Linear spline mapping in Normal Distribution S. BOONTHIEM, S. BOONTA, W. KLONGDEE Khon Kaen University, THAILAND +669-1059-0942, [email protected]

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1 Introduction

The inferential statistics can describe the behavior of data by parametric statistics and non-parametric statistics depending upon trend of the data. The trend of data indicates the shape of the distribution. Distribution theory is closely related to probability theory and important in order to calculate and approximate a probability distribution. Nowadays, many researchers are interested in the approximation of the standard cumulative normal distribution function, due to the integral of normal distribution does not have closed form theoretical expression. However, it give only approximation as the work of many researchers. For instances, Bowling, Khasawneh, Kaewkuekool, and Cho (2009) developed a logistic approximation to the cumulative normal distribution. The proposed algorithm has a simpler function form and gave higher accuracy with maximum error of less than 0.00014 for the entire range [1]. Vazquez-Leal et al. (2012) provided an approximate solution to the normal distribution integral by using the homotopy perturbation method (HPM). The HPM method have a high level of accuracy [2].

Choudhury (2014) proposed a very simple approximation formula to the standard cumulative normal distribution function. The formula could be implemented in any hand-held calculator. The maximum absolute error of the approximation was 0.00019 [3].

This article interested in the case study of the approximation of the cumulative normal distribution using linear spline mapping. We expect that this method can use to other symmetrical distributions. Linear spline is widely method for estimation in many fields such as

statistics, mathematics, industrial engineering and so forth. In 1999, Chen studied the estimation of the probability density function and the cumulative distribution function of random variable. There were three spline density estimators consisting of a quantile regression spline estimator and two maximum likelihood spline estimators. These methods proposed for variable observed with measurement error [4]. In the same year, Lindstrom improved the computational and estimation properties of free-knot splines while retaining their adaptive smoothing properties and proposed estimator for the knots defined as the optimizer of a slightly penalized residual sum-of-squares [5]. In 2002, Zhang and Lin investigated the optimal models for building histograms based on linear spline techniques and presented efficient algorithms to achieve these proposed optimal models. The experimental results showed that these techniques could greatly improve the approximation accuracy comparing to the existing techniques [6]. In 2012, Holland employed penalized B- splines in the context of the partially linear model to estimate the non-parametric component, when both the number of knots and the penalty factor vary with the same size [7]. In 2013, Valenzuela, Pasadas, Ortuño, and Rojas presented a novel methodology for optimal placement and selected of knots, for approximating or fitting curves to data, using smoothing splines [8]. In 2015, Kang, Chen, Li, Deng, and Yang proposed a computationally efficient framework to calculate knots for splines fitting via sparse optimization [9]. In the same year, Tjahjowidodo, Dung, and Han proposed a fast method for knots calculation in a B-spline fitting based on the second derivative [10]. In 2017, Hussain, Abbas, and Irshad proposed a new quadratic

Linear spline mapping in Normal Distribution

S. BOONTHIEM, S. BOONTA, W. KLONGDEE Khon Kaen University, THAILAND

+669-1059-0942, [email protected]

Abstract— This article presents a construction of a new distribution by using linear spline mapping based on probability density function of normal distribution where two end points have a value of probability density function as zero. In addition, we propose the cumulative distribution function and the inverse of cumulative distribution function of the distribution. Furthermore, we illustrate the parameter estimation of 77 data of the student’s average intelligent quotient (IQ) for Grade 1 in Thailand by method of moments and propose minimum KS-statistics of the distribution by difference of the node width.

Keywords— Linear spline mapping, method of moments

WSEAS TRANSACTIONS on MATHEMATICS S. Boonthiem, S. Boonta, W. Klongdee

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trigonometric B-spline with control parameters was constructed to address the problems related to two dimentional digital image interpolation [11]. Our study is to construct a new distribution using linear spline mapping on probability density function of normal distribution and obtain the cumulative distribution function after that. Furthermore, this study proposes the inverse of cumulative distribution function that have useful for generating the random variable of some data.

2 Main Results

In this section, we interpret the principle of linear spline mapping through on the following definitions, propositions, and theorems. Throughout this article, we use the following notations,

  

1

   

1

    

,

D i D i D i

T f x_ T f x_ T f x

   x_i_₁x_i_₁x_i,

1

2

i i

i

x x

x  ^

 , f x

 

_i_1  f x

 

_i_1 f x _i ,

  

1



 

1 ,

2

i i

i

f x f x

f x_ ^ 



     1

2 .

k k

k

x x

x  

  ^



Definition 1. Given

  

^: is continuous and bounded



CB   f  f ,

0, 1, , _n ,

x x  x  for each x₀x₁x_n and let

 0, ,1 , _n

D x x  x be a partition which corresponding to the function T_D:C_B  C_B  which is given by

1. T_D



f x

 

0



T_D



f x

 

_n



0,

2. TD



f x

 

k



 f x

^{ }

k , for all k1, 2,,n1.

TD is called a linear spline mapping which corresponding to the partition Dx x0, ,1,x_n. Remark 1. From the Definition 1, we obtain that for all

, x  D

  



^{ }



_ _ _ _{ }_

 

1

1 1

,

0 1

1

, .

i i

n D i

D i D i x x

i i

T f x

T f x x x T f x x

x x x x

 _

 

  



  

  

 

      

 



Proposition 2. Let Dx x0, ,1 ,x_n be a partition which corresponding to the

functionT_D:C_B  C_B  . Then TD is a linear operator.

Proof. To show that TD is a linear operator, we prove two properties as the following:

1. TDf g x TD  f x TD  g x , for all

 

, _B .

f gC 

2. T_D  f x T_D  f x , for all   and

 . f CB 

Let f g, C_B  x. Consider

     

D D

T f x T g x

 

_ _ _{  }

 

1

1 1

,

0 1

i i

n D i

i D i x x

i i

T f x

x x T f x x

x  _

 

  

  





     

 

_ _ _{  }

 

1

1 1

,

0 1

i i

n D i

i D i x x

i i

T g x

x x T g x x

x  _

 

  

  





     

       _ _ 

0 1 1

0 0 ,

1 D

D x x

T f x

x x T f x x

x 

  

       

        _ _ 

1 1 1,

n n

D n

n D n x x

n

T f x

x x T f x x

x 

  

    

   

 

 

        _ _ 

0 1 1

0 0 ,

1

T^D g x _D _{x x}

x x T g x x

x 

  

       

        _ _ 

1 1 1,

n n

D n

n D n x x

n

T g x

x x T g x x

x 

  

    

   

 

 

  _ _ _ _{ }

 

1 1

1

,

0 1

i i

n

D i

i D i x x

i i

T f g x

x x T f g x x

x  _

 

  

   





      

T_Df g x . Let  .Consider

   TD f x



 

_ _ _{  }

 

1

1 1

,

0 1

i i

n D i

i D i x x

i i

T f x

x x T f x x

 ^ x ^ _ _

 

  





     

 

_ _ _{  }

 

1

1 1

,

0 1

i i

n D i

i D i x x

i i

T f x

x x T f x x

x

   _

 

  

  





     

 

_ _ _{ }

 

1 1

1

,

0 1 ⁱ ⁱ

n

i

i i x x

i i

f x x x f x x

x

   _

 

  

  





     

 

_ _ _{  }

 

1 1

1

,

0 1 ⁱ ⁱ

n

D i

i D i x x

i i

T f x

x x T f x x

x

   _

 

  

  





     

  . TD f x



Q.E.D.

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Proposition 3. Let



CB

 

 ^{, .}_



be a normed space for allx, Dx x0, ,1 ,x_n be a partition which corresponding to the function T_D:C_B  C_B  and

TD be a linear spline mapping. Then TD is bounded linear operator.

Proof. Let f C_B  . Consider

D

 

T f _ ^^max



^{f x}

 

⁰ ^, ^{f x}

 

¹ ^,^^, ^{f x}



ⁿ^¹



^, ^{f x}

 

ⁿ



^sup



 



x

f x







 f _.

Q.E.D.

Remark 4. Let ^L¹^B^



^f ^:



 ^{f dx}^{ }



^,^f ^^L¹^B^{ } ,

 0, ,1 , _n

D x x  x be a partition which corresponding to the function T_D:C_B  C_B  and IT_D:L¹_B   such that ITD

 

f ^



T fD

 

x dx^.





 ^Then^IT^Dis a linear operator.

Proposition 5. LetDx x0, ,1 ,x_n be a partition which corresponding to the

functionT_D:C_B  C_B  ,



L¹B

 

 ^{, .}_



be a normed space for all x, and IT_D:L¹_B   be a linear spline mapping. Then IT_Dis bounded.

Proof. Let f L¹B  . Consider

 

ITD f _

       



⁰ ¹ ¹



max n f x ,n f x , ,n f x_n_ ,n f x_n

 

       



0 1 1



max , , , _n , _n

n f x f x f x_ f x

 

  sup

x

n f x



 

n f _

 .

Q.E.D.

Theorem 6. Let Dx x0, ,1,x_n be a partition corresponding to a linear spline mapping

   

D: B B

T C  C  . Then



 

  

0

1

1 1

0

n n .

x

D i i

x i

T f x dx f x x



 









Proof. Consider



 



0 xn

x TD f x dx



 

    _ 

1

1 1 2

1

,

0 1 2

i

i i

i x x n

i i

i x x

i i

x x

f x x x

f x x x

x 





 

  



  

 





   

  

² _{ }

1 1 1

1

0 1 2

n i i i

i i

f x x x

f x x x

  



 

  

 





   

 

   

1 2

1

0 1 2

n

i i i

i i

f x x x

f x x x

 

 

  

 





   

 

_{ }

1 1

0 2

n i

i i i

i

f x x f x x

 

 



 

 





    

 

1

1 1

0

.

n

i i

i

f x x



 







Q.E.D.

Definition 2. Let Dx x0, ,1,x_n be a partition corresponding to a linear spline mapping

   

D: B B

T C  C  . The partition D is called a Ppartition corresponding to T_D for a probability density function f C_B  , if

   

^1.

TD f x dx



 



In addition, T f_D is a probability density function.

Theorem 7. Let x, Dx x₀, ,₁ ,x_n,

0 1 n

x x x where

, 2, 3, , 1 , 1 or ,

i

d i n

x i i n

  

    



 and d0,

  ¹  

1 2

1 1

n i i

x d

x 







 

 

  





 ,

   

D: B B

T C  C  be a linear spline mapping and

 x

 be any probability density function. Then D is Ppartition for the probability density function .

Proof. Consider

   

TD  x dx









 



0 xn

x TD  x dx







 

 

 

 

 



1 1

0 1 1

ⁿ ⁿ

n

x x x

D D D

x T  x dx x ^T  x dx x T  x dx















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 1 ¹    

2

n

i n

i

x x d x

 ^  



 ^



 ^

 1 ¹  

2 n

i i

x x d

 ^ 



 ^



 1   ¹   ¹  

2 2

1

1 1

n n

i i

x x d x d

 x  



 

 

 

 

  







1.

Thus D is Ppartition for a probability density

function

.

Q.E.D.

Corollary 8. Let Dx x0, ,1 ,x_n be a Ppartition corresponding to a linear spline mapping

   

D: B B

T C  C  where x₀x₁x_n, , 2, 3, , 1 , 1 or ,

i

d i n

x i i n

  

    



 0,

d

  ¹  

1 2

1 1

n i i

x d

x 







 

 

  





 and  x be any

probability density function. Then D is a partition corresponding to a T__D__for a probability density function of normal distribution^N



^{ }^, ²



^.

Proof. For each x_iD, we consider

 _i ; ,  f x   

1 2

2

xi

e

  



 

   

 

  



1 2

2

xi

 e

 

¹

 

xi



such that  is a probability density function of a standard normal distribution. Consider

 



^{; ,}



D i

T_ _ f x dx

    



  



1  

T_D _ x dx

 





 

 





 



 



1

T_D _ x dx

 





 





 

    _ 

1

1 1 2

1

,

0 1 2

i

i i

i x x n

i i

i x x

i i

x x

x x x

x x x

x

  





 

  



  

 





   

 ^{ } ^       

2 2

1

1 1 1

1

0 1 2 1 2

n

i i i i i

i i i i

i i i

x x x x x

x x x x

x x

 

   



  

    

 

    

 

   

 



 

_{ }

1

1 1

0 2

n

i

i i i

i

x x x x

 

 

 



 

 







    

 

1

1 1

0 n

i i

i

x x

 

 







1.

Q.E.D.

Next, we illustrate to support our theorems that help reader to understand the approach. Firstly, we consider

 

0

1 0 1

1 2

2 2 3

3

0 ;

5 5 ;

5 ;

5 5 ;

0 ; , x x

x x x x x

f x x x x

x x x x x

x x

 



    



     

 





and set ¹

d6, we calculate  value as the following equation:

 

 1  2 1

1 1 5 5 1 1

1 1

2 5 2 6 30

f x f x f x d

       

    

       

 ,

and then we get

 

3

0 x

x

f x dx



^¹₂ ^{f x}^{ }¹ ^^ ^{f x}^{ }¹ ^₂ ^{f x}^{ }² ^d^¹₂ ^{f x}^{ }² ^

¹ ⁵ ¹ ⁵ ⁵ ¹ ¹ ⁵ ¹

2 30 2 6 2 30

        

   

           1.

Theorem 9. Let Dx x0, ,1 ,x_n be a Ppartition corresponding to a linear spline mapping

   

D: B B

T C  C  , x0x1xn where , 2, 3, , 1 , 1 or ,

i

d i n

x i i n

  

    



 0,

d

  ¹  

1 2

1 1

n i i

x d

x 







 

 

  





 ,  be probability

density function of normal distribution and xi  f x i . If ¹  

2

0 1

n i i

 x d









 ^,

  min : _k N x  k xx ,

 



 



_i,_i₁_ x

D D x x

CT f x T f s  ds

 





 , and given