CONSTRUCTING A NEW FAMILY DISTRIBUTION WITH METHODS OF ESTIMATION

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1 International Journal of Management (IJM) Volume 7, Issue 6, September October 2016, pp , Article ID: IJM_07_06_021 Available online at Journal Impact Factor (2016): (Calculated by GISI) ISSN Print: and ISSN Online: IAEME Publication CONSTRUCTING A NEW FAMILY DISTRIUTION WITH METHODS OF ESTIMATION Rawa M. Saleh Department of Statistics, Economic and Administration College, Al Mustansryia University, Iraq. ASTRACT 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 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 editor@iaeme.com

2 Rawa M. Saleh 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()() (4) ()=()(1+) 2() ()= (1 ) $(1+) 2%1 (1 ) &' = (1 ) $1+ 2+2(1 ) ' The new transmuted.. of Kumarasmay distribution is; And its C.D.F is; ()= (1 ) $1 +2(1 ) ' (5) ()=(1+)$1 (1 ) ' $1 (1 ) ' =$1 (1 ) '(1+ $1 (1 ) ') =$1 (1 ) '$1+(1 ) ' (6) To derive the moments about origin * +, ; *, + =-( + )=. + () (7) =0 +1 (1 ) $1 +2(1 ) ' =(1 )0 +1 After some steps using transformation, we find; (1 ) (1 ) Let 2= =2 4 5 = *, + =(1 ) (1 2) (1 2) editor@iaeme.com

3 Constructing a New Family Distribution with Methods of Estimation *, + =(1 ) (1 2) (1 2) 2 *, + = (1 )9:;< ,7+2 9:;< 6+ +1,27 (8) Equation (8) shows the moments of this transmuted Kumarasmay distribution. If 1 we can form two equations ( =1,2) from (*, + ) and equating (*, + >? 8 ) to find CD EFG,H EFG I. When () is unknown, we can find three moment estimators of CD EFG,D EFG,H EFG I from solving (*, + >? 8 ) for ( =1,2,3) Maximum Likelihood Estimator Let (,,, ) be a random variables from.. in (5), then; L =M( N ) = M N M(1 N ) M$1 +2(1 N ) ' (9) logl =Tlog+Tlog+( 1)Ulog N +( 1)Ulog(1 N ) +Ulog$1 +2(1 N ) ' (10) Where; VlogL V = T +Ulog N+( 1)U ( N )log( N ) C1 +U 1 VW N I WV VW V =2(1 N ) ( N )log( N ) = 2 N log( N )(1 N ) VlogL V = T H +Ulog N ( 1)U C N X Ilog( N ) C1 X U 2 N log( N )(1 N ) N I (1 +2C1 N I =0 (11) ) Solved numerically to obtain (H EYZ ). VlogL V = T +Ulog(1 N ) +U 2(1 N )log(1 N ) (1 +2C1 N I =0 (12) ) Equation (12) can also be solved numerically to find (D EYZ ). Now we can restricted 1 to estimate (,) only Proposed Regression Estimators (PRE) Let,, [. \ be a random sample from P.D.F defined in (5), than editor@iaeme.com

4 Rawa M. Saleh 2 N = 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^ The model can be written as: N =(1 ^_) log2^ =log()+( 1)log ^+( 1)log N+log(1 λ+2 N a )+b N Using least square procedure and according to D cd =() e 2 We can use D cd to estimate H,H,H After defining the variables N =log^ N =(1 ^_) Since λ 1 we can restricted [N =(1 +2( N) ) y 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 2 g h =log + log N +( 1)log N+log [N +b N (15) [N is implicit function of (λ, N,β) and then (D cd ) we use: D =() e 2 T U N U N U [N k n j U N U N N U N [N m () e = j m j j U N U N m [N m i U [N l editor@iaeme.com

5 Constructing a New Family Distribution with Methods of Estimation U2 N k n ju N 2 N m 2 = j m j j U m N2 N m i U [N 2 N l According to the values of observation N and generated values of 2 N (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) editor@iaeme.com

6 Rawa M. Saleh Table 2 Estimating values ( =1.5, =1.5, = 0.5) Table 3 Estimating values ( =1.5, =2,= 0.5) editor@iaeme.com

7 Constructing a New Family Distribution with Methods of Estimation Table 4 Estimating values ( =2, =1.5,= 1) Table 5 Estimating values ( =1.5, =2,= 1) MO M editor@iaeme.com

8 Rawa M. Saleh Table 6 Estimating values ( =1.5, =1.5, =0.5) Table 7 Estimating values ( =1.5, =2,=0.5) editor@iaeme.com

9 Constructing a New Family Distribution with Methods of Estimation Table 8 Estimating values ( =4, =1.5,= 0.5) Table 9 Estimating values ( =4, =2, = 0.5) editor@iaeme.com

10 Rawa M. Saleh Table 10 Estimating values ( =2, =2,= 0.5) 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 αx xy,βz xy,λz xy. 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. REFERENCE [1] A. Ahmad, S. P. Ahmad, and A. Ahmed, (2015), Characterization and estimation of transmuted Kumaraswamy distribution, 9, vol. 5, no. 9, pp ,. [2] Aryal, G.R. and C.D. Tsokos, (2009), "On the Transmuted Extreme Value Distribution with Application", Nonlinear Analysis Theory, Methods and Applications, 71: [3] Aryal G.R., Tsokos Ch. P., (2011), Transmuted Weibull Distribution: A Generalization of the Weibull probability distribution, European journal of pure and applied mathematics Vol. 4, No. 2, editor@iaeme.com

11 Constructing a New Family Distribution with Methods of Estimation [4] Aryal, R. G. (2013), Transmuted Log-Logistic Distribution. Journal of Statistics Applications & Probability. No. 1, 11-20,. [5] [5] Ashour, S.K. and M.A. Eltehiwy, (2013), "Transmuted Exponentiated Lomax Distribution", Australian Journal asic and Applied sciences, 7(7): [6] I. Elbatal, (2013) Transmuted modified inverse Weibull distribution: a generalization of the modified inverse Weibull probability distribution, International Journal of Mathematical Archive, vol. 4, no. 8, pp ,. [7] I. Elbatal and G. R. Aryal, (2013), On the transmuted additive Weibull distribution, Austrian Journal of Statistics, vol. 42, no. 2, pp [8] [8]Gokarna R. Aryal1, Chris P. Tsokos. (2011), "Transmuted Weibull distribution: A Generalization of the Weibull Probability Distribution". European Journal of Pure and Applied Mathematics, Vol. 4, No. 2, [9] Khan, M. Shuaib, King Robert, (2013) "Transmuted Generalized Inverse Weibull Distribution", Journal of Applied Statistical Sciences, Nova Science. Vol. 20 (3), 15-32,. [10] Marcelo ourguignon, Indranil Ghosh, and Gauss M. Cordeiro, (2016), "General Results for the Transmuted Family of Distributions and New Models", Journal of Probability and Statistics Volume 2016, Article ID , 12 pages. [11] Merovic, F. (2013), "Transmuted Rayliegh Distribution", Australian Journal of statistics, 42(1); [12] M. R. Mahmoud and R. M. Mandouh, On the transmuted Fréchet distribution, Journal of Applied Sciences Research, vol. 9, no. 10, pp [13] M. S. Khan and R. King, (2013), Transmuted modified weibull distribution: a generalization of the modified weibull probability distribution, European Journal of Pure and Applied Mathematics, vol. 6, no. 1, pp ,. [14] Muhammad Shuaib Khan, Robert King, (2015), "Transmuted Modified Inverse Rayleigh Distribution", Austrian Journal of Statistics, Vol 44, No. 3. [15] Yuzhu Tiana, Maozai Tiana &QianqianZhua, (2014),"Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution", Communications in Statistics - Simulation and Computation Volume 43, Issue 10. [16] Dhwyia S. Hassun, Constructing a New Family Distribution from Three Parameters Weibull using Entropy Transformation. International Journal of Advanced Research in Engineering and Technology (IJARET), 5(6), 2014, pp [17] Faris M. Al-Athari, Moment Properties of Two Maximum Likelihood Estimators of the Mean of Truncated Exponential Distribution. International Journal of Advanced Research in Engineering and Technology (IJARET), 4(7), 2014, pp editor@iaeme.com