408 

© 2024 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

PISSN: 1609-4042, EISSN: 2521-3407 

 IHJPAS. 2024, 37(4) 

 

A Comparison of some non-parametric kernel estimators using simulations 

 

Rabab Abdulrida Saleh1,*   and  Iqbal Mahmood Alwan2   
1,2Department of Statistics, College of Administration and Economics, University of Baghdad, Baghdad, 

Iraq. 

*Corresponding Author.  
                      

Received: 16 December 2023                Accepted: 6 February 2024               Published: 20 October 2024 

doi.org/10.30526/37.4.3805 

Abstract 

         In this research, we conducted a comprehensive investigation of various kernel functions 

employed in the estimation of nonparametric regression functions. In particular, we investigated 

the Nadaraya-Watson method and local polynomial regression techniques involving linear, 

quadratic and cubic forms. These methods were evaluated using five different kernel functions: 

Gaussian, Epanechnikov, uniform, triangular and quartic, collectively referred to as GEUTQ. 

The main objective of the work was to determine the best estimators for non-parametric kernel 

functions. To achieve this, we performed a rigorous comparison with simulation methods, different 

regression models and different sample sizes. The evaluation of the performance of the estimators 

was based on the mean absolute percentage error (AMAPE) assuming a standard normal 

distribution with a mean of zero and a variance of one. 

Our simulation results and plots clearly show that the quadratic estimator (LP2) using the kernel 

function (G, E) consistently has the lowest (AMAPE) across all sample sizes and two models. 

Similarly, the local linear estimator (LP1) in the functions (U, T, Q) has the lowest (AMAPE) for 

all sample sizes and two models. As for the optimal functions, function is identified as the most 

effective kernel function among the options considered, leading to the lowest average values. 

Furthermore, it is observed that as the sample size increases, the average values for the following 

methods decrease: Nadaraya, the linear method and the quadratic method. Conversely, the LP3 

estimators, especially the linear cubic regression, turn out to be the least favorable and have 

relatively high values compared to the other estimators. These results provide valuable insights 

into the performance of different estimators and kernel functions in nonparametric regression 

models and thus contribute to future research and decision-making processes. 

Keywords: nonparametric regression, Nadaraya-Watson estimator, kernel functions, Quadratic, 

local polynomial estimator. 

 

1.  Introduction  

       In 1988, [1] introduced and studied Kernel Smoothing for estimating partial linear models, [2] 

conducted a study titled "A Comparison of Some Semiparametric Estimators for Consumption 

Function Regression". In 2013, [3] presented a paper titled "Performance of Nonparametric 

Regression Estimation with Diverse Covariates Pak." In 2014, [4] introduced a paper comparing 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0009-0004-8147-217X
mailto:rabab.saleh@coadec.uobaghdad.edu.iq
https://orcid.org/0000-0002-6656-7662
mailto:Iqbal.alwan@coadec.uobaghdad.edu.iq


  IHJPAS. 2024, 37(4) 

409 
 

semiparametric estimators models using different smoothing methods. Additionally, [5] 

contributed to the literature with a paper titled "Hazard Rate Estimation Using Varying Kernel 

Function for Censored Data Type I.". In 2020, [6] employed wavelet and kernel smoothers to 

estimate stock return rates. Additionally, in 2020, [7] introduced a comparison between various 

nonparametric methods, utilizing the methodology of quantile regression models. 

In 2021, [8] presented a paper titled 'Estimation of Nonparametric Autoregressive Curves Using 

the Smoothing Spline Method.' Furthermore, in the same year, [9] made their own introduction. 

(Linear Regression Model to Study the Effects of Weather Variables) In the past, numerous 

researchers have incorporated the Nadaraya-Watson technique into their studies. Some notable 

examples include [10-12], all of whom conducted research involving this technique. 

Regression analysis aims to uncover the relationship between the dependent variable and the 

explanatory variables. It involves incorporating these variables into a mathematical model to 

predict future values of the dependent variable. Regression models can be categorized into three 

types: 

The first model is a parametric regression model, which requires knowledge of the distribution and 

has assumptions and limitations [13]. The second model is when data analysis is challenging in 

this model, we use the non-parametric regression models, which offer greater flexibility for 

analyzing variable relationships, do not require knowledge of the distribution, and do not 

necessitate restrictions or special conditions as in parametric models. This flexibility makes non-

parametric models attractive to many researchers. However, they are susceptible to the problem of 

the curse of dimensionality, which arises when the number of variables increases. The last model 

combines both parametric and non-parametric elements [14,13]. 

The objective of this study is to compare non-parametric estimators, including Nadaraya-Watson 

and local linear, quadratic, and cubic estimators, using multiple kernel functions (GEUTQ). We 

employed Monte Carlo simulation to identify the best estimator based on mean absolute percentage 

errors (MAPE). Simulation results have demonstrated that the LP1 estimator for the quartic (Q) 

function consistently outperforms other estimators across various sample sizes and models. 

Section 2 provides the theoretical foundation, encompassing methods for estimating non-

parametric regression models, kernel functions, the Local Polynomial Regression Estimator, and 

bandwidth. Section 3 We studied five of the kernel functions. Section 4 presents the results of our 

simulation study, conducted using MATLAB 19. In Section 5, we present the results and engage 

in discussion. Section 6 summarizes the key findings of this study, and Section 7 offers 

recommendations for future research. 

 

2. Methods for Estimating Non-parametric Regression Models 

Non-parametric regression, initially proposed by researcher Jacob Wolfowib in 1942, focuses on 

estimating the curve of the regression function rather than estimating specific parameters. This is 

achieved through the application of non-parametric regression functions, ultimately leading to the 

identification of the most suitable method for non-parametric regression models [8]. 

( )  + iY i m Xi =                                       (1)                  

 where   i = 1, 2, … ,n  

Yi is the variable of interest, m(Xi)  is an unknown function to be determined using sample data  

ei is an error term assumed to be N (0, 𝜎2) under the model.[16] 

The methodology of estimating non-parametric regression models is widely employed to 

determine the average treatment effect when implementing regression discontinuity designs. 



  IHJPAS. 2024, 37(4) 

410 
 

Instead of simply calculating averages, this method helps reduce bias, and any remaining bias can 

be treated as normal [17,18]. For a more in-depth exploration of non-parametric regression, refer 

to the works of Some´[19-32], among others. 

kernel function (Nadaraya-Watson- estimator NW) 

In 1988, Robinson constructed an estimate of the non-parametric component using the least 

squares method in conjunction with the Nadaraya-Watson estimator. Subsequently, in 1997, 

Hamilton, Genentech, and Truong applied the Local Linear Smoother, which is based on the 

Speckman method [33]. 

Partial Linear Model Definition: 

The partial linear model encompasses two types of Nadaraya-Watson estimators. 

 The first type employs Nadaraya and Watson with a fixed smoothing parameter. This estimator is 

among the most widely used nonparametric regression estimators. It is also known for utilizing a 

fixed smoothing parameter. An essential characteristic of this estimator is its ability to provide a 

continuous estimate of the regression when a continuous kernel function is used. The mathematical 

representation of this estimator is as follows [22]. 

 

�̑�𝐹(𝑥) =
∑ 𝐾 (

𝑋 − 𝑥
ℎ

) 𝑌𝑖
𝑛
𝑖=1

∑ 𝐾 (
𝑋 − 𝑥
ℎ

)𝑛
𝑖=1

 

  =
1

𝑛
∑[

𝐾ℎ(𝑥𝑖)
(𝑋𝑖 − 𝑥)

𝑛−1𝐾ℎ(𝑥𝑖)
(𝑋𝑖 − 𝑥)

]

𝑛

𝑖=1

𝑌𝑖  

   =
1

2
∑𝑊ℎ(𝑋𝑖)

𝑛

𝑖=1

𝑌𝑖 

The second is Nadaraya and Watson with variable smoothing parameters’ 

This estimator relies on the utilization of variable smoothing parameters as opposed to fixed 

smoothing parameters. It earns its name "variable" due to its adaptability, wherein the smoothing 

parameter varies at each point. This variability is achieved by employing large windows in areas 

with low data density and small windows in areas with high data density. The formula for this 

estimator is given as follows [3]: 

�̑�𝑣(𝑥) =
∑ 𝐾ℎ(𝑥𝑖)

(𝑋𝑖 − 𝑥)𝑌𝑖
𝑛
𝑖=1

∑ 𝐾ℎ(𝑥𝑖)
(𝑋𝑖 − 𝑥)𝑛

𝑖=1

 

  =
1

𝑛
∑[

𝐾ℎ(𝑥𝑖)
(𝑋𝑖 − 𝑥)

𝑛−1𝐾ℎ(𝑥𝑖)
(𝑋𝑖 − 𝑥)

]

𝑛

𝑖=1

𝑌𝑖 

This modification ensures that the estimator adjusts its smoothing parameter based on the local 

data density, contributing to its adaptability in various contexts. 

 

 The Local Polynomial Regression Estimator: 

 Consider a random sample 1 1 2 2,  ,  ,( ) ( ) ( ) , ,  ,  n nX Y X Y X Y  of bivariate data taken from a finite 

population. From the model in (1), to estimate the unknown function m(Xi), the procedure below 

follows [20]. For more on Local Polynomial Regression Estimator [34-39]: 

 

 

 



  IHJPAS. 2024, 37(4) 

411 
 

( )   /m X E Y X x= =   

The approximation for this can be achieved using Taylor's series, as outlined in [46] 

( ) ( )
( )

( )
( )

( )
( )

2

0 0( )
2! !

p
po o

o o

m x m x
m x m x m x x x x x

p


 + + − + + −  .            (2) 

When utilizing the local polynomial X0, and minimizing the Weighted Least Square Regression, 

we arrive at the following result. For more comprehensive information [34].  

 

for further details: 

( ) ( )  ( )
2

0 1 0 0 0

1

min
n

p

i K p K h K
B

k

y B B X x B X x K X x
=

− − − − − − −                                          (3) 

Where: ( )0 1, ,..,..
T

PB B B B=    

K=kernel function 

h: Represents a bandwidth   

Then we obtain the estimator of Weighted Least Square (WLS) in form; [50] 

( ) ( )
1

B x X WX X WY
−

 =                                                                                                              (4) 

Where 

( ) ( )

( ) ( )

( ) ( )

1 0 1 0

2 0 2 0

0 0

1

1

1

p

p

p

n n

X x X x

X x X x
X

X x X x

 − −
 
 − −

=  
 
 

− − 

,

1

2

n

y

y
Y

y

 
 
 =
 
 
 

  

( )

( )

( )

( )( )

0 1

0 2

0

0 0

0 0

0 0

h

h

h

h n

K X x

K X x
W diag K X x

K X x

 −
 

− 
= = − 
 
 − 

 

So, the local polynomial regression estimator for the regression function (smoothing) m will be 

[39]: 

( ) ( ), 0p hm x B X=   

And  ( ) ( )0m x B X  

• When the degree of the polynomial (p=0), we obtain the local constant estimator 

(Nadaraya-Watson). 

• When the degree of the polynomial (p=1), we obtain the local linear estimator. 

• When the degree of the polynomial (p=2), we obtain the local quadratic estimator. 

• When the degree of the polynomial (p=3), we obtain the local cubic estimator. 

• When the degree of the polynomial is p, we obtain the local polynomial regression. 

In order to emphasize the importance of linked kernels in multiple regression, various types have 

been used in numerous studies. These include discrete univariate kernels, continuous univariate 

kernels and bivariate kernels. Further information on the different kernel types can be found in the 

work of [5,31,40,41]. 

 

 

3. kernel functions 



  IHJPAS. 2024, 37(4) 

412 
 

1-Epanechnikov kernel 

The interconnected kernel defines the kernel (Epan) Epanechniko [42]   

( ) ( )  
2

1

3
1 ,

4 X
K x x I


= −                                                                                                       (5) 

2- Gaussian kernel   

the Gaussian kernel for multiple regression is, 

 

21 1
( ) exp ,

22 x

k x x I
 

 
= − 

 
                         (6) 

3-Uniform kernel  

( )  1
1

,
2 x

K x I


=                                                 (7) 

4- Triangular kernel, 

( ) ( )  1
1 ,

x
K x x I


= −                                       (8) 

5-Quartic kernel,  

( ) ( )  

2
2

1

15
1 ,

16 X
K x x I


= −                              (9) 

Bandwidth has been a subject of study by numerous researchers in various studies, 

including[43,44]. 

In the context of multivariate analysis, the bandwidth is denoted by the symbol H, while in 

univariate analysis it is denoted by the symbol h. It is often referred to as 'bandwidth' or 'window 

size' A small bandwidth is usually associated with areas of dense data, while a large bandwidth is 

preferred in regions of sparse data. The value of bandwidth can be determined using the cross-

validation (CV) method [21,44]. The bandwidth matrix selection was originally proposed by [15]. 

Further results can be found in [45, 46]. For our study, we employed the leave-one-out cross-

validation (CV) method proposed by Scott and Terrell in1987 [47]. 

The basic idea of this method is to choose a value that reduces the value of CV(h) and delete all 

controls xi and yi from the data set and prepare the prediction for the remaining n-1 controls 

according to the following formula: 

( ) ( )( )
2

1
1

1
( )

n

i i

i

lcv h m x y
n

−
=

= −                                      (10) 

( ) ( )1
.m

−
Denotes the smoothing estimator when the odd data point xi,yi is deleted from the dataset, 

and only the remaining data n-1 is used to calculate this estimate, This calculation depends on 

leaving out of one for regression estimates ( ) ( )1
.m

−
which can be illustrated by the following 

equation:  

( ) ( )
( ) ( ) ( )

( )
1

1
1

i i i i

i

i i

m x L X Y
m x

L X

−

−

−
=

−
 

Substituting the above equation into equation (10) becomes a criterion (cv)as follows: 

( )
( ) ( )( )
( )( )

2

1

2
1

1

2 1

n
i

i
i i

Y m x
lcv h

L X

−

=

−
=

−
  

Naturally, we choose the bandwidth that minimizes cv(h), which is known to minimize the 

Kullback-Leibler distance between ( )m x  and   ( )m x  



  IHJPAS. 2024, 37(4) 

413 
 

 

4. Simulation study 

The primary objective of this research was to conduct several simulation experiments to select the 

best estimators. We considered two regression functions taken from previously published research, 

namely: 

M(x)=x-3              [31] linear smoothing function 

M(x)=x+2exp(-16x2)       nonlinear smoothing function 

The simulation was performed on the following two models: 

Y= M(x-3) + e   

y=exp+2exp(-16x2) +e 

The illustrative variable x was generated using a uniform distribution based on the Box-Muller 

method, while the random error followed a standard normal distribution with a mean of zero and 

a variance of one. For our simulation experiments, we employed five different sample sizes: 

n=10,30,50,70,100n=10,30,50,70,100 for each model. The experiments were repeated R=500 

times to ensure accurate results. 

In the estimation of nonparametric regression functions, we utilized the following kernel functions: 

Gaussian (G), Epanechnikov (E), uniform (U), Triangular (T), and Quartic (Q). To compare the 

models, we employed the Mean Absolute Percentage Error (MAPE) [48]. 

This rigorous approach allowed us to make meaningful comparisons between the different models 

and assess their performance effectively." 

  

1

1

1

1

R

i

n

i

AMAPE MAPE
R

where

y y
MAPE

n y

=

=

=

−
=





  

 n= sample size, y=real values   

�̂�=estimator values, R=is the replication   

and used (cv) to find the smoothing bandwidth for the nonparametric estimators  

 

 5. Results and Discussion  

          In our research, we utilized MATLAB 19 for the thorough analysis of results obtained from 

each model. 

First model  

We utilized a simple linear model for the smoothing NW   Nadaraya-Watson- Estimator, LP1 

Local linear Estimator, LP2 Local quadratic Estimator, and LP3 Local cubic Estimator and the 

results are in the table below:  

Table 1. MAPE for nonparametric estimators’ size (n=10,30,50,70,100) for the first model 

            Methods 

kernel function 

NW LP1 LP2 LP3 Best 

G                       10 0.642346 1.003498 0.581124 0.665476 LP2(G) 

 E  0.634757 0.558149 0.574258 0.665494 LP1(E) 

 U 0.613096 0.442113 0.554662 0.667669 LP1(U) 

 T 0.621812 0.341636 0.562547 0.668542 LP1(T) 

 Q 0.619036 0.318724 0.560036 0.669462 LP1(Q) 

 Best  NW(U)  Lp1(Q)  LP2(U) LP3(G)  LP1(Q) 



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414 
 

G                       30 0.545729 0.93469 0.519102 0.818688 LP2(G) 

 E  0.548701 0.556902 0.521929 0.823759 LP2(E) 

 U 0.555327 0.442176 0.528232 0.81483 LP1(U) 

 T 0.552742 0.350689 0.525774 0.817275 LP1(T) 

 Q 0.55185 0.327352 0.524925 0.8121 LP1(Q) 

 Best  NW(G)  LP1(Q)  LP2(G) LP3(Q)  LP1(Q) 

G                       50  0.532882 0.920043 0.515409 0.875729 LP2(G) 

 E  0.534376 0.555453 0.516854 0.874997 LP2(E) 

 U 0.535663 0.442501 0.518098 0.876267 LP1(U) 

 T 0.539932 0.348913 0.522227 0.872257 LP1(T) 

 Q 0.534709 0.326183 0.517176 0.8794 LP1(Q) 

 Best  Nw(G)  LP1(Q)  LP2(G)  LP3(T) LP1(Q) 

G                      70 0.526177 0.913865 0.513184 0.903238 LP2(G) 

 E  0.527437 0.554494 0.514414 0.903668 LP2(E) 

 U 0.525973 0.444323 0.512985 0.904481 LP1(U) 

 T 0.526167 0.353313 0.513174 0.905332 LP1(T) 

 Q 0.529208 0.326448 0.51614 0.904708 LP1(Q) 

 Best  NW(U)  LP1(Q)  LP2(U) LP3(E) LP1(Q) 

G                       100 0.522963 0.908445 0.512607 0.923784 LP2(G) 

 E  0.524939 0.552334 0.514544 0.923153 LP2(E) 

 U 0.522031 0.443607 0.511694 0.922888 LP1(U) 

 T 0.522288 0.353527 0.511945 0.92167 LP1(T) 

 Q 0.521718 0.328334 0.511387 0.924304 LP1(Q) 

Best    NW(Q) LP1(Q) LP2(Q) LP3(T) LP1(Q) 

 

The results of the first model showed that in the table (1) we note: 

1- Each function will be compared with the methods at size 10 

For function G the LP2 method is better. For functions (E, U, T, Q) the LP1 method is better. 

Each function will be compared with the methods at size 30,50,70,100 

For function (G, E) The LP2 method is better. For function (U, T,Q) the LP1 method is better 

2- We find that the best for all methods, for all functions, and for all sizes is: LP1(Q) 

3- MAPE values decrease when the sample size increases for most estimators. 

4- To compare the best functions for each estimator, for the estimator Nadaraya Watson, the lowest 

value of MAPE was when N=100 at the function Quartic (Q) 

5- for the Local linear LP1 estimator, the lowest value of MAPE for all sizes was the Quartic (Q) 

6-For the Local quadratic LP2 estimator, the lowest value of MAPE when N=100 is the function 

Quartic (Q) 

7-For the LP3 estimator, the lowest value of MAPE was when N = 10, which is the function 

Gaussian (G)   

8-The worst estimators were LP3 for all functions and all size 

Figures 1-5 illustrate what was mentioned above 

 



  IHJPAS. 2024, 37(4) 

415 
 

 
Figure 1. AMAPE for the parametric methods at the function G for the first model 

 
   Figure 2. AMAPE for the parametric methods at the function E for the first model 

 

 



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416 
 

  Figure 3. AMAPE for the parametric methods at the function U for the first   model 

 

 

 
  Figure 4. AMAPE for the parametric methods at the function T for the first   model 

 
  Figure 5. AMAPE for the parametric methods at the function Q for the first   model 

Table 2. MAPE for nonparametric estimators’ size (n=10,30,50,70,100) for the second model 

            Methods 

kernel function 

NW LP1 LP2 LP3 Best 

G                   10 0.624293 0.995722 0.564791 0.667828 LP2(G) 

 E  0.6297 0.555418 0.569683 0.674076 LP1(E) 

 U 0.616122 0.445208 0.557399 0.662098 LP1(U) 

 T 0.624639 0.341685 0.565105 0.664842 LP1(T) 

 Q 0.613763 0.324112 0.555265 0.656937 LP1(Q) 

 Best  NW(Q)  LP1(Q)  LP2(Q)  LP3(Q) LP1(Q) 

G                     30 0.55542 0.93609 0.528321 0.816388 LP2(G) 

 E  0.553955 0.55382 0.526928 0.823293 LP2(E) 

 U 0.552651 0.442828 0.525687 0.818546 LP1(U) 

 T 0.552655 0.350887 0.52569 0.817904 LP1(T) 

 Q 0.554171 0.323977 0.527132 0.817301 LP1(Q) 

 Best  NW(U) LP1(Q)  LP2(U)  LP3(G) LP1(Q) 

G                    50 0.532262 0.921187 0.514809 0.873312 LP2(G) 



  IHJPAS. 2024, 37(4) 

417 
 

 

The results of the second model showed that in the table (2) we note: 

1. Each function will be compared with the methods: 

At size 10 For function G the LP2 method is better. For function (E, U, T, Q) the LP1 method 

is better. 

At size 30,50,70,100 

For function (G, E) The LP2 method is better. For functions (U, T, Q) the LP1 method is better 

2. decrease in MAPE values when increasing sample size for most estimators 

3. To compare the best functions for each estimator, for the estimator Nadaraya Watson, the 

lowest value of MAPE was when N=100 at the function the Triangular (T)  

4. FOr the Local linear LP1 estimator, the lowest value of MAPE for all sizes was the quartic 

(Q) 

5. For the Local quadratic LP2 estimator, the lowest value of MAPE when N=100 is the 

Triangular (T) 

6. For the LP3 estimator, the lowest value of MAPE was when N = 10, which is the quartic (Q) 

7. The worst estimators were LP3 for all functions and all size 

 

Figures 6-10 illustrate what was mentioned above 

 E  0.539639 0.552673 0.521944 0.874695 LP2(E) 

 U 0.536898 0.442926 0.519294 0.873117 LP1(U) 

 T 0.539634 0.350005 0.521939 0.872166 LP1(T) 

 Q 0.531799 0.32971 0.514362 0.872262 LP1(Q) 

 Best  NW(Q)  LP1(Q)  LP2(Q)  LP3(T) LP1(Q) 

G                    70 0.527445 0.912154 0.514421 0.906164 LP2(G) 

 E  0.526648 0.555864 0.513643 0.900793 LP2(E) 

 U 0.528657 0.4429 0.515603 0.903672 LP1(U) 

 T 0.525111 0.353748 0.512144 0.906952 LP1(T) 

 Q 0.527096 0.327854 0.51408 0.904502 LP1(Q) 

 Best  NW(T)  LP1(Q)  LP2(T)  LP3(E) LP1(Q) 

G                 100  0.520777 0.909646 0.510464 0.922083 LP2(G) 

 E  0.524435 0.55284 0.514049 0.922157 LP2(E) 

 U 0.526603 0.441297 0.516174 0.921811 LP1(U) 

 T 0.519077 0.355318 0.508798 0.924151 LP1(T) 

 Q 0.521945 0.328704 0.511609 0.921277 LP1(Q) 

Best  NW(T) LP1(Q) LP2(T) LP3(Q) LP1(Q) 



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418 
 

 
Figure 6. AMAPE for the parametric methods at the function G for the second model 

 

 
Figure 7. AMAPE for the parametric methods at the function  E for the second model 

 



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419 
 

 
Figure 8. AMAPE for the parametric methods at the function U for the second model 

 

 
Figure 9. AMAPE for the parametric methods at the function T for the second model 

 

 
Figure 10. AMAPE for the parametric methods at the function Q for the second model 



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420 
 

6. Conclusions 

Based on the results of the simulation experiments, the following conclusions can be drawn: 

1. The LP1 estimator is preferred when using the quartic function across all sample sizes and 

for two models. This is closely followed by the LP2 estimator, which performs well when 

quartic and triangular kernels  are used. 

2. The quartic function proves to be the best kernel function among the options considered. 

3. In contrast, the LP3 estimators (linear cubic regression estimators), which have relatively 

large values compared to the other estimators, are the worst. 

These results provide valuable insights into the performance of different estimators and kernel 

functions in nonparametric regression models and will help in future research and decision-making 

processes. 

 

7. Recommendations 

1. We recommend considering the use of cubic smoothing estimators, in addition to kernel 

and local polynomial regression functions, for future research. 

2. It is advisable to investigate methods for detecting outliers and assessing linear correlations 

among variables when conducting future studies. 

3. Exploring alternative functions beyond those employed in this research is encouraged for 

a more comprehensive analysis. 

 

Acknowledgment 

         The authors are very grateful to the executive manager and editorial board members of the 

Ibn AL-Haitham Journal of Pure and Applied Sciences. 

 

Conflict of Interest  

         The authors declare that they have no conflicts of interest. 

 

Funding 

        There is no funding for the article. 

 

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