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.
International Journal of Management (IJM) Volume 7, Issue 6, September–October 2016, pp.189–191, Article ID: IJM_07_06_021 Available online at http://www.iaeme.com/ijm/issues.asp?JType=IJM&VType=7&IType=6 Journal Impact Factor (2016): 8.1920 (Calculated by GISI) www.jifactor.com ISSN Print: 0976-6502 and ISSN Online: 0976-6510 © IAEME Publication 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 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 http://www.iaeme.com/IJM/index.asp 189 editor@iaeme.com Rawa M Saleh more distributional flexibility It will be shown that the analytical results are applicable to model real world data THEORETICAL ASPECT The two parameters ( , ), p.d.f of Kumarasmay distribution is giving by; 2.1 Transmuted Kumarasmay Distribution ( ; , )= (1 − ) 0< ? B ) to find When ( ) is unknown, we can find three moment estimators of C D EFG , DEFG , HEFG I from solving (*+, = ∑@ ?A4 >? B Let ( , ) for ( = 1,2,3) ,…, be a random variables from in (5), then; 2.2 Maximum Likelihood Estimator B) B L = M ( N) = NO B B B M NO + (1 − log L = T log + T log B B N ) ' N M(1 − B + ( − 1) U log NO + U log$1 − + (1 − NO B V log L T = + U log V N NO Where; VW =2 V B V log L T = + U log V H NO N − ( − 1) U Solved numerically to obtain ( HEYZ ) B V log L T = + U log(1 − V NO N NO B )+U NO N N ) (− (− C1 − N N (1 − + C1 − N […… \ N N I B +U NO VW WV ) log( N ) (1 − (1 − + C1 − N ) I ) Equation (12) can also be solved numerically to find ( DEYZ ) Now we can restricted | | ≤ to estimate ( , ) only , ) ) log( N ) log( N ) (1 − ) log(1 − N N (10) ) log( N ) N B X N I log( N ) −U X C1 − N I NO (1 − (9) NO ) ' NO N NO + ( − 1) U log(1 − + ( − 1) U (1 − C N M$1 − B B = −2 B ) N NO B =0 N N ) I ) = (11) (12) 2.3 Proposed Regression Estimators (PRE) Let , be a random sample from P.D.F defined in (5), than http://www.iaeme.com/IJM/index.asp 183 editor@iaeme.com Rawa M Saleh 2N = N = −1 (1 − N ) (1 − + (1 − N ) ) Since| | ≤ 1, using this restriction on λ, we can estimate the two parameters (α, β) by regression estimators as follows: Let / = N = log =β−1 ^ N The model can be written as: ) + ( − 1) log log 2^ = log( Using least square procedure and according to Dcd = ( e ) ^ = (1 − ^ _) + ( − 1) log N + log(1 − λ + N a ) + bN 2́ We can use Dcd to estimate H/ , H , H After defining the variables N N = log = (1 − ^ ^ _) Since |λ|≤1 we can restricted [N = (1 − + ( When =1 N) ) By [N = 2( a N) ) [N = (1 − λ + 2λ( (13) Similarly for any value of λ we can compute a N) ) (14) Now the final form of Regression Model is [N is 2gh = log / + implicit function of (λ, D =( e ) k j ( e )=j j j i T 2́ U U N N U N N +( − 1) log and then ( Dcd ) we use: N ,β) U U log N N N U U U U [N [N N N N + log [N + bN (15) n [N m m m [N m http://www.iaeme.com/IJM/index.asp l 184 editor@iaeme.com Constructing a New Family Distribution with Methods of Estimation U 2N k n jU N 2N m m 2́ = j jU m N Nm j iU [N 2N l 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 To find the estimator's (oL- , opq:T;,