (4.10) and (4.11) allow application of the Fourier inversion to the function. As a result, we arrive at

.

Here, however, we have, and

(4.12)

Therefore, using (4.6)-(4.8), we can proceed as in the proof of Lemma 11. As a final result we get bounds similar to those of Theorem 2, with the additional summand.

5. References

[1]    V. Ulyanov and F. Götze, “Uniform Approximations in the CLT for Balls in Euclidian Spaces,” 00-034, SFB 343, University of Bielefeld, 2000, p. 26. http://www.math.uni-bielfeld.de/sfb343/preprints/pr00034.pdf.gz

[2]    V. Bentkus and F. Götze, “Optimal Bounds in NonGaussian Limit Theorems for U-Statistics,” The Annals of Probability, Vol. 27, No.1, 1999, pp. 454-521. doi:10.1214/aop/1022677269

[3]    S. A. Bogatyrev, F. Götze and V. V. Ulyanov, “NonUniform Bounds for Short Asymptotic Expansions in the CLT for Balls in a Hilbert Space,” Journal of Multivariate Analysis, Vol. 97, 2006, pp. 2041-2056.

[4]    T. A. Zubayraev, “Asymptotic Analysis for U-Statistics: Approximation Accuracy Estimation,” Publications of Junior Scientists of Faculty of Computational Mathematics and Cybernetics, Moscow State University, Vol. 7, 2010, pp. 99-108. http://smu.cs.msu.su/conferences/sbornik7/smu-sbornik-7.pdf