Constructing a new family distribution with methods of estimation

The estimation of the model parameters is performed by maximum likelihood method. We hope that the new distribution proposed here will serve as an alternative model to the other models which are available in the literature for modeling positive real data in many areas.

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CONSTRUCTING A NEW FAMILY DISTRIBUTION

WITH METHODS OF ESTIMATION

Rawa M Saleh

Department of Statistics, Economic and Administration College,

Al – Mustansryia University, Iraq

ABSTRACT

A new parameter ( ) is introduced to expand the family of two parameters Kumarasmy to obtain new generated transmuted Kumarasmay distribution The , C.D.F and moment of this distribution are studied, parameters ( , , ) were obtained by moment and maximum likelihood method, and regression estimator

Key words: Transmuted Kumarasmay Distribution, Moment Estimators, Maximum likelihood

Estimator and regression estimator

Cite this Article: Rawa M Saleh, Constructing a New Family Distribution with Methods of

Estimation International Journal of Management, 7(6), 2016, pp 189–191

http://www.iaeme.com/IJM/issues.asp?JType=IJM&VType=7&IType=6

1 INTRODUCTION

We can expand family of any distribution by introducing new parameter to the given p.d.f In this paper we work on expanding Kumarasmay distribution with two parameters ( , ) to another family using the parameter ( ) from some quadratic transformation on the given C.D.F [ ( )] to obtain a new Cumulative distribution function [ ( )], then new generated transmuted [ ( )] Many researchers work on this new mathematical formulation like Ashour and Eltehiwy (2013)[5], studied a generalization of the Lomax distribution so-called the transmuted Lomax distribution is proposed and studied Various structural properties including explicit expressions for the moments The estimation of the model parameters is performed by maximum likelihood method We hope that the new distribution proposed here will serve as

an alternative model to the other models which are available in the literature for modeling positive real data in many areas Merovic (2013)[11], generalize the Rayleigh distribution using the quadratic rank transmutation map studied by Shaw et al (2009) to develop a transmuted Rayleigh distribution We provide a comprehensive description of the mathematical properties of the subject distribution along with its reliability behavior The usefulness of the transmuted Rayleigh distribution for modeling data is illustrated using real data Aryal, G.R and C.D Tsokos (2009)[2], studieda functional composition of the cumulative distribution function of one probability distribution with the inverse cumulative distribution function of another is called the transmutation map In this article, we will use the quadratic rank transmutation map (QRTM) in order to generate a flexible family of probability distributions taking extreme value distribution as the base value distribution by introducing a new parameter that would offer

more distributional flexibility It will be shown that the analytical results are applicable to model real world data

2 THEORETICAL ASPECT

2.1 Transmuted Kumarasmay Distribution

The two parameters ( , ), p.d.f of Kumarasmay distribution is giving by;

( ; , ) = (1 − ) 0 < < 1 (1) And the cumulative distribution function C.D.F is;

( ; , ) = 1 − (1 − ) (2)

We can obtain the new p.d.f called transmuted distribution by introducing parameter ( ) using quadratic transformation on the cumulative distribution function;

( ) = (1 + ) ( ) − ( ) | | ≤ 1 (3)

Then;

( ) = ( ) (1 + ) − 2 ( ) ( ) = (1 − ) $(1 + ) − 2 %1 − (1 − ) &'

The new transmuted of Kumarasmay distribution is;

( ) = (1 − ) $1 − + 2 (1 − ) ' (5) And its C.D.F is;

( ) = (1 + )$1 − (1 − ) ' − $1 − (1 − ) '

= $1 − (1 − ) ' (1 + − $1 − (1 − ) ')

= $1 − (1 − ) '$1 + (1 − ) ' (6)

To derive the moments about origin *+,;

*+, = -( +) = +

/ ( ) (7)

= (1 − ) 0 +1

After some steps using transformation, we find;

Let 2 = → = 254 = 254 2

*+, = (1 − ) 0 62457+1

/ (1 − 2) 1254 2 + 2 0 62457+1

*+, = (1 − ) 0 258

/ (1 − 2) 2 + 2 0 285

/ (1 − 2) 2

*+, = (1 − )9:;< 6+ + 1, 7 + 2 9:;< 6+ + 1,2 7 (8) Equation (8) shows the moments of this transmuted Kumarasmay distribution

If | | ≤ 1 we can form two equations ( = 1,2) from (*+,) and equating (*+, = ∑@?A4>?8

B ) to find

C DEFG, HEFGI

When ( ) is unknown, we can find three moment estimators of C DEFG, DEFG, HEFGI from solving (*+, =∑@?A4>?8

B ) for ( = 1,2,3)

2.2 Maximum Likelihood Estimator

Let ( , , … , B) be a random variables from in (5), then;

L = M ( N)

B NO

= B BM N

B NO

M(1 − N )

B NO

M$1 −

B

NO

+ 2 (1 − N ) ' (9) log L = T log + T log + ( − 1) U log N

B NO

+ ( − 1) U log(1 − N )

B NO

+ U log$1 − + 2 (1 − N ) '

B NO

(10)

V log L

V = T+ U log N

B NO

+ ( − 1) U(− N ) log( N)

C1 − N I

B NO

+ U1W

B NO

VW V Where;

VW

V = 2 (1 − N ) (− N ) log( N)

= −2 N log( N) (1 − N )

V log L

V =TH + U log N

B NO

− ( − 1) UC NXI log( N)

C1 − NXI

B NO

− U2 N log( N) (1 − N )

(1 − + 2 C1 − N I )

B NO

= 0 (11)

Solved numerically to obtain (HEYZ)

V log L

V = T+ U log(1 − N )

B NO

+ U2 (1 − N ) log(1 − N ) (1 − + 2 C1 − N I )

B NO

= 0 (12)

Equation (12) can also be solved numerically to find ( DEYZ)

Now we can restricted | | ≤ 1 to estimate ( , ) only

2.3 Proposed Regression Estimators (PRE)

Let , , [… … \ be a random sample from P.D.F defined in (5), than

2N = N (1 − N ) (1 − + 2 (1 − N ) )

Since| | ≤ 1, using this restriction on λ, we can estimate the two parameters (α, β) by regression

estimators as follows:

Let

/ =

= − 1

= β − 1

N = log ^

N = (1 − ^_) The model can be written as:

log 2^ = log( ) + ( − 1) log ^+ ( − 1) log N+ log(1 − λ + 2 Na) + bN Using least square procedure and according to

Dcd = ( )e 2́

We can use Dcd to estimate H/, H , H

After defining the variables

N = log ^

N = (1 − ^_)

Since |λ|≤1 we can restricted

[N = (1 − + 2 ( N) ) By

When = 1

[N = 2( N)a) (13) Similarly for any value of λ we can compute

[N = (1 − λ + 2λ( N)a) (14) Now the final form of Regression Model is

2gh = log /+ log N+ ( − 1) log N+ log [N+ bN (15)

[Nis implicit function of (λ, N,β) and then ( Dcd) we use:

D = ( )e 2́

( )e =

i

j

j

j

j

U N U N N U N [N

U N U N [N

m m m m n

2́ =

i

j

j

j

j

k U 2N

U N2N

U N2N

U [N2Nl

m m m m n

According to the values of observation N and generated values of 2N (from C.D.F of this transmuted probability distribution, we estimate the parameters by using regression estimator

3 SIMULATION PROCEDURES

To find the estimator's (oL- , opq:T;, <T r: :sstpT) we perform simulation experiments using Monte Carlo assuming that;

Table 1 Estimating values ( = 2, = 1.5, = −0.5)

30

60

90

100

Table 2 Estimating values ( = 1.5, = 1.5, = −0.5)

30

60

90

100

Table 3 Estimating values ( = 1.5, = 2, = −0.5)

30

60

90

100

Table 4 Estimating values ( = 2, = 1.5, = −1)

30

60

90

100

Table 5 Estimating values ( = 1.5, = 2, = −1)

30

60

90

100

Table 6 Estimating values ( = 1.5, = 1.5, = 0.5)

30

60

90

100

Table 7 Estimating values ( = 1.5, = 2, = 0.5)

30

60

90

100

Table 8 Estimating values ( = 4, = 1.5, = −0.5)

30

60

90

100

Table 9 Estimating values ( = 4, = 2, = −0.5)

30

60

90

100

Table 10 Estimating values ( = 2, = 2, = −0.5)

30

60

90

100

4 CONCLUSION

• The new generated family obtained from expanding two parameters Kumamasmay, to anew transmuted family with third parameters (α, β, λ), where (λ) extend the flexibility and help in finding the better description to data without effecting the parametric Model

• When | λ |≤ 1 we need only to estimate (α, β)

• The formula of moments about origin is derived, and applied in the method of moments estimators

• The proposed method of estimation is Regression estimators which depend on least square method estimators is explained in this research and any statistical program can be used to find the parameters estimators αXxy , βzxy, λzxy

• We find the best estimators, first were maximum likelihood estimators, then regression estimators, while the third was moment estimator, this indicate that we can use the methods of linear estimation (least square method) as well as parametric method used in inference, for the purpose of estimation

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