Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution

Transcription

1 Global Joural of Pure ad Appled Mathematcs. ISSN Volume 3, Number 9 (207), pp Research Ida Publcatos Comparg Dfferet Estmators of three Parameters for Trasmuted Webull Dstrbuto Dr. Saad Ahmed Abdulrahma Assstat Professor, Departmet of Statstcs, College of Admstrato ad Ecoomcs, Baghdad Uversty, Baghdad 207, Iraq. Abstract Ths paper deals wth expadg the famly of Webull dstrbuto, through troducg aother parameter (), to the orgal Webull famly whch have oe scale parameter (), ad oe shape parameter (). Ths expaso ecessary to costruct dfferet model whch wll be add flexblty represetg the data, the famly s called trasmuted Webull, here we expla the cumulatve dstrbuto fucto ad the probablty desty fucto, ad derver th momets about org, ad the estmatg parameters by dfferet methods lke maxmum lkelhood method, ad momet estmators. The comparso betwee estmators s doe through smulato procedure usg dfferet sample sze =00, 50, ad dfferet sets of hal values for (,,), through applyg the geerato formula whch s derved ths paper. The comparso betwee estmators s doe usg statstcal measure mea square error (MSE), ad all result s explaed tables. Keywords: Trasmuted Webull dstrbuto, Maxmum lkelhood method ad Momet estmators, Three Parameters.. INTRODUCTION The modfcato o dstrbutos are troduced, lke usg weghted dstrbuto ad trasmuted dstrbuto [2], through troducg a ew parameter, to obta a ew famly dstrbuto (or modfed formula) as compared wth orgal probablty dstrbuto.

2 56 Dr. Saad Ahmed Abdulrahma The modfcato o models helps the researcher to have aother model that deals wth estmato, ad computg cofdece terval for estmated parameters, also for modelg lfetme data. Also the geeralzed Webull dstrbutos, was studed by Murthy.D.N.P,Xe.M, & Jag.R. [0] ad dcate to varous propertes ad varous extesos whch ca be appled o ths dstrbuto we kow that Webull dstrbuto have a potetal cotrbuto to equpmet mateace ad lfe polces. We cotue the research about the dstrbuto (Webull) due to ts beefts, may statstcal applcato ad work o fdg a formula for the p.d.f ad C.D.F. of trasmuted Webull dstrbuto ad the work o estmatg ts three parameters (,.) usg dfferet methods of mouts ad maxmum lkelhood. 2. THE THEORETICAL SIDE Let T be A radom varable dstrbuted Webull dstrbuto wth two parameters (,) defed by equato bellow : 0= 0.w t<0 : s shape parameter, : s scale parameter wth C.D.F. gve equato (2) Apply equato (3) to the C.D.F gve (2), to obta G(t): G(t)= (+) F(t)- (F(t)) 2 (3) Usg equato (3), so that the ew C.D.F of trasmuted Webull(),Whch s G(t)= (+) (F(t))- (F(t)) 2 Ths lead to the ew C.D.F of trasmuted

3 Comparg Dfferet Estmators of three Parameters for Trasmuted ) ( 2 t t t e e e G t Dervg equato (4), gves us the p.d.f g(t) od the ew model, whch s gve equato(5) The we ca smplfy the formula equato (5), for the p.d.f of trasmuted Webull > 0, t > 0, Ad the C.D.F of ths ew geerated trasmuted webull s: Whle the p.d.f of ths ew geerated trasmuted Webull dstrbuto whch s gve equato (5)

4 58 Dr. Saad Ahmed Abdulrahma

5 Comparg Dfferet Estmators of three Parameters for Trasmuted GENERATION OF RANDOM NUMBERS The geerato of radom umbers for estmato: Normally we solve the equato of ths type: Let Equato (8) ca be used for geeratg values of observatos t, from ths ew trasmuted Webull.

6 520 Dr. Saad Ahmed Abdulrahma 4. METHODS OF SOLVING FORMULA.The r th momets: To derve a formula about org,.e After some steps of smplfcato, we have: ( ) Therefore the formula for E(tr) ca be used for obtag momet estmators from solvg So that the r, th momet about org s Hece

7 Comparg Dfferet Estmators of three Parameters for Trasmuted 52 Ad from From the equato () Also we have The thrd equato whch s eeded to obta: These three equato whch s obtaed from ( are solved order to fd Maxmum lkelhood method. The estmators From the ew trasmuted Webull (equ.6) are obtaed from Maxmzg log L= L = ( t ) ( t ). e e ( t )... (4) Let k = e ( t ) log L = log log log ( ) log t (t ) log k... (5) If we dervatve ths equato three tmes by, ad order to obtaed three equatos as bellow:

8 522 Dr. Saad Ahmed Abdulrahma (6)... 2 log t 2 log t ) (t log t t t e e 0 ()log t. 2 () log t ) (t log t = L log t e k Whch s a mplct fucto of. The: 0 = L log 2 k = t e Solved umercally to fd ad whle = We obta: Solved umercally to fd.

9 Comparg Dfferet Estmators of three Parameters for Trasmuted CONCLUSION From the results of smulato, we fd the best estmator for dfferet sets of tal values (,,Q )ad also dfferet set of sample sze (=00, 50, ), are momet estmators, frst ad maxmum lke lkelhood secod (. e) MLE Momet BEST Mom Mom MLE MLE Momet BEST Mom Mom MLE MLE Momet BEST Mom MLE MLE MLE Momet MLE Momet MLE Momet

10 524 Dr. Saad Ahmed Abdulrahma MLE Momet MLE Momet MLE Momet MLE Momet MLE Momet MLE Momet

11 Comparg Dfferet Estmators of three Parameters for Trasmuted 525 MLE Momet MLE Momet MLE Momet MLE Momet MLE Momet MLE Momet

12 526 Dr. Saad Ahmed Abdulrahma MLE Momet BEST Mom Mom MLE MLE Momet BEST Mom Mom MLE MLE Momet BEST Mom Mom MLE MLE Momet MLE Momet MLE Momet

13 Comparg Dfferet Estmators of three Parameters for Trasmuted 527 MLE Momet MLE Momet MLE Momet % 22.2% % 22.2% % 22.2% 00 00% 66.7% 77.8% 50 00% 66.7% 77.8% 00% 55.6% 77.8% We fd the percetage of preface of (momet estmator) s hgher tha percetage of (maxmum lkelhood estmator) ths s due to closed form of momet estmators, whle MLE estmators eed teratos

14 528 Dr. Saad Ahmed Abdulrahma REFERENCES [] A. M. Sarha ad M. Zad. Modfed Webull dstrbuto, Appled Sceces, Vol., 2009, pp [2] C.-C. Lu. A Comparso betwee the Webull ad Logormal Models used to Aalyze Relablty Data. PhD thess Uversty of Nottgham, UK [3] Carrasco M, Ortega EM, Cordero GM. A geeralzed modfed Webull dstrbuto for lfetme modelg. Computatoal Statstcs ad Data Aalyss 2008;53(2): [4] Cordero GM, Ortega EM, Nadarajah S. The Kumaraswamy Webull dstrbuto wth applcato to falure data. joural of the Frakl Isttute 200:347: [5] G. Mudholkar, D. Srvastava, ad G. Kolla. A geeralzato of the Webull dstrbuto wth applcato to the aalyss of survval data. Joural of the Amerca Statstcal Assocato, 9(436): , 996. [6] G. R. Aryall ad C. P Tsokos. O the trasmuted extreme value dstrbuto wth applcatos. Nolear Aalyss: Theory, s ad applcatos, Vol. 7, [7] H. Pham ad C.-D. La. O Recet Geeralzatos of the Webull Dstrbuto. IEEE Trasactos o Relablty, 56(3): , [8] Iferece for Webull Dstrbuto Stat 498B Idustral Statstcs, Frtz Scholz, May 22, [9] Joural of appled sceces research, 203 q(0): , ISSIV 89 - Suux, orgal Artcles o the Trasmuted Fre'chet Dstrbuto [0] M.R. Mahmoud ad R.M. ad others, Isttute of statstcal studes ad research, Caro Uversty, Egypt, 203. [] Murthy.D.N.P,Xe.M, & Jag.R.(2004).Webullmodels.Hoboke:Joh Wley & Sos,Ic.Nadaraah.S.(2005).O the momets of the modfed dstrbuto.relablty Egeerg ad System Safety 90,4-7. [2] Pha ICIC. A ew modfed Webull dstrbuto fucto. Commucatos of the Amerca Ceramc Socety 987;70(8):82-4. [3] Sarha AM, Zad M. Modfed Webull dstrbuto. Appled Sceces 2009;: [4] Thoma, D.R. Ba, L.J. ad Atle, C. E. (970)"Exact cofdece tervals for relablty, ad tolerace lmts the Webull dstrbuto", Techometrcs, Vol. 2, 2,