Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

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DOI: https://doi.org/10.5281/zenodo.11540455

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Boundary Estimation with the Fuzzy Set

Regression Estimator

Estimaci´on Frontera con el Estimador de Regresi´on con Conjunto Difuso

Jes´us A. Fajardo

(jfajardogonzalez@gmail.com;jfajardo@udo.edu.ve)

Departamento de Matem´atica del N´ucleo de Sucre

Universidad de Oriente

Cuman´a 6101, Rep´ublica Bolivariana de Venezuela

Abstract

In order to extend the properties of the fuzzy set regression estimation method and provide

new results related to the nonparametric regression estimation problems not based on ker-

nels, this paper analyzes the possible boundary eﬀects, if any, of the fuzzy set regression

estimator and presents a criterion to remove it. Moreover, a boundary fuzzy set estimator

is proposed which is deﬁned as a particular class of fuzzy set regression estimators, where

the bias, variance, mean squared error and function that minimizes the mean squared er-

ror of the proposed estimator are given. Finally, these theoretical ﬁndings are illustrated

through some numerical examples, and with one real data example. Simulations show that

the proposed estimator has better performance at points near zero in a spread neighborhood

of the smoothing parameter, when it is compared with a general boundary kernel regression

estimator for the two regression models and two density functions considered. The previ-

ously exposed represents the natural extension of the recent results to the boundary fuzzy

set density estimator case.

Palabras y frases clave: Fuzzy set regression estimator, boundary estimation.

Resumen

Con el ﬁn de ampliar las propiedades del m´etodo de estimaci´on de regresi´on con conjunto

difuso y proporcionar nuevos resultados relacionados con los problemas de estimaci´on no

param´etrica de la regresi´on no basados en n´ucleos, este art´ıculo analiza los posibles efectos

frontera, si los hay, del estimador de regresi´on con conjunto difuso y presenta un criterio para

eliminarlo. Adem´as, se propone un estimador frontera con conjunto difuso el cual se deﬁne

como una clase particular de estimadores de regresi´on con conjunto difuso, donde el sesgo,

la varianza, el error cuadr´atico medio y la funci´on que minimiza el error cuadr´atico medio

del estimador propuesto son presentados. Finalmente, estos resultados te´oricos se ilustran

a trav´es de algunos ejemplos num´ericos y con un ejemplo de datos reales. Las simulaciones

muestran que el estimador propuesto tiene un mejor desempe˜no en los puntos cercanos a

cero en un entorno disperso del par´ametro de suavizado, cuando se compara con un esti-

mador general frontera de la regresi´on con n´ucleo para los dos modelos de regresi´on y las

dos funciones de densidad consideradas. Lo expuesto anteriormente representa la extensi´on

Recibido 04/05/20023. Revisado 14/7/2023. Aceptado 12/12/2023.

MSC (2010): Primary 62G99; Secondary 62G05.

Autor de correspondencia: Jes´us Fajardo.

Boundary estimation 83

natural de los resultados recientes al caso del estimador frontera de la densidad con conjunto

difuso.

Key words and phrases: Estimador de regresi´on con conjunto difuso, estimaci´on frontera.

1 Introducci´on

Let (X

, Y

), . . . , (X

, Y

) be n independent copies of a random vector (X, Y ). The regression

model is given by

= r(X

) + ε

, i = 1, . . . , n,

where the observation errors ε

, . . . , ε

are random variables such that

E[ε

|X] = 0, V ar[ε

|X] = σ

< ∞, i = 1, . . . , n,

and unknown regression function r(t) = E[Y |X = t], for t ∈ R. The main goal of this paper is

to introduce a new method to estimate the regression function r at points near zero in a spread

neighborhood of the smoothing parameter.

Numerous nonparametric methods have been developed in the literature to estimate r, with

independent pairs of data. Nevertheless, most of those estimation methods are not consistent

when the support of r has ﬁnite endpoints, seriously aﬀecting the overall performance of the

implemented estimators. The lack of consistency is a consequence of the so called “boundary

eﬀects,” where the connection between the estimation methods and boundary eﬀects is reﬂected

in the performances of the proposed estimators for each method. This makes their performances

at boundary points usually to diﬀer from the performances at interior points. Theoretically, the

convergence rates of the estimators at boundary points are slower than the convergence rates at

interior points of the support of r. Strictly speaking, the bias of the estimators is of order O(b

)

instead of O(b

) at boundary points, where b

is the smoothing parameter or bandwidth (b

→ 0

as n → ∞). This imposes the need to carry out a study to detect whether the boundary eﬀects

are present or not in the estimator of any function, since it is not obvious that the behavior of the

estimator is the same at both the boundary and the interior points. To remove those boundary

eﬀects in kernel regression estimation, a variety of methods have been developed in the literature.

An excellent summary of some well-known methods is given in [24]. Finally, it is important to

remark here that the above phenomenon, called “boundary eﬀects” in the estimation theory, also

aﬀects the fuzzy set regression estimator introduced in [9].

In this paper, a criterion to remove the boundary eﬀects, without boundary modiﬁcations,

in the nonparametric fuzzy set regression estimator is proposed, obtaining a natural extension

of the approach introduced in [8] to the regression case. For this, at each point near 0 in a b

spread neighborhood, the proposed boundary estimator is deﬁned as a particular class of fuzzy

set regression estimators, where the bias, variance and optimal mean squared error (MSE) are

given. Moreover, the function that minimizes the M SE is obtained. Simultaneously, extensive

simulations are carried out to compare the local MSE of the proposed boundary estimator with

the local MSE of the general boundary estimator given in [24] at points near 0 in a b

spread

neighborhood, observing that the local MSE of the proposed boundary estimator is the smallest

for the two regression models and the two density functions considered. This reduction shows that

the boundary fuzzy set regression estimator has better performance than the estimator studied

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

84 Jes´us A. Fajardo

in [24]. Besides, it is appropriate to note that the above results extend the properties of the

fuzzy set regression estimation method, providing new properties related to the nonparametric

regression estimation problems not based on kernels.

The particular choice above was based mainly on the results of the simulations obtained

in [24], for the two regression models and the two density functions considered in this work,

which showed that the general boundary kernel regression estimator deﬁned in the above paper

performed quite well when it was compared with both local linear and classical kernel regression

estimators. Among other reasons that supported the above particular choice, the theoretical

properties that are shared by the boundary estimator deﬁned in [24] and the proposed boundary

estimator are highlighted: non-negativity, “natural boundary continuation” and they improve the

bias but holding on to the low variances. Moreover, it is worth pointing out that the paper [24]

extends the approach introduced in [17] to the regression case, by deﬁning the popular Nadaraya-

Watson estimator, [20, 27], in terms of the boundary kernel density estimator given in [17]. It is

worth noting that the results of the simulations presented in [17] for the four shapes of densities

considered showed that the boundary kernel density estimator introduced in the above work

performed quite well when it was compared with the estimators boarded in [16, 29] and its

simple modiﬁcation which allows obtaining the local linear ﬁtting estimator [13,30]. Nonetheless,

the results of the simulations obtained in [8], for the four shapes of densities considered in [17],

showed that the boundary fuzzy set density estimator performed quite well when it was compared

with the boundary kernel estimator deﬁned in [17]. Now, combining this last result with the

idea established in [24], it is reasonable to deﬁne an estimator of the Nadaraya-Watson type

regression function in terms of the boundary fuzzy set density estimator, in order to achieve the

objectives emphasized in this paper and to solve the problem proposed in [8]. On the other hand,

a literature review on the proposed topic revealed that there is not evidence of publications

with respect to the comparison of the performance between other methods and the method

introduced in [24]. Besides, the author guarantees a conclusion analogous to the previous one for

the fuzzy set regression estimator case. Finally, it is necessary to point out that in the recent

works [1–4, 15, 18, 19, 25] the problems of nonparametric regression estimation are studied under

speciﬁc conditions and new regression estimators are introduced through the approach of each

previous work. It should be noted that the method introduced in [15] combines the smoothing

spline and kernel functions. Nonetheless, in the papers [1, 3, 4, 18] and [2,19,25] both Nadaraya-

Watson and local linear estimators are the main actors, respectively. This last point suggests

the combination of the approaches in the works [2, 19, 25] and [7] to future research, since in [7]

was shown that the fuzzy set regression estimator has better performance than the local linear

regression smoothers.

This paper is organized as follows. In Section 2, the boundary eﬀects in the fuzzy set regre-

ssion estimator are studied and the criterion to remove such eﬀects is presented. Moreover, the

boundary fuzzy set regression estimator is deﬁned and its asymptotic properties are introduced.

Besides, the function that minimizes the M SE of the proposed boundary estimator is calculated.

The simulation studies and data analysis are introducen in Sections 3 and 4, respectively. Final

comments are given in Section 5.

2 Fuzzy set regression estimator and boundary eﬀects

A study to detect the presence or not of the boundary eﬀects in the estimator of any function

is necessary since it is not obvious that the behavior of the estimator can be the same at the

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 85

boundary points as in the interior points. Consequently, this section analyzes the possible bounda-

ry eﬀects, if any, of the fuzzy set regression estimator given in [9], and introduces a criterion to

remove it, without boundary modiﬁcations. Moreover, the deﬁnition of the boundary fuzzy set

regression estimator and its asymptotic properties are given. Also, the function that minimizes

the MSE of the proposed boundary estimator is obtained.

2.1 Fuzzy set estimator of the regression function

It is important to emphasize that the fuzzy set regression estimation method introduced in [9]

is based on deﬁning an estimator of the Nadaraya-Watson type for independent pairs of data in

terms of the fuzzy set density estimator given in [10]. In order to familiarize the reader with the

above method, a general summary of the details that allowed to deﬁne the estimator introduced

in [10] will be presented.

For independent copies X

, . . . , X

of a random variable X in R, whose distribution L(X)

has a Lebesgue density f

near some ﬁxed point x

∈ R, a fuzzy set estimator of f

, at the point

∈ R, is deﬁned in [11], by means of thinned point processes N

, a process framed inside the

theory of the point processes, as follows

) =

(t)

(R)

n a

i=1

, t ∈ R,

where

n a

i=1

(t) = P (U

= 1| X

= t), ε

is the random Dirac measure, a

> 0 is a smoothing parameter

or bandwidth such that a

→ 0 as n → ∞, and the random variables U

, 1 ≤ i ≤ n, are

independent with values in {0, 1}, which determines whether X

belongs to the neighborhood of

or not. See e.g. [21], for more details on the theory of the point processes and thinned point

processes. In [11], only the asymptotic eﬃciency within the class of all estimators that are based

on randomly selected points from the sample X

, . . . , X

was proved. Eﬃciency was established

using LeCam’s LAN theory. Although the almost sure and uniform convergence properties over

a compact subset on R are not studied, the pointwise convergence in law whose distribution limit

has an asymptotic variance that depends only of f

) is proposed. On the other hand, it is

opportune to point out 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. Thus, the

simulations to estimate the density function will be more complicated. However, to overcome the

diﬃculties presented by the estimator introduced in [11], a particular case of the above estimator

was deﬁned in [10].

Let X be a real random variable whose distribution L(X) has a Lebesgue density f

near some

ﬁxed point x

∈ R. For each measurable Borel function ϕ : R → [0, 1] and each random variable

V , uniformly distributed on [0, 1] and independent of X, the random variable 1I

[0,ϕ(X)]

(V ) satisﬁes

ϕ(t) = P(1I

[0,ϕ(X)]

(V ) = 1|X = t). This simple observation allows us to construct a fuzzy set

estimator of f

to estimate f

), which satisﬁes the conditions required in [11]. Moreover, as

the local behavior of the distribution of X will be evaluated, it is obvious that only observations

in a neighborhood of x

can reasonably contribute to the estimation of f

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

86 Jes´us A. Fajardo

Let X

, . . . , X

be an independent random sample of X. Let V

, . . . , V

be independent ran-

dom variables uniformly distributed on [0, 1] and independent of X

, . . . , X

. Let 1I

0,ϕ



−x

i

)

be random variables where b

> 0 is a smoothing parameter or bandwidth such that b

→ 0 as

n → ∞. For each t ∈ R, one obtains that



t − x



= P



0,ϕ



−x

i

) = 1| X

= t



where ϕ

(t) = ϕ



t−x



is a Markov kernel (see [21], Section 1.4). Thus, for independent copies

, V

), 1 ≤ i ≤ n, of (X, V ), the thinned point process is deﬁned as follows

(·) =

i=1

0,ϕ



−x

i

) ε

(·),

with underlying point process N

(·) =

i=1

(·) and a thinning function ϕ

(see [21], Section

2.4), where ε

is the random Dirac measure.

On the other hand, observe that the set of observations or the events {X

= t}, t ∈

R, in a neighborhood of x

can now be described by the thinned point process N

, where

0,ϕ



−x

i

) decides, whether X

belongs to the neighborhood of x

or not. Precisely, ϕ

(t)

is the probability that the observation X

= t belongs to the neighborhood of x

. Note that this

neighborhood is not explicitly deﬁned, but it is actually a fuzzy set in the sense of the paper [28],

given its membership function ϕ

. The thinned process N

is therefore a fuzzy set representation

of the data (see [11], Section 2).

Next, the fuzzy set density estimator introduced in [10] is presented, which is a particular

case of the estimator proposed in [11].

Deﬁnition 1. Let X

, . . . , X

be an independent random sample of X. Let V

, . . . , V

be inde-

pendent random variables uniformly distributed on [0, 1] and independent of X

, . . . , X

. Let ϕ

be such that a

= b

ϕ(u) du and 0 <

ϕ(u) du < ∞. Then, the fuzzy set estimator of the

density function f

at the point x

∈ R is deﬁned as

) =

i=1

0,ϕ



−x

i

) =

). (1)

Observe that estimator (1) can be written in terms of a fuzzy set representation of the data, since

) = (na

)

−1

(R). This equality justiﬁes the fuzzy set term of estimator (1), where the

thinning function ϕ

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. Moreover,

it is important to highlight that (1) is of easy practical implementation and the random variable

) is binomial B (n, α

)) distributed with

) = E



0,ϕ



X−x

i

(V )



= P



0,ϕ



X−x

i

(V ) = 1



= E





X − x



. (2)

In what follows, it will be assumed that α

) ∈ (0, 1). Besides, throughout this article ϕ is the

thinning function that characterizes to estimator (1), and the abbreviation “w.t.f.” will be used

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 87

to the phrase “with thinning function.” See [6] and [10], for more details on the asymptotic and

convergence properties of estimator (1).

Next, the regression estimator of the Nadaraya-Watson type introduced in [9], for independent

pairs of data, is presented, which it is deﬁned in terms of estimator (1).

Deﬁnition 2. Let ((X

, Y

), V

), . . . , ((X

, Y

), V

) be independent copies of a random vector

((X, Y ), V ), where V

, . . . , V

are independent random variables uniformly distributed on [0, 1],

and independent of (X

, Y

), . . . , (X

, Y

). Let ϕ be such that a

= b

ϕ(u) du and 0 <

ϕ(u) du < ∞. Then, the fuzzy set estimator of the regression function r at the point x

∈ R is

deﬁned as

ˆr

) =











i=1



0,ϕ



−x



)

if τ

) 6= 0

0 if τ

) = 0,

(3)

where τ

is given in (1).

As in the case of estimator (1), the function ϕ characterizes estimator (3). See [5] and [9], for

more details on the asymptotic and convergence properties of estimator (3).

2.2 The boundary problem of the fuzzy set regression estimator and

its boundary estimator

In order to provide new results related to the nonparametric regression estimation problems not

based on kernels, and to ensure the natural extension of the approach to the boundary fuzzy

set density estimation case, this section studies the behavior of estimator (3), at the particular

points 0 and x ∈ (0, b

], under the following conditions:

C1 Functions f

and r are at least twice continuously diﬀerentiable in a neighborhood of

z ∈ [0, ∞).

C2 f

(z) > 0.

C3 Sequence b

satisﬁes: b

→ 0, nb

→ ∞, as n → ∞.

C4 Function ϕ is symmetrical regarding zero, has compact support on [−B , B], B > 0, and it

is continuous at 0 with ϕ(0) > 0.

C5 There exists M > 0 such that |Y | < M a.s.

C6 Function φ(z) = E[Y

|X = z] is at least twice continuously diﬀerentiable in a neighborhood

of z ∈ [0, ∞).

Next, the results associated with the behavior of the estimator

ˆg

(z) =

i=1

0,ϕ



−z

i

), z ∈ [0, ∞), (4)

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

88 Jes´us A. Fajardo

at the particular points 0 and x ∈ (0, b

], are presented. Here, g(z) = r(z)f

(z), a

= b

ϕ(u) du

and 0 <

ϕ(u) du < ∞. Moreover, the function ψ will be deﬁned as ψ(u) =

ϕ(u)

ϕ(u) du

[−B, B]

(u),

and as each x ∈ (0, b

] has the form x = c b

, 0 < c ≤ 1, in that which follows, to simplify the

notation, x will be written instead of c b

Theorem 1. Under the conditions (C1) − (C4), we have

E [ˆg

(0)] = g(0) +

(0)

ψ(u) du + o





Proof. Replacing z with 0 into (4), given that V and (X, Y ) are independent, and taking the

conditional expectation with respect to X = 0 and V = v, we obtain

E [ˆg

(0)] = a

−1

∞

−∞





g(u)du.

Combining (C1) and (C4), we can write the above equality as

E [ˆg

(0)] =

−

ψ(u) g (ub

) du.

As (C1) allows us to make a the Taylor expansion of function g in the neighborhood of 0, we can

rewrite the above equality as

E [ˆg

(0)] =

−



g(0) + b

(0)u +

(λub

)



ψ(u) du. (5)

Moreover, (C3) allows us to suppose, without loss of generality, that b

< 1. Now, we can

guarantee that for B > 0

> B. (6)

Combining (C3), (5) and (6), we have

E [ˆg

(0)] = g(0) +

(0) +



(λub

) − g

(0)

i

ψ(u) du. (7)

Now combining (C1), (C3) and (C4), we obtain

(λub

) − g

(0)

ψ(u) du = o(1). (8)

The result now follows by combining (7) and (8).

Corollary 1. Under the conditions (C1) − (C4), we have

(0)

= f

(0) +

(0)

ψ(u) du + o





Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 89

Proof. Setting Y = 1, we have g(0) = f

(0) and E[ˆg

(0)] = E[

(0)]. From Theorem 1, we obtain

the result.

Theorem 2. Under the conditions (C1) − (C4), we have

E [ˆg

(x)] =

−c



g(0) + b

(0)u +

(0)u



ψ(u) du + o





Proof. Replacing z with x into (4), using that V and (X, Y ) are independent, and taking the

conditional expectation with respect to X = x and V = v, we obtain

E [ˆg

(x)] = a

−1

∞

−∞



u − x



g(u) du.

Combining (C1) and (C4), we can write the above equality as

E [ˆg

(x)] =

−

ψ(u) g (x + ub

) du.

As (C1) allows us to make a Taylor expansion of function g in the neighborhood of x, we can

rewrite the above equality as

E [ˆg

(x)] =

−c



g(x) + b

(x)u +

(x + λub

)



ψ(u) du. (9)

Now combining (C1) with (C3), and (C3) with (C4), we obtain

h(x) = h(0) + o (1) , for h = g, g

, (10)

and

−c

(x + λub

) − g

(0)] ψ(u)du = o (1) . (11)

The result now follows by combining (9), (10) and (11).

Corollary 2. Under the conditions (C1) − (C4), we have

(x)

−c



(0) + b

(0)u +

(0)u



ψ(u) du + o





Proof. Setting Y = 1, we have g(x) = f

(x) and E[ˆg

(x)] = E[

(x)]. From Theorem 2, we

obtain the result.

The above results allow us to guarantee that both (1) and (4) estimators do not present the

boundary eﬀects at 0, and they will present the boundary eﬀects at x when h

(0)

−c

u ψ(u) du 6=

0, for h = f

, g and for each c ∈ (0, 1]. Moreover, it is important to remark here that Corollaries

1 and 2 are Theorems 1 and 2 in [8].

On the other hand, it is probable enough that estimator (3) presents the above behavior at

the particular points 0 and x ∈ (0, b

]: estimator (3) does not present the boundary eﬀects at

0 and it will present the boundary eﬀects at x when r

(0)

−c

u ψ(u) du 6= 0, for each c ∈ (0, 1].

The following two theorems allow to conﬁrm the above suspicion.

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

90 Jes´us A. Fajardo

Theorem 3. Under the conditions (C1) − (C5), we have

E [ˆr

(0)] = r(0) +



(0) + 2

(0)f

(0)



ψ(u) du + o





Proof. Let us consider the following decomposition (see, e.g., [12])

ˆr

(t) =

ˆg

(t)





1 −

(t) − E

(t)





(t) − E

(t)

[ˆr

(t)]

−1

. (12)

Replacing z with 0 into (12) and taking the expectation, we obtain

E [ˆr

(0)] =

E [ˆg

(0)]

(0)

−

(0)

where

= E

h

(0) − E

(0)

i

ˆg

(0)

and A

= E





(0) − E

(0)

i

ˆr

(0)



Combining (C3), (7) and (8), we have

E [ˆg

(0)] = g(0) + o (1) . (13)

Now setting Y = 1, we can write (4) as

(0)

= f

(0) + o (1) . (14)

As the random vectors ((X

, Y

), V

), 1 ≤ i ≤ n, are identically distributed, we have

= Cov

ˆg

(0),

(0)

(na

)

i=1

Cov [Y

, U

] =



Y U



−



Y U









ˆg

(0)



−



ˆg

(0)





(0)



where U = 1I

[0,ϕ(

)]

(V ) = U

. Combining (C3), (13) and (14), we obtain

[g(0) + o (1)] −

[g(0) + o (1)] [f

(0) + o (1)] =

g(0) + o





On the other hand, by (C5) there exists C > 0 such that |ˆr

(0)| ≤ C. Thus, we can write

| ≤ C E



(0) − E[

(0)]









− (E [U])







(1 − E [U]) .

Now setting Y = 1 and combining with (7), we can write





= E

(0)

= f

(0) +

(0)



(λub

) − f

(0)



ψ(u) du.

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 91

Moreover, (C1) implies that f

is bounded in the neighborhood of 0, and (C3) allows us to

suppose that b

< 1. Then we can bound E[

]. Besides, by (2) we can bound (1 − E[U ]).

Therefore, A

= O(

). Now, we can write



(0)

i

(0))

+ o(1)





g(0) + o





= o(1), by (C3),

and



(0)

i

(0))

+ o(1)





= O





The last two equalities, imply that

E [ˆr

(0)] =

E [ˆg

(0)]

(0)

+ O





Now combining (C1), (C3) and (C4), we obtain

(λub

) − g

(0)

ψ(u) du = o(1). (15)

Moreover, combining (7) and (15) we can write

E [ˆg

(0)] = g(0) +

(0)

ψ(u) du,

whence

(0)] = f

(0) +

(0)

ψ(u)du.

Then

E [ˆr

(0)] =

g(0) +

(0)

ψ(u) du

(0) +

(0)

ψ(u) du

+ O





= H

(0) + O





Next, we will obtain an equivalent expression for H

(0). Taking the conjugate, we have

(0) =

g(0)f

(0) +

ψ(u) du



(0)f

(0) − f

(0)g(0)



−





(0)g

(0)



ψ(u) du



(0)]

−1

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

92 Jes´us A. Fajardo

where D

(0) = (f

(0))

−



(0)

ψ(u) du



. From (C2) and (C3), we obtain [D

(0)]

−1

(0))

−2

+ o(1). Thus,

(0) =

g(0)f

(0) +



(0)f

(0) − f

(0)g(0)



ψ(u) du

−





(0)g

(0)



ψ(u) du



(0))

−2

+ o(1)

= r(0) +

(0)f

(0) − f

(0)g(0)

(0))

ψ(u) du + o





Then

E [ˆr

(0)] = r(0) +



(0) + 2

(0)r

(0)



ψ(u) du + o(b

) + O





By (C3), we have (nb

)

−1

= o(1). The result now follows combining the last two equalities.

Theorem 4. Under the conditions (C1) − (C5), we have

E [ˆr

(x)] = r(0) + r

(0) b

−c

u ψ(u) du +



(0) + 2

(0)f

(0)



−c

ψ(u) du

−

(0)f

(0) b

(0)]

−c

u ψ(u) du

+ o





. (16)

Proof. Replacing z with x into (12) and taking the expectation, we obtain

E [ˆr

(x)] =

E [ˆg

(x)]

(x)

−

(x)

where

= E

h

(x) − E

(x)

i

ˆg

(x)

and

= E



(x) − E

(x)

i

ˆr

(x)

Combining (9) and (10), both hold true under the hypotheses of Theorem 4, we can write

E [ˆg

(x)] =

−c

g(0) + b

(0)u +

(0) + [g

(x + λub

) − g

(0)])

ψ(u)du. (17)

Remember that, (C3) allows us to suppose that b

< 1. Now, combining (C1), (C3), (C4) with

(17), we have that E[ˆg

(x)] = g(0)

−c

ψ(u) du + o (1), whence E[

(x)] = f

(0)

−c

ψ(u) du +

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 93

o (1). Moreover, combining the fact that ((X

, Y

), V

), 1 ≤ i ≤ n, are identically distributed,

with the previous equalities and (C3), we have

(na

)

i=1

Cov [Y

, U

] =



ˆg

(x)



−



ˆg

(x)





(x)



g(0)

−c

ψ(u) du + o





where in this case U = 1I

0,ϕ



X−x

i

(V ) = U

. On the other hand, by (C5) there exists C > 0

such that |ˆr

(0)| ≤ C. Thus, we can write

| ≤ CE



(x) − E[

(x)]









− (E [U])







(1 − E [U]) .

Now setting Y = 1 and combining with (17), we can write

(x)] =

−c



(0) + b

(0)u +

(x + λub

)



ψ(u) du.

Moreover, (C1) implies that f

is bounded in the neighborhood of x, and (C3) allows us to

suppose that b

< 1. Thus, we can bound E

(x)

. Besides, by (2) we can bound (1 − E[U]).

Then

= O(

). Therefore

(E[

(x)])

= o(1) and

(E[

(x)])

= O(

). In consequence,

E [ˆr

(x)] =

E [ˆg

(x)]

(x)

+ O





Now combining (C1), (C3) and (C4), we obtain

−c

(x + λub

) − g

(0)

ψ(u) du = o(1).

The previous equality allows us to rewrite (17) as

E[ˆg

(x)] =

−c

[g(0) + b

(0)u +

(0)u

]ψ(u) du,

whence

(x)] =

−c

(0) + b

(0)u +

(0)]ψ(u) du.

Then

E [ˆr

(x)] =

g(0)C

+ C

n,g

(0)

(0)C

+ C

n,f

(0)

+ O





(0) + O





, (18)

where

−c

ψ(u) du

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

94 Jes´us A. Fajardo

and

n,q

(0) = b

(0)

−c

u ψ(u) du +

(0)

−c

ψ(u) du, for q = g, f

Next, an equivalent expression for

(0) will be obtained. Taking the conjugate and combining

the following equalities

(i) C

n,f

(0) = o(1), by (C3),

(ii)

(0) C

)

−



n,f

(0)



(0) C

)

+ o(1), by (i),

(iii) C

n,g

(0)C

n,f

(0) = b

(0)f

(0)

−c

uψ(u)du

+ ø





, by(C3),

(iv) g(0)f

(0) = r(0) [f

(0)]

, since g(x) = r(x)f

(x),

(v) f

(0)g

(0) − g(0)f

(0) = r

(0) [f

(0)]

, given that r(x) =

g(x)

(x)

(vi) g

(0)f

(0) − f

(0)g(0) = r

(0) [f

(0)]

+ 2f

(0)r

(0)f

(0),

we can write

(0) = r(0) + r

(0) b

−c

u ψ(u) du + b



(0) + 2

(0)f

(0)



−c

ψ(u) du

(0)f

(0) b

(0)]

−c

u ψ(u) du

+ o





By (C3), we have that (nb

)

−1

= o(1). The result now follows combining (18) and the last two

equalities.

Next, the results that will allow us to eliminate the boundary eﬀects in estimator (3) are

presented, which were used in [8], Theorems 3 and 4, to eliminate the boundary eﬀects in estimator

(1). The following theorem in particular will allow us to control suitably the constants that deﬁne

the bias of estimator (3) and it justiﬁes a condition in the criterion introduced to eliminate the

terms with coeﬃcients b

in all the expressions above.

Theorem 5. Under condition (C4), we have that for M > 0 there exists B

> 0 such that

v =

−B

ψ(u) du ≤ M.

Proof. Similar to the proof of Theorem 3 in [8].

Observe that combining (C4) and Theorem 5 we obtain

v =

−B

ψ(u) du =

ψ(u) du ≤ M,

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 95

for B

> B. Now, we can redeﬁne ψ as

ψ(u) =

ϕ(u)

ϕ(u) du

[−B

]

(u), B

≤ B. (19)

To give a simple and eﬀective solution to the boundary problem, without boundary corrections,

a criterion will be implemented which will allow us to eliminate the terms with coeﬃcients b

all above expressions, making

−c

u ψ(u) du = 0 for each c ∈ (0, b

]. Such criterion is based on

deriving a thinning function ϕ as the solution to the following variational problem:

Maximizing :

ϕ(u) du.

Subject to :

(u) du = k,

u ϕ(u) du = 0, (VP)



− v



ϕ(u) du = 0,

k > 0, ϕ(u) = 0 for u ∈ (−B, B)

, ϕ(0) > 0, ϕ(u) ∈ [0, 1], v ∈ (0, M].

Theorem 6. The solution of (VP) is given by

(u) =

1 −



15 k



[

−

]

(u), k > 0. (20)

In particular, for each c ∈ (0, b

] we have

(u) =

1 −



15 c



[

−

]

(u). (21)

Proof. Similar to the proof of Theorem 4 in [8].

From (16) w.t.f. ϕ

we obtain

E [ˆr

(x)] = r(0) +



(0) + 2

(0)f

(0)



−B

(u) du + o





, (22)

where 0 < c ≤ 1, 0 < B

≤

c, ψ

is given by (19) w.t.f. ϕ

, and ϕ

is given by (21).

Thus, estimator (3) does not present the boundary eﬀects at x when the thinning function is ϕ

Moreover, combining Theorem 3 with Theorem 3.1 in [5], we obtain

E [ˆr

(z)] = r(z) +



(z) + 2

(z)f

(z)



−B

(u) du + o





for each k > 1 and z ∈ {0} ∪ (b

, ∞), where B

≤

k, ψ

is given by (19) w.t.f. ϕ

, and ϕ

given by (20). Thus, estimator (3) does not present the boundary eﬀects at z ∈ {0} ∪ (b

, ∞)

when the thinning function is ϕ

, for each k > 1.

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

96 Jes´us A. Fajardo

On the other hand, denoting the Epanechnikov kernel by K

, and replacing k with

and

M with M

(u) du into (V P ), we have that estimator (3) w.t.f. ϕ

, is the estimator

studied in [5]. That is, the estimator introduced in [5] is a particular case of the class of estimators

deﬁned by (3) w.t.f. ϕ

, for each k > 1. Moreover, the results obtained in [5] allow to guarantee

that ϕ

minimize M SE[ˆr

(z)], for each k > 1 and z ∈ {0} ∪ (b

, ∞). Particularly in [5], it was

shown that MSE [ˆr

(t)] ≤ MSE

ˆr

(t)

for each t ∈ R, where ˆr

is given by (3) w.t.f. ϕ

and

ˆr

is the classical kernel regression estimator.

Next, the boundary estimator introduced in [8] and the main result of the above article are

presented. The inclusion of the above results is based on the following two reasons. First,

because the boundary regression estimator will be deﬁned in terms of the estimator proposed

in [8]. Second, to highlight two asymptotic properties of the boundary estimator deﬁned in [8].

Deﬁnition 3. Let X

, . . . , X

be an independent random sample of X. Let V

, . . . , V

be inde-

pendent random variables uniformly distributed on [0, 1] and independent of X

, . . . , X

. Then,

the boundary fuzzy set estimator of the density function f

at the point x ∈ (0, b

] is deﬁned as

(x) =

n a

i=1

0, ϕ



−x

i

) =

(c)

(x), (23)

where 0 < c ≤ 1, a

= b

(u) du, and ϕ

is given by (21).

Theorem 7. Under conditions (C1)-(C3), we have

(x) − f

(0)

−B

(u) du + o





and

V ar

(x)

n b

(u) du

(0) + o



n b



where 0 < c ≤ 1, 0 < B

≤

c, ψ

is given by (19) w.t.f. ϕ

, and ϕ

is given by (21).

Next, the boundary fuzzy set estimator of r and the main result of this paper are presented.

Deﬁnition 4. Let ((X

, Y

), V

), . . . , ((X

, Y

), V

) be independent copies of a random vector

((X, Y ), V ), where V

, . . . , V

are independent random variables uniformly distributed on [0, 1],

and independent of (X

, Y

), . . . , (X

, Y

). Then, the boundary fuzzy set estimator of the regressi-

on function r at the point x ∈ (0, b

] is deﬁned as

˜r

(x) =











i=1



0,ϕ



−x



)

(c)

(x)

if τ

(c)

(x) 6= 0

0 if τ

(c)

(x) = 0,

(24)

where 0 < c ≤ 1, 0 < B

≤

c, τ

(c)

is given in (23), and ϕ

is given by (21).

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 97

Lemma 1. Under the conditions (C1) − (C6), we have

E [˜r

(x) − r(0)] =



(0) + 2

(0) r

(0)



−B

(u)du + o





and

V ar [˜r

(x)] =

n b

(u) du



φ(0) − r

(0)



+ o



n b



where 0 < c ≤ 1, 0 < B

≤

c, ψ

is given by (19) w.t.f. ϕ

, and ϕ

is given by (21).

Proof. From (3) w.t.f. ϕ

, we have ˜r

(x) = ˆr

(x). Now, combining the above equality with (22),

we obtain the expression for E [˜r

(x) − r(0)]. For the proof of V ar [˜r

(x)], follow [5] considering

as thinning function, keeping in mind that q(x) = q(0) + o(1), for q = f

, r, φ.

For z ≥ b

, ˜r

(z) reduces to the fuzzy set regression estimator ˆr

(z) given by (3) w.t.f. ϕ

Thus, ˜r

(z) is a natural boundary continuation of estimator (3) w.t.f. ϕ

. Moreover, the results

obtained in [5] allow us to guarantee that the thinning function ϕ

locally minimizes MSE[˜r

(x)],

for each c ∈ (0, 1].

Next, asymptotic formulae for the optimal smoothing parameter and optimal mean square

error of (24), b

∗

and MSE

∗

, are presented, which are an immediate consequence of Lemma 1, and

they guarantee that estimator (24) is locally optimal in terms of rates of convergence (see [26]).

Corollary 3. Under the conditions (C1) − (C6), we have

∗

= n

−1/5









(u) du



−1



φ(0)−r

(0)





(0) + 2

(0) r

(0)





−B

(u)du









1/5

and

MSE

∗

[˜r

( ˙x)] =

−4/5









φ(0)−r

(0)





(0) + 2

(0) r

(0)





−B

(u)du



−2



(u) du









1/5

where 0 < c ≤ 1, ˙x = c b

∗

, 0 < B

≤

c, ψ

is given by (19) w.t.f. ϕ

, and ϕ

is given by (21).

3 Simulations results

In this section some simulation results are presented, which are only designed to illustrate the

performance of (24) at points near 0 in a b

spread neighborhood. For purposes of comparison,

the estimator introduced in [24] was also considered. It is important to remark here that, the

particular choice above was based mainly on the results of the simulations obtained in [24], for

the two regression models and two densities considered in this section, which showed that the

boundary kernel regression estimator performed quite well when it was compared with the local

linear and ˆr

estimators. Among other reasons that sustained the above particular choice,

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

98 Jes´us A. Fajardo

the theoretical properties that are shared by the boundary kernel regression estimator deﬁned

in [24] and the proposed boundary fuzzy set regression estimator are highlighted: non-negativity,

“natural boundary continuation” and they improve the bias but holding on to the low variances.

Properties not justiﬁed will be veriﬁed in this section. The author considers that the previous

comments and discussion motivated by the literature review results, justify considering only the

estimator given by [24] to develop the simulations.

The general boundary kernel estimator introduced in [24] is deﬁned as

˜r

(l) =

i=1



l+w

)



+ K



l−w

)

o

i=1



l+w

)



+ K



l−w

)

o

, (25)

where l = s h

, 0 ≤ s ≤ 1, h

is the smoothing parameter and K is a kernel function of order 2.

Moreover, w

is a transformation deﬁned as

(y) = y +

+ λ

)

, k = 1, 2,

where

= w

(0) =

(0)

g(0)

K,s

, d

= w

(0) =

(0)

K,s

and λ

is a constant such that 12λ

> 1, with

K,s

(u − s)K(u) du

(u − s)K(u) du + s

To assess the eﬀect of the boundary estimators at points near 0 in a b

spread neighborhood, the

following models are studied:

Model 1 : r

(z) = 2 z + 1 and Model 2 : r

(z) = 2 z

+ 3 z + 1,

where z ∈ [0, ∞) and errors ε

, are assumed to be standard normally distributed independent

random variables. Likewise, consider two cases of density f

with support [0, ∞):

Density 1 : f

(z) = exp(−z) and Density 2 : f

(z) =

π(1 + z

)

It is important to emphasize that, for z ≥ h

the estimator (25) reduces to the classical kernel

estimator ˆr

. Thus, (25) is a natural boundary continuation of ˆr

(see [24], Section 2).

On the other hand, the following optimal global smoothing parameters were implemented as

the smoothing parameters of both (24) and (25) estimators (see Theorems 3.1 and 2.4.1 in [5]

and [12], respectively):









(u) du



−1



(u) du









1/5

(26)

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 99

and

(u)du



(u)du



1/5

, (27)

where

C =



φ(u)−r

(u)





(u) + 2

(u) r

(u)



, (28)

is given by (20), ψ

is given by (19) w.t.f. ϕ

, and v =

(u)du ≤

(u) du = M

(see [5], Section 3).

It is important to emphasize that the estimator (3) w.t.f. ϕ

has better performance than the

estimator ˆr

(see [5]). Moreover, the reason for using an optimal global smoothing parameter

as the smoothing parameter is that the comparisons based on the optimal smoothing parameter

are more convincing than comparisons based on approximated smoothing parameters, which—

because of the quality or otherwise of the smoothing parameter selection method—might be

misleading. Also, a global rather than local smoothing parameter choice is made, because this is

much more likely to be used in applications.

The following formulas and values are used in all simulations, K

(u) =

(1 − x

)1I

[−1,1]

(u),

= 0.0933, v = M

/4, s = 0.1, 0.2, 0.3, 0.4, 0.5 and c = (h

) s, where the smoothing parame-

ters b

and h

are given by (26) and (27), respectively. Moreover, the simulated sample sizes are

n = 50 and n = 500, and all results are calculated by averaging over 1000 trials. Simultaneously,

for each regression model and each density, the absolute bias (absolute value of the estimated

value minus the true value) and the MSE of both estimators are calculated, at the points x = c b

and l = s h

of the following boundary regions (0, b

) and (0, h

). Nonetheless, the comparison

of both (24) and (25) estimators through the mean integrated squared error is not convenient,

since to the two regression models and two densities considered the boundary regions satisfy the

following property (0, h

] ⊂ (0, b

]. The results are shown in Tables 3 and 3. Furthermore, a

close examination of Tables 3 and 3 allows us to see that for the two regression models and two

densities considered the proposed boundary estimator is the best in terms of MSE at points near

0 in a b

spread neighborhood, since it has low bias and extremely low variance, guaranteeing that

MSE [˜r

(x)] ≤ M SE[˜r

(x)] at the points x near 0 in a b

spread neighborhood. Additiona-

lly, these properties are apparently preserved over the rest of the boundary regions. Thus, the

proposed boundary estimator outperforms the estimator introduced in [24].

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

100 Jes´us A. Fajardo

Table 1: Bias and MSE of the indicated regression estimators at boundary, case of density 1.

˜r

Model 1 ˜r

= 3.5045 n = 50 h

= 2.1047

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0011 0.0560 0.1332 0.2454 0.3812 — 0.1726 0.4102 0.5823 0.7552 0.9440

MSE — MSE

0.0000 0.0031 0.0177 0.0602 0.1453 — 0.0298 0.1682 0.3391 0.5703 0.8911

= 2.2112 n = 500 h

= 1.3280

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0101 0.0240 0.0528 0.0963 0.1515 — 0.2347 0.2842 0.3228 0.3807 0.4496

MSE — MSE

0.0001 0.0006 0.0028 0.0093 0.0230 — 0.0551 0.0807 0.1042 0.1449 0.2022

˜r

Model 2 ˜r

= 1.2978 n = 50 h

= 0.7794

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0383 0.0200 0.0225 0.0408 0.0579 — 0.1667 0.2058 0.2433 0.2762 0.3045

MSE — MSE

0.0015 0.0004 0.0005 0.0017 0.0034 — 0.0278 0.0423 0.0592 0.0763 0.0927

= 0.8189 n = 500 h

= 0.4918

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0036 0.0012 0.0062 0.0111 0.0230 — 0.0727 0.0907 0.1047 0.1133 0.1162

MSE — MSE

0.0000 0.0000 0.0000 0.0001 0.0005 — 0.0053 0.0082 0.0110 0.0128 0.0135

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 101

Table 2: Bias and MSE of the indicated regression estimators at boundary, case of density 2.

˜r

Model 1 ˜r

= 2.1614 n = 50 h

= 1.2980

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0201 0.0148 0.0359 0.0734 0.1233 — 0.0996 0.2173 0.3078 0.3800 0.4398

MSE — MSE

0.0004 0.0002 0.0013 0.0054 0.0152 — 0.0099 0.0472 0.0947 0.1444 0.1934

= 1.3637 n = 500 h

= 0.8190

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0050 0.0017 0.0042 0.0179 0.0378 — 0.0652 0.1161 0.1502 0.1702 0.1801

MSE — MSE

0.0000 0.0000 0.0000 0.0003 0.0014 — 0.0042 0.0135 0.0226 0.0290 0.0324

˜r

Model 2 ˜r

= 1.0466 n = 50 h

= 0.6285

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0710 0.0309 0.0284 0.0088 0.0132 — 0.0720 0.1305 0.1670 0.1806 0.1708

MSE — MSE

0.0050 0.0010 0.0008 0.0001 0.0002 — 0.0052 0.0170 0.0279 0.0326 0.0292

= 0.6603 n = 500 h

= 0.3966

c — s

0.0601 0.1201 0.1802 0.2402 0.3003 — 0.1000 0.2000 0.3000 0.4000 0.5000

| Bias | — | Bias |

0.0064 0.0040 0.0013 0.0028 0.0088 — 0.0333 0.0579 0.0697 0.0678 0.0544

MSE — MSE

0.0000 0.0000 0.0000 0.0000 0.0001 — 0.0011 0.0033 0.0049 0.0046 0.0030

4 Data analysis

The proposed estimator was tested over one well-known real dataset to demonstrate its usefulness

in practical applications. For purposes of comparison, the particular classical kernel regression

estimator introduced in [22] was considered. With respect to this last point, a brief discussion

will be developed at the end of this section. The real dataset is the Motorcycle data found in

Appendix 2-Table 1 of [14], where the experiment, a simulated motorcycle crash, is described in

detail in [23]. It has n = 133 observations of X, where the X-values denote time (in milliseconds)

after a simulated impact with motorcycles and the response variable Y is the head acceleration

(in g) of a post mortem human test object. To facilitate the comparison of the eﬀectiveness of

estimators (24), ˆr

, (3) w.t.f. ϕ

, and (3) w.t.f. ϕ

, the performance of each estimator will

be graphed over the dataset through an average of 1000 trials. Moreover, for the ﬁxed random

sample, X

, . . . , X

, a sample of independent random variables V

(d)

, . . . , V

(d)

will be used, where

(d)

is uniformly distributed on [0, 1] for each i, d, 1 ≤ i ≤ n and 1 ≤ d ≤ 1000.

In order to simulate the performance of the estimator ˆr

, the idea in [22] will be adopted

with smoothing parameter h

= 2.40, and quartic kernel K

(u) = (16/25)(1 − u

)

[−1,1]

(u)

instead of K

. Simultaneously, for the fuzzy set estimation the smoothing parameter (26) will

be implemented, with v = M

/2, where the approximate value of constant (28) was calculated

through (27) using h

= 2.40 and K

instead of K

. Here M

(u) du. Figures 4 and 4

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

102 Jes´us A. Fajardo

show the simulation results together with the data points on [0, 57.60] and [0, 6.50], respectively.

From Figures 4 and 4, the estimator (3) w.t.f. ϕ

does not appear to oﬀer an appropriate

approximation on large part of the region [2.92, 57.60], when compared with the similar results

produced by both (3) w.t.f. ϕ

and ˆr

estimators. At the same time, Figure 4 shows that the

presence of the boundary problem in (3) w.t.f. ϕ

was removed by (24), where both (24) and

ˆr

estimators fail miserably over their respective boundary regions, and only the estimator (24)

follows the data very closely on the left side of the picture, by making the curve be constant,

equal zero, over the region [0, 1.305].

Finally, it is necessary to point out that [24] does not present a study of simulations with real

data, and within the context of the above paper an appropriate criterion for the estimation of

the parameters d

and d

was not proposed. The previously exposed explains the absence, in this

section, of a study related to the graphical comparison of estimators (25) and (24). Remember

that, within the proposed objectives in this paper only the estimation of the parameters associated

with the fuzzy set estimation method will be considered. Consequently, the subjective choice of

the values of the parameters d

and d

would be incorrect, since it could beneﬁt the behavior of

(24) and impair the behavior of (25), which would produce unreliable results.

Nueva_Completa-eps-converted-to.pdf

—— ˜r

-.-.-.-. ˆr

pppppppppppp ˆr

w.t.f. ϕ

··· · · ˆr

w.t.f. ϕ

Figure 1: Regression estimates to the motorcycle data at the points inside the region [0, 57.60]

using smoothing parameters h

= 2.40 and b

= 2.92, where v =

. The circles indicate the

raw data.

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

Boundary estimation 103

Nueva_Boundary-eps-converted-to.pdf

—— ˜r

-.-.-.-. ˆr

pppppppppppp ˆr

w.t.f. ϕ

· · · · · ˆr

w.t.f. ϕ

Figure 2: Regression estimates to the motorcycle data at the points inside the region [0, 6.5] using

smoothing parameters h

= 2.40 and b

= 2.92, where v =

. The circles indicate the raw

data.

5 Final remarks

In this paper, a new contribution in the area of regression estimation not based on kernels

is presented, obtaining a natural extension of the results introduced in [8]. In particular, the

boundary eﬀects in the fuzzy set regression estimator are studied and removed, where it does not

require boundary modiﬁcations to eliminate such eﬀects, unlike most well-known kernel regression

estimators. Moreover, among the desirable properties of the boundary fuzzy set estimator the

non-negativity of the estimator is highlighted, as well as its performance which is generally very

robust with respect to various shapes of regression and density functions, since it allows important

reductions of the bias, while maintaining low variance. It is clear that no single existing estimator

in the literature dominates all the others for all shapes of regression and density functions. Each

estimator has certain advantages and works well at certain times. However, for the two regression

models and the two density functions considered, the boundary fuzzy set regression estimator

has better performance than the boundary regression estimator introduced in [24] at points near

zero in a spread neighborhood of the smoothing parameter.

On the other hand, it is appropriate to highlight the important role that the thinning function

plays in the results obtained, since its adequate construction allowed to eliminate the boundary

eﬀects in the fuzzy set regression estimator, giving each boundary point the least approximation

between the boundary fuzzy set estimator and regression function. Finally, it is necessary to

point out that through the thinning point process that describes the fuzzy set density estimation

method, the set of observations considered can be characterized in a neighborhood of the esti-

mation point, where the indicator functions that deﬁne the fuzzy set density estimator decide

whether observation belongs on the neighborhood of the estimation point or not, and the thinning

functions that deﬁne each indicator function are used to select the sample points with diﬀerent

Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 82–106

104 Jes´us A. Fajardo

probabilities, in contrast to the kernel estimators which assign equal weight to all points of the

sample.

6 Acknowledgements

This research has been supported by a grant from the Academia de Ciencias de Am´erica Latina-

ACAL.

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