Constructing a new family distribution with methods of estimation

Chia sẻ: Huỳnh Ngọc Toàn | Ngày: | Loại File: PDF | Số trang:11

Thêm vào BST

Báo xấu

13
lượt xem 0
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

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.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Constructing a new family distribution with methods of estimation

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 obtain new generated transmuted Kumarasmay distribution. The . . , C.D.F and A new parameter ( ) is introduced to expand the family of two parameters Kumarasmy to this distribution are studied, parameters ( , , ) were obtained by moment and maximum moment of 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 work on expanding Kumarasmay distribution with two parameters ( , ) to another family using the We can expand family of any distribution by introducing new parameter to the given p.d.f. In this paper we 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. 2. THEORETICAL ASPECT The two parameters ( , ), p.d.f of Kumarasmay distribution is giving by; 2.1. Transmuted Kumarasmay Distribution ( ; , )= (1 − ) 0<
Constructing a New Family Distribution with Methods of Estimation *+, = (1 − ) 0 2 5 (1 − 2) 2+2 0 2 5 (1 − 2) 2 8 8 / / *+, = (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 (*+, = ) to find ∑@ 8 ?A4 >? B C DEFG , HEFG I. When ( ) is unknown, we can find three moment estimators of C D EFG , DEFG , HEFG I from solving (*+, = ) for ( = 1,2,3). ∑@ 8 ?A4 >? B Let ( , ,…, be a random variables from . . in (5), then; 2.2. Maximum Likelihood Estimator B) B B B B L = M ( N) = B B M N M(1 − N ) M$1 − NO NO NO NO + 2 (1 − N ) ' (9) B B log L = T log + T log + ( − 1) U log N + ( − 1) U log(1 − N ) NO NO B + U log$1 − + 2 (1 − N ) ' (10) NO B B B V log L T (− ) log( N ) 1 VW = + U log + ( − 1) U +U N V N C1 − N I WV NO NO NO Where; VW =2 (1 − ) (− ) log( N ) V N N = −2 N log( N ) (1 − N ) B B B V log L T C N I log( N ) 2 log( N ) (1 − ) X = + U log − ( − 1) U −U = 0 (11) N N V H N C1 − N IX (1 − + 2 C1 − I ) NO NO NO N Solved numerically to obtain ( HEYZ ). B B V log L T 2 (1 − ) log(1 − ) = + U log(1 − )+U N N =0 (12) V N (1 − + 2 C1 − I ) NO NO N 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 http://www.iaeme.com/IJM/index.asp 183 editor@iaeme.com
Rawa M. Saleh 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 N a ) + 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) [N is implicit function of (λ, N ,β) and then ( Dcd ) we use: D =( e ) 2́ T U N U N U [N k n j U N U N N U N [N m ( e )=j m j U U m j N N [N m i U [N l http://www.iaeme.com/IJM/index.asp 184 editor@iaeme.com
Constructing a New Family Distribution with Methods of Estimation U 2N k n jU N 2N m 2́ = j m jU 2 m j N Nm 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;,
Rawa M. Saleh Table 2 Estimating values ( = 1.5, = 1.5, = −0.5). T Method MSE( ) MSE( ) MLE 3.541205 2.363131 2.360808 0.001200 30 MOM 3.472286 4.712435 2.390151 0.001450 REG 4.562740 1.263481 2.368512 0.014466 REG MLE MLE 3.285041 0.251843 2.372088 0.000253 60 MOM 3.101814 0.488171 2.372817 0.001210 REG 3.745441 2.648731 2.357205 0.020588 MLE MLE MLE 2.820057 0.276607 2.380338 0.000108 90 MOM 2.804727 0.332006 2.388014 0.000451 REG 3.745341 0.514051 2.420118 0.008864 MLE MLE MLE 3.066625 0.270557 2.383362 0.000032 100 MOM 2.804073 0.214553 2.382121 0.000263 REG 2.801511 0.208033 2.403278 0.004153 REG MLE Table 3 Estimating values ( = 1.5, = 2, = −0.5). T Method MSE( ) MSE( ) MLE 4.465840 1.010705 1.880043 0.000176 30 MOM 7.360268 5.480611 2.015071 0.002060 REG 4.452554 1.103444 1.864645 0.002406 MLE MLE MLE 4.182831 0.526341 1.884337 0.000020 60 MOM 3.546686 1.120651 1.871271 0.000742 REG 4.063306 0.470311 2.036651 0.012122 REG MLE MLE 3.817211 0.570218 1.882077 0.000084 90 MOM 4.430338 1.120651 2.002011 0.000681 REG 4.058814 0.470311 1.876382 0.001558 REG MLE MLE 4.021001 0.570318 1.883804 0.000062 100 MOM 4.430338 1.674141 2.010111 0.000166 REG 4.058814 0.725708 1.887471 0.001561 MLE MLE http://www.iaeme.com/IJM/index.asp 186 editor@iaeme.com
Constructing a New Family Distribution with Methods of Estimation Table 4 Estimating values ( = 2, = 1.5, = −1). T Method MSE( ) MSE( ) MLE 4.027128 1.107352 2.368262 0.000744 30 MOM 4.607525 4.187281 2.404153 0.001541 REG 4.168044 1.010241 2.325303 0.011273 REG MLE MLE 3.760715 0.712121 2.380222 0.000167 60 MOM 3.671584 1.300363 2.381074 0.001851 REG 4.014352 1.337082 2.401168 0.013433 MLE MLE MLE 4.152061 0.708062 2.386506 0.000008 90 MOM 4.135852 2.212133 2.383281 0.001511 REG 4.160414 1.712333 2.414108 0.005566 MLE MLE MLE 4.335627 0.584063 2.386596 0.0000103 100 MOM 4.212356 1.420017 2.381112 0.0010611 REG 3.860625 0.637308 2.431274 0.0055281 MLE MLE Table 5 Estimating values ( = 1.5, = 2, = −1). T Method MSE( ) MSE( ) MLE 3.071613 1.678141 1.868181 0.825093 30 MOM 3.678881 5.031015 2.016325 0.027561 REG 3.427461 2.552882 1.880805 0.501455 MLE MOM MLE 3.562102 0.441706 1.877067 0.065227 60 MOM 4.048478 4.771012 2.004703 0.101741 REG 4.226181 2.751861 1.862103 0.110303 MLE MLE MLE 3.307454 0.161717 1.881017 0.702176 90 MOM 3.601384 0.706741 2.006521 0.367756 REG 3.482035 0.400307 1.870463 0.605406 MLE MO M MLE 3.235781 0.263236 1.884206 0.052807 100 MOM 3.348531 0.258632 2.005756 0.017014 REG 3.352501 0.201578 1.860743 0.080777 REG MOM http://www.iaeme.com/IJM/index.asp 187 editor@iaeme.com
Rawa M. Saleh Table 6 Estimating values ( = 1.5, = 1.5, = 0.5). T Method MSE( ) MSE( ) MLE 3.628867 1.547051 2.356625 0.001861 30 MOM 3.685208 2.301319 2.406525 0.001821 REG 3.704271 1.555103 2.423868 0.003311 MLE MOM MLE 3.100128 0.600406 2.364541 0.000872 60 MOM 3.623682 5.051811 2.408386 0.001642 REG 3.757016 1.766308 2.316868 0.011452 MLE MLE MLE 3.508505 0.052092 2.381381 0.000084 90 MOM 4.066258 1.040842 2.402052 0.000816 REG 3.480604 0.607481 2.406335 0.005115 MLE MLE MLE 3.725817 0.230002 2.401561 0.000112 100 MOM 3.741207 0.380621 2.401561 0.000530 REG 3.687313 0.208887 2.403285 0.001738 REG MLE Table 7 Estimating values ( = 1.5, = 2, = 0.5). T Method MSE( ) MSE( ) MLE 4.355840 1.010605 1.760043 0.000166 30 MOM 7.240268 4.350611 2.014071 0.002050 REG 4.342554 1.103214 1.754645 0.002306 MLE MLE MLE 3.650715 0.612121 2.270222 0.000157 60 MOM 3.561584 1.200363 2.251074 0.001851 REG 4.012252 1.237082 2.301168 0.013233 MLE MLE MLE 3.021613 1.568141 1.758181 0.725093 90 MOM 3.568881 5.021015 2.015325 0.026561 REG 3.317461 2.432882 1.670805 0.501455 MLE MOM MLE 3.107454 0.151717 1.761017 0.602176 100 MOM 3.401384 0.606741 2.006521 0.267756 REG 3.352035 0.300307 1.750463 0.405406 REG REG http://www.iaeme.com/IJM/index.asp 188 editor@iaeme.com
Constructing a New Family Distribution with Methods of Estimation Table 8 Estimating values ( = 4, = 1.5, = −0.5). T Method MSE( ) MSE( ) MLE 3.518867 1.327051 2.246625 0.001761 30 MOM 3.575208 2.201319 2.306525 0.001721 REG 3.604271 1.435103 2.313868 0.003211 MLE MOM MLE 4.172831 0.416341 1.674337 0.000020 60 MOM 3.436686 1.120251 1.761271 0.000642 REG 4.053306 0.360311 2.026651 0.012121 REG MLE MLE 4.142061 0.608062 2.276506 0.000007 90 MOM 4.125852 2.112133 2.283281 0.001411 REG 4.150414 1.612333 2.314108 0.004566 MLE MLE MLE 3.100127 0.500406 2.254541 0.000772 100 MOM 3.523682 5.041811 2.308386 0.001542 REG 3.647016 1.656308 2.216868 0.011352 MLE MLE Table 9 Estimating values ( = 4, = 2, = −0.5). T Method MSE( ) MSE( ) MLE 3.175041 0.141843 2.372088 0.000243 30 MOM 3.101614 0.378171 2.372817 0.001210 REG 3.535441 2.538731 2.357205 0.020488 MLE MLE MLE 3.054625 0.160557 2.733621 0.000031 60 MOM 2.704073 0.114553 2.282121 0.000262 REG 2.701511 0.108033 2.303278 0.004143 REG MLE MLE 3.717211 0.570218 1.762077 0.000083 90 MOM 4.230338 1.120651 2.003011 0.000671 REG 4.048814 0.470311 1.676382 0.001458 REG MLE MLE 3.125781 0.263236 1.654206 0.052607 100 MOM 3.238531 0.258632 2.004756 0.016014 REG 3.242501 0.201578 1.507431 0.080677 REG MOM http://www.iaeme.com/IJM/index.asp 189 editor@iaeme.com
Rawa M. Saleh Table 10 Estimating values ( = 2, = 2, = −0.5). T Method MSE( ) MSE( ) MLE 2.727422 0.231005 0.0000241 0.101811 30 MOM 3.078274 0.570246 0.0004770 0.228567 REG 2.717832 0.355094 0.0021070 0.076605 MLE REG MLE 3.462102 0.341706 1.677067 0.055227 60 MOM 4.038478 4.671012 2.003703 0.101641 REG 4.216181 2.651861 1.762103 0.110203 MLE MLE MLE 3.625817 0.130002 2.301561 0.000112 90 MOM 3.641207 0.280621 2.301551 0.000430 REG 3.587313 0.108887 2.303285 0.001738 REG MLE MLE 4.031001 0.470318 1.783804 0.000052 100 MOM 4.230338 1.574141 2.010111 0.000156 REG 4.048814 0.625708 1.787471 0.001461 MLE MLE 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 Xαxy , βzxy , λzxy . estimators is explained in this research and any statistical program can be used to find the parameters • 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. 168–174,. [2] Aryal, G.R. and C.D. Tsokos, (2009), "On the Transmuted Extreme Value Distribution with Application", Nonlinear Analysis Theory, Methods and Applications, 71: 1401 – 1407. [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, 89- 102. http://www.iaeme.com/IJM/index.asp 190 editor@iaeme.com
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 Basic and Applied sciences, 7(7): 658 – 667. [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. 117–129,. [7] I. Elbatal and G. R. Aryal, (2013), “On the transmuted additive Weibull distribution,” Austrian Journal of Statistics, vol. 42, no. 2, pp. 117–132. [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, 89-102. [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 Bourguignon, 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 7208425, 12 pages. [11] Merovic, F. (2013), "Transmuted Rayliegh Distribution", Australian Journal of statistics, 42(1);21 – 31. [12] M. R. Mahmoud and R. M. Mandouh, “On the transmuted Fréchet distribution,” Journal of Applied Sciences Research, vol. 9, no. 10, pp. 5553–5561 [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. 66–88,. [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. 136–143. [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. 258–265. http://www.iaeme.com/IJM/index.asp 191 editor@iaeme.com