Journal of Probability and Statistics

Volume 2012 (2012), Article ID 593036, 18 pages

doi:10.1155/2012/593036

Research Article

A Criterion for the Fuzzy Set Estimation of the Regression Function

Jesús A. Fajardo

Departamento de Matemáticas, Universidad de Oriente, Cumaná 6101, Venezuela

Received 1 May 2012; Accepted 30 June 2012

Academic Editor: A. Thavaneswaran

Copyright © 2012 Jesús A. Fajardo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We propose a criterion to estimate the regression function by means of a nonparametric and fuzzy set estimator of the Nadaraya-Watson type, for independent pairs of data, obtaining a reduction of the integrated mean square error of the fuzzy set estimator regarding the integrated mean square error of the classic kernel estimators. This reduction shows that the fuzzy set estimator has better performance than the kernel estimations. Also, the convergence rate of the optimal scaling factor is computed, which coincides with the convergence rate in classic kernel estimation. Finally, these theoretical findings are illustrated using a numerical example.

1. Introduction

The methods of kernel estimation are among the nonparametric methods commonly used to estimate the regression function , with independent pairs of data. Nevertheless, through the theory of point processes (see e.g, Reiss [1]) we can obtain a new nonparametric estimation method, which is based on defining a nonparametric estimator of the Nadaraya-Watson type regression function, for independent pairs of data, by means of a fuzzy set estimator of the density function. The method of fuzzy set estimation introduced by Falk and Liese [2] is based on defining a fuzzy set estimator of the density function by means of thinned point processes (see e.g, Reiss [1], Section 2.4); a process framed inside the theory of the point processes, which is given by the following:

where is a scaling factor (or bandwidth) such that as , and the random variables , , are independent with values in , which decides whether belongs to the neighborhood of or not. Here is the point of estimation (for more details, see Falk and Liese [2]). On the other hand, we observe that the random variables that define the estimator do not possess, for example, precise functional characteristics in regards to the point of estimation. This absence of functional characteristics complicates the evaluation of the estimator using a sample, as well as the evaluation of the fuzzy set estimator of the regression function if it is defined in terms of .

The method of fuzzy set estimation of the regression function introduced by Fajardo et al. [3] is based on defining a fuzzy set estimator of the Nadaraya-Watson type, for independent pairs of data, in terms of the fuzzy set estimator of the density function introduced in Fajardo et al. [4]. Moreover, the regression function is estimated by means of an average fuzzy set estimator considering pairs of fixed data, which is a particular case if we consider independent pairs of nonfixed data. Note that the statements made in Section in Fajardo et al. [3] are satisfied if independent pairs of nonfixed data are considered. This last observation is omitted in Fajardo et al. [3]. It is important to emphasize that the fuzzy set estimator introduced in Fajardo et al. [4], a particular case of the estimator introduced by Falk and Liese [2], of easy practical implementation, will allow us to overcome the difficulties presented by the estimator and satisfy the almost sure, in law, and uniform convergence properties over compact subsets on .

In this paper we estimate the regression function by means of the nonparametric and fuzzy set estimator of the Nadaraya-Watson type, for independent pairs of data, introduced by Fajardo et al. [3], obtaining a significant reduction of the integrated mean square error of the fuzzy set estimator regarding the integrated mean square error of the classic kernel estimators. This reduction is obtained by the conditions imposed on the thinning function, a function that allows to define the estimator proposed by Fajardo et al. [4], which implies that the fuzzy set estimator has better performance than the kernel estimations. The above reduction is not obtained in Fajardo et al. [3]. Also, the convergence rate of the optimal scaling factor is computed, which coincides with the convergence rate in classic kernel estimation of the regression function. Moreover, the function that minimizes the integrated mean square error of the fuzzy set estimator is obtained. Finally, these theoretical findings are illustrated using a numerical example estimating a regression function with the fuzzy set estimator and the classic kernel estimators.

On the other hand, it is important to emphasize that, along with the reduction of the integrated mean square error, the thinning function, introduced through the thinned point processes, can be used to select points of the sample with different probabilities, in contrast to the kernel estimator, which assigns equal weight to all points of the sample.

This paper is organized as follows. In Section 2, we define the fuzzy set estimator of the regression function and we present its properties of convergence. In Section 3, we obtain the mean square error of the fuzzy set estimator of the regression function, Theorem 3.1, as well as the optimal scale factor and the integrated mean square error. Moreover, we establish the conditions to obtain a reduction of the constants that control the bias and the asymptotic variance regarding the classic kernel estimators; the function that minimizes the integrated mean square error of the fuzzy set estimator is also obtained. In Section 4 a simulation study was conducted to compare the performances of the fuzzy set estimator with the classical Nadaraya-Watson estimators. Section 5 contains the proof of the theorem in the Section 3.

2. Fuzzy Set Estimator of the Regression Function and Its Convergence Properties

In this section we define by means of fuzzy set estimator of the density function introduced in Fajardo et al. [4] a nonparametric and fuzzy set estimator of the regression function of Nadaraya-Watson type for independent pairs of data. Moreover, we present its properties of convergence.

Next, we present the fuzzy set estimator of the density function introduced by Fajardo et al. [4], which is a particular case of the estimator proposed in Falk and Liese [2] and satisfies the almost sure, in law, and uniform convergence properties over compact subset on .

Definition 2.1. Let be an independent random sample of a real random variable with density function . Let be independent random variables uniformly on distributed and independent of ,. Let be such that and , . Then the fuzzy set estimator of the density function at the point is defined as follows:

where

Remark 2.2. The events , , can be described in a neighborhood of through the thinned point process

where

and decides whether belongs to the neighborhood of or not. Precisely, is the probability that the observation belongs to the neighborhood of . Note that this neighborhood is not explicitly defined, but it is actually a fuzzy set in the sense of Zadeh [5], given its membership function . The thinned process is therefore a fuzzy set representation of the data (see Falk and Liese [2], Section 2). Moreover, we can observe that and the random variable is binomial , distributed with

In what follows we assume that .

Now, we present the fuzzy set estimator of the regression function introduced in Fajardo et al. [3], which is defined in terms of .

Definition 2.3. Let be independent copies of a random vector , where are independent random variables uniformly on distributed, and independent of ,. The fuzzy set estimator of the regression function at the point is defined as follows:

Remark 2.4. The fact that , , , is a kernel when is a density does not guarantee that is equivalent to the Nadaraya-Watson kernel estimator. With this observation the statement made in Remark by Fajardo et al. [3] is corrected. Moreover, the fuzzy set representation of the data is defined over the window with thinning function , where denotes the neighborhood of . In the particular case , , the fuzzy set representation of the data comes given by .

Consider the following conditions.

 (C1) Functions and are at least twice continuously differentiable in a neighborhood of . (C2).  (C3) Sequence satisfies: . (C4) Function is symmetrical regarding zero, has compact support on , , and it is continuous at with . (C5) There exists such that   . (C6) Function is at least twice continuously differentiable in a neighborhood of . (C7), as . (C8) Function is monotone on the positives. (C9) and , as . (C10) Functions and are at least twice continuously differentiable on the compact set . (C11) There exists such that .

Next, we present the convergence properties obtained in Fajardo et al. [3].

Theorem 2.5. Under conditions (C1)–(C5), one has

Theorem 2.6. Under conditions (C1)–(C7), one has

The “” symbol denotes convergence in law.

Theorem 2.7. Under conditions (C4)–(C5) and (C8)–(C11), one has

Remark 2.8. The estimator has a limit distribution whose asymptotic variance depends only on the point of estimation, which does not occur with kernel regression estimators. Moreover, since we see that the same restrictions are imposed for the smoothing parameter of kernel regression estimators.

3. Statistical Methodology

In this section we will obtain the mean square error of , as well as the optimal scale factor and the integrated mean square error. Moreover, we establish the conditions to obtain a reduction of the constants that control the bias and the asymptotic variance regarding the classic kernel estimators. The function that minimizes the integrated mean square error of is also obtained.

The following theorem provides the asymptotic representation for the mean square error () of . Its proof is deferred to Section 5.

Theorem 3.1. Under conditions (C1)–(C6), one has

where

with

Next, we calculate the formula for the optimal asymptotic scale factor to perform the estimation. The integrated mean square error of is given by the following:

From the above equality, we obtain the following formula for the optimal asymptotic scale factor

We obtain a scaling factor of order , which implies a rate of optimal convergence for the of order . We observe that the optimal scaling factor order for the method of fuzzy set estimation coincides with the order of the classic kernel estimate. Moreover,

where

with

Next, we will establish the conditions to obtain a reduction of the constants that control the bias and the asymptotic variance regarding the classic kernel estimators. For it, we will consider the usual Nadaraya-Watson kernel estimator

which has the mean squared error (see e.g, Ferraty et al. [6], Theorem 2.4.1)

where

Moreover, the of is given by the following:

From the above equality, we obtain the following formula for the optimal asymptotic scale factor

Moreover,

where

The reduction of the constants that control the bias and the asymptotic variance, regarding the classic kernel estimators, are obtained if for all kernel

Remark 3.2. The conditions on allows us to obtain a value of such that

Moreover, to guarantee that

we define the function

with compact support on . Next, we guarantee the existence of . As

we have

Observe that for each exists

such that

Combining (3.21) and (3.23), we obtain

In our case we take .

On the other hand, the criterion that we will implement to minimizing (3.6) and obtain a reduction of the constants that control the bias and the asymptotic variance regarding the classic kernel estimation, is the following

subject to the conditions

with , , and , where is the Epanechnikov kernel

The Euler-Lagrange equation with these constraints is

where , , and the three multipliers corresponding to the three constraints. This yields

The new conditions on , allows us to affirm that for all kernel

Thus, the fuzzy set estimator has the best performance.

4. Simulations

A simulation study was conducted to compare the performances of the fuzzy set estimator with the classical Nadaraya-Watson estimators. For the simulation, we used the regression function given by Härdle [7] as follows:

where the were drawn from a uniform distribution based on the interval . Each has a normal distribution with mean and variance. In this way, we generated samples of size , , and . The bandwidths was computed using (3.5) and (3.13). The fuzzy set estimator and the kernel estimations were computed using (3.29), and the Epanechnikov and Gaussian kernel functions. The values of the fuzzy set estimator and the kernel estimators are given in Table 1.

Table 1: IMSE* values of the estimations for the fuzzy set estimator and the kernel estimators.

*Minimum IMSE* in each row.

As seen from Table 1, for all sample sizes, the fuzzy set estimator using varying bandwidths have smaller values than the kernel estimators with fixed and different bandwidth for each estimator. In each case, it is seen that the fuzzy set estimator has the best performance. Moreover, we see that the kernel estimation computed using the Epanechnikov kernel function shows a better performance than the estimations computed using the Gaussian kernel function.

The graphs of the real regression function and the estimations of the regression functions computed over a sample of , using points and , are illustrated in Figures 1 and 2.

Figure 1: Estimation of with and .

Figure 2: Estimation of with and .

5. Proof of Theorem 3.1

Proof. Throughout this proof will represent a positive real constant, which can vary from one line to another, and to simplify the annotation we will write instead of . Let us consider the following decomposition

Next, we will present two equivalent expressions for the terms to the right in the above decomposition. For it, we will obtain, first of all, an equivalent expression for the expectation. We consider the following decomposition (see e.g, Ferraty et al. [6])

Taking the expectation, we obtain

where

The hypotheses of Theorem 3.1 allow us to obtain the following particular expressions for and , which are calculated in the proof of Theorem in Fajardo et al. [3]. That is

Combining the fact that , , are identically distributed, with condition , we have

On the other hand, by condition there exists such that . Thus, we can write

Note that

Thus, we can write

Note that by condition the density is bounded in the neighborhood of . Moreover, condition allows us to suppose, without loss of generality, that and by (2.5) we can bound . Therefore,

Now, we can write

The above equalities, imply that

Once more, the hypotheses of Theorem 3.1 allow us to obtain the following general expressions for and , which are calculated in the proofs of Theorem in Fajardo et al. [3, 4], respectively. That is

By conditions and , we have that

Then

Next, we will obtain an equivalent expression for . Taking the conjugate, we have

where

By condition , we have

So that,

Now, we can write

By condition , we have

where

Therefore,

Next, we will obtain an expression for the variance in (5.1). For it, we will use the following expression (see e.g., Stuart and Ord [8])

Since that are i.i.d and the are i.i.d, , we have

the last equality because is binomial distributed. Remember that

Moreover, the hypothesis of Theorem 3.1 allow us to obtain the following expression

which is calculated in the proof of Lemma in Fajardo et al. [3]. By condition , we have

Remember that

Thus,

for . Finally, we saw that

Therefore,

Thus,

where

We can conclude that,

where

Acknowledgment

The author wants to especially thank the referees for their valuable suggestions and revisions. He also thanks Henrry Lezama for proofreading and editing the English text.

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