The Research on Interpolation Methods and Fitting Models for the Lorenz Curve45302

The Research on Interpolation Methods and Fitting Models for the Lorenz Curve Li Zhang 1 ,2 , Kien Nguyen The 3,* , Youjian Qi [4] , Lizhen Wei [5] ，Khac Lich Hoang [3] , Manh Hung Le

The Research on Interpolation Methods and Fitting Models for the Lorenz Curve

Li Zhang (1) ,(2) , Kien Nguyen The (3),(*) , Youjian Qi [4] , Lizhen Wei [5] ，Khac Lich Hoang [3] , Manh Hung Le (6)

(1) School of Mathematics and Statistics, Changshu Institute of Technology, Suzhou, China

(2) Business School, Nanjing Normal University, Nanjing, China

(3) VNU University of Economics and Business, Vietnam National University, Hanoi, Vietnam

(4) Yangzhou High School of Jiangsu Province, Jiangsu, China

(5) Jiangsu Province Changshu Vocational Educational Center School, Suzhou, China

(6) Ministry of Education & Training, Hanoi, Vietnam

* Correspondence: thekien.edu@gmail.com

Abstract: The Lorenz curve is very important to show the income distribution situations for a country Based on the work of other scholars, the paper at first discusses some exiting interpolation methods and points out shortcomings among them In order to overcome the existing of higher order derivatives for Lagrange interpolation, this paper sets up one method and makes some analysis about the methods Then this paper puts forward a new family of Lorenz curve and discusses the corresponding property With some collected income dates of some regions in China, statistical indexes shows the better results compared with other existing curves

Keywords: Interpolation Method, Lorenz curve

1 Introduction

From the last century since the reform and opening, with the continuous development of economy, the overall strength of our country and economic conditions have been improved greatly But some provinces rural poverty still exist in remoted areas The party and the government have put great importance to poor problem in our country with the special economic development policy and fiscal subsidy (Bai and Cao 2007; Guo 2007) and so on These efforts by the government have achieved obvious results In the spring of

2014, Prime Minister Li Keqiang, on behalf of the central people government to do the work report, pointed out last year to reduce rural poverty population 16.5 million and the gap between urban and rural residents income continues to shrink At the same time, the United Nations say yes for China in recent stage poverty alleviation work results According to the wealth of the credit suisse report in October 2014, the phenomenon of the unequal distribution of wealth more intensified, the Gini coefficient rising around the world Chinese Gini coefficient has gradually rise, cordon and to secure the red line approximation (Mei and Fan 2005; Wang and Fan 2005; Hu 2004; Liu 2006) This is must attach great importance

to the problem, involving the stability and prosperity

In economics, the Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth, and was developed

by Max O Lorenz in 1905 for representing inequality of the wealth distribution The Lorenz curve is used to compare and analysis of a country in a different age or wealth inequality of different countries at the same time, the curve as the convenience of a summary of income and wealth distribution information of graphic method is widely used Through the Lorenz curve, we can visually see a national income distribution equality or inequality Draw a rectangle, rectangle high measure the percentage of social wealth, will be divided into five equal parts, each class is divided into 20 social total wealth In rectangular to nod, 100 families from the poor to the very richest arranged from left to right, is divided into five parts, the first equal parts on behalf of 20 of the lowest income families

This paper is organized as follows In section 2, we discusses some interpolation methods and analyze the characteristics of them Then one kind of function is advocated to get any higher order derivatives for Lagrange interpolation In section 3, describe the basic property about the Lorenz curve and Gini coefficient Some literature on it is given out in recent years In section 4, we put forward the new construction for Lorenz Curves and new expressions of the new family of Lorenz curve are given out In section 5, based on our new function for Lorenz Curve and other functions, the computations about some statistical indexes such as MAS, MAE, MES are done and the results are reported In section 6, some concluding remarks are addressed at last

2 The Interpolation Methods About Lorenz Curve

A large number of engineering problems involving the unknown function approximation Usually a set of observations, then use the appropriate interpolation method can calculate the function approximation But when making the error analysis, the function must have good properties with 1 to n  1order derivative In the practical engineering problems, however, the function can ensure continuity, but it is very difficult to obtain higher derivative and guarantee the existence of the higher derivative

2.1 Analysis of Some Interpolation Methods about Lorenz Curve

Since the Lorenz curve was put forward in 1905 and the collected data are discrete, so some scholars have paid attentions to the interpolation method for it Gastwarth gives out the Hermite interpolation method for Lorenz curve Other scholars have put forward linear interpolation method and Newton interpolation method In this part, we gives out the concrete formulas then discussed advantages and shortcomings of these method It is known that interpolation method is to construct the Lorenz curve using the collected data

)

,

( xi yi ,i0,1,2,,n about the income distribution The simple way of interpolation

method is linear interpolation That is to use a line to connect two adjacent points( x yk, k) and ( xk1, yk1) with the formula as following:

L1( x )  l0( x ) yk  l1( x ) yk 1,

Where:

1

1











k k

k

x x

x x x

l

k

x x

x x x l







1

The basic thought of Hermite interpolation is the piecewise interpolation nodes, but need to connect the piecewise function curve smooth, forming a smooth curve The most commonly used cubic spline interpolation For the nodes ( xi, yi) , i0,1,2,,n the

concrete expression in the interval[ , x xi i1] of Hermite interpolation is following:

( ) , 0,1, , 1

3





x

Then through some conditions it is possible to determine the coefficients Newton interpolation N x( )is following:

Where:

) ( ) ( )

(x L x R x

,

0

( ) [ ] [ , ]( ) [ , , , ] ( ) ( )

n

i



0

( [ , , , ] [ , , , ])

k

k











so the error estimation is R xn( ).

The piecewise linear interpolation method is very simple, but it can't guarantee to

be differentiable at the nodes, more far from the second order differentiable For the Lorenz curve, the piecewise linear interpolation method is on the curve, so it is certain to make the Gini coefficient smaller So the method is not limited The Hermite interpolation method is used in some fields but it has some shortcomings The first one is the computations are complicated when the numbers of nodes is large The second one is the error estimation Because the defined curve is not expressed clearly with discrete data and the higher order

of derivative (more than 3) is unattainable So the error estimation about Hermite interpolation method is not too accurate

2.2 Analysis of New Method for Lagrange Interpolation

The function ( )x is defined as following:

2 2

1

, ( )

0,

x













 



， (1)

Where is a constant and 2 2

1







  The constant a is positive It is

obvious that the function ( )x is continuous and differentiable if x  a

Now we consider that expression that

lim ( ) lim 0 0,

    (2)

At the same time, it is also true that

1

lim ( ) lim a a x 0

a









     (3)

So the function is continuous at the point xa It is similar that the function is

continuous at the point x a So the function ( )x is continuous in R, and it is

differentiable in the open interval(a a, ) So the function( )x is continuous in R, and it

is differentiable in the open interval(a a, ) where a isa positive constant On the other

hand, the integration of the function is on R is 1, it is expressed as following:

( ) ( ) a ( ) 0 1 1

   (4) Theorem 2.1

The function ( )x is the one as in before and the functionm( ) x is defined as

following :

m( ) x  m mx m  ( ),  1, 2, , (5)

So the following conclusions are true

(1) m( ) 1.

 (6)

(2) the compact support set of the function m( ) x is [ a a, ]

m m

 (7)

Proof

The proof of The Theorem 2.1 is following

m( ) ( ) ( ) 1.

R x dx  Rm  mx dx  R u du 

   (8)

When x a ,mx a,

m

  Thus m( ) x  m mx  ( )  0

Theorem 2.2

The function f x( )is continuous in R and the function m( ) x is the one as before

And the function fm( ) x is defined as following:

m( ) ( ) m( ) , 1, 2,

R

f x   f y  x  y dy m   (9)

So the following conclusions are true for anyx  R

（1） lim m( ) ( )

  (10)

（2） fm( ) x  C( ), R m  1, 2,  (11) Proof

Let us proof the first conclusion in this theorem

Set y   v x , so it is clear that

f x   f y  x  y dy   f v  x   v dv   f v  x  v dv (12)

Thus we consider the absolute value of two functionf x( ), fm( ) x which is in the following:

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

( ) ( ) ( )

( ) ( ) ( )

sup ( ) ( ) ( )

sup ( ) ( )

m

m R

v m

v m

a

v

m m

a v m





























(13)

Because the function is continuous in R, the following is true:

lim sup a ( ) ( ) 0

m

(14) Now we give out the proof about the second conclusion It is known thatf y( ) is

continuous on R and the function fm( ) x  C( ) R on the interval(x a,x a)

  So we see the following express:

( ) ( ) ( ) ( ) ( ), 1, 2,

a m a

m



      (15)

make the p derivative about the function fm( ) x , then we have the following

expression as below:

( )p ( ) ( ) ( ) ,( 1, 2, )

x





  (16) The functionsfm( ) x , f x ( ) are defined in the former part Now we consider the

interpolation polynomial for fm( ) x But That function fm( ) x , we have not any observation

points There are one group of observation points ( xi, yi),i0,1,2,,n By the related

theory, for any  0, there exists N 0, when m  N ,for any k 0,it is clear that

fm( ) x  f x ( )   (17)

Set y mk( )  fm( xk),yk  f x ( k),If y mk( )  fm( xk) is substituted byyk  f x ( k),

this kind of error estimation is under control for the conclusion above Set the function Rn m,

as following:

( )

( 1)!

f

n





 (18)

For any x  R, It is obtained whenm   that

,

( ) ( ) ( ( ) ( )) ( ( ) ( )) ( ) ( ) 0

R x R x  f x P x  f x P x  f x  f x  (19)

3 The Research on Fitting Models for The Lorenz Curve

Income distribution is related to the broad masses of people’s standard of living, the degree of distribution justice is the key point for ordinary people To measure the distribution of the residents¡¯ income level, we often adopt the Lorenz curve Here, the function L p( ) is equal to p low-income population share end has a share of the total, is

defined in the function on the interval pF x( )said the proportion of people earning less

than or equal to x, where F x( ) is the distribution function of income distribution The

function f x( ) income distribution density function where

3.1 Basic Theory and Related Research

Empirical analysis of income distribution, income distribution curve generally is what is called the forward bias, the peak point to the left and the right end, dragging a long tail Point x0is called the modal point, mis the median number,  is the average income

It is clear that x0 m  under this circumstance Conditions by above knowable, L p( )

can be expressed as:

0

1 ( ) x ( ) ( )



   (20)

There are the relation between the function L p( )and the function f x( )

'( ) , ( ) ''1

( )





  (21)

Because pF x( ), known of inverse function for x  F1( ), p and L p( ) can be expressed as the Lorenz curve

1 0

1 ( ) q ( )





  (22)

And because China statistical yearbook published on so-called packet data in the form of:

( , i), 1, 2, ,

i

i

p

x



  (23)

( , ), p L ii i  1, 2,  , n (24)

It is well known that L p'( )

x



( )

(24) expresses the points in the curve of L p( )

3.2 The Construction of New Lorenz Curve

It is vital to set up the necessary conditions for Lorenz curve which is the definition

of the Lorenz curve, we call full enough under the condition of the curve

is a lorenz curve The curve which is satisfied with the following conditions is called

a Lorenz curve

(1) L(0)0, (1)L 1; (25)

(2) L p( )0,p[0,1]. (26)

(3) L p( ) is the increasing function about p , which satisfy the L

'

L p 

represents that the greater the population share of low-income has the corresponding group has the greater the share of the total income

(4) L p( ) is a convex function about p, which satisfies

''

L p  said is when p

increases, the L p( ) to a larger proportion increase

A number of parametric models that satisfy the basic properties of a LC have been proposed in the literature See, for example, Kakwani and Podder (1973, 1976), Rasche et al.(1980), Gupta (1984), Rossi (1985), Arnold (1986), Rao and Tam (1987), Basmann et al (1990), Ortega et al (1991), Chotikapanich (1993), Ogwang and Rao (1996, 2000), Sarabia

(1997) and Wang et al (2009) In those papers, the scholars advocates the different models for LC Some of the concrete results is listed in the Table 1

Table 1 Some Models for A Lorenz Curve

Time Authors Model

L p  p e  

1980 Rasche

L p    p 

1991 Ortega et al

L p  p   p  

1991 Ortega et al

L p  p   p   

1999 Sarabia et al

L p  p   p  

2000 Ogwang

1 ( )

1

p

e

L p

e











2009 Wang et al

L p  p   p e 

From the character of an arch function, we set the function expression is following:

I   Ap  p  (27)

This kind of the function can show the different arches with the different values of parameterand  When , the arch of the function tend to right When, the

arch of the function tend to left Combined with the 45 degree line and the curveI0, So we can get the new function for Lorenz curves which is shown in the following:

I  p  Ap(1  p ) , (28)

Where the parameters meet the following conditions:

0,0 1,0 1

A    (29)

From the definition of Lorenz curve, we can find that the function I is satisfied with the following conditions:

L (0)  0, (1) L  1, ( ) L p'  0, ( ) L p''  0. (30)

So it is clear that the function I is the new expression for a Lorenz curve Compared with others’work, this expression is out of the usual GP model with more generality to be suitable for the real world Studying the model given by Sarabia et al (1999), we put forward the new family for Lorenz curve which is shown in the following expression

L p( )  p[ p  Ap(1  p ) ], (31)

Since most models for Lorenz curve are dependent on the classical Pareto curve which form is 1 (1   p ) In this expression of Pareto curve we can see that there is nop Based on the related theory, L p( ) is a Lorenz curve Combined the demand for parameters

in the function L(p) , the conditions for parameters in the function

( )

L p is following:

A0,01, 0 1, 0 1. (32)

3.3 Parameters Fitting and Model Comparison With the collected data which reflects the income of some province in China which is in the

following table:

Table 2 The Data

j

x

xj1 fj

pj

Lj