Global Joural of Pure ad Appled Mathematcs. ISSN 0973-768 Volume 3, Number 9 (207), pp. 55-528 Research Ida Publcatos http://www.rpublcato.com 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.
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
Comparg Dfferet Estmators of three Parameters for Trasmuted 57...4 ) ( 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)
58 Dr. Saad Ahmed Abdulrahma
Comparg Dfferet Estmators of three Parameters for Trasmuted 59 3. 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.
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
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:
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 = 0 2 2 t e Solved umercally to fd ad whle = We obta: Solved umercally to fd.
Comparg Dfferet Estmators of three Parameters for Trasmuted 523 5. 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.0797 0.020978 2.355.3037 0.78522 0.697366 00 Momet 0.995733 0.000666 2.02724.045673 0.0453 0.802544 BEST Mom Mom MLE MLE.066099 0.07574 2.068542.209383 0.5402 0.728907 50 Momet 0.993644 0.000559 2.025385.052958 0.04507 0.809 BEST Mom Mom MLE MLE.03296 0.00909.99473.032243 0.3254 0.76422 Momet 0.989989 0.00495 2.03662.08667 0.04889 0.80234 BEST Mom MLE MLE MLE.067264 0.02357.842773 0.792 0.32642 0.76349 00 Momet 0.99697 0.000504 2.09676.0447 0.52892 0.22865 MLE.059098 0.0658.779604 0.640207 0.49966 0.73799 50 Momet 0.98452 0.002706 2.02775.05933 0.52677 0.22425 MLE 0.99593 0.0229.720249 0.54472 0.92939 0.67278 Momet 0.987433 0.000909 2.022297.046595 0.52565 0.27
524 Dr. Saad Ahmed Abdulrahma MLE.063256 0.022799.488076 0.278039 0.8045 0.692978 00 Momet 0.9860 0.00407 2.023028.04902 0.839828 0.025824 MLE.024093 0.0723.433803 0.2545 0.25876 0.586024 50 Momet 0.98922 0.00276 2.020627.043486 0.840077 0.02578 MLE.00586 0.089.4620 0.23206 0.3999 0.50796 Momet 0.99224 0.000607 2.0743.035808 0.839065 0.02597 MLE.480902 0.030336.45202 0.097 0.30595.52937 00 Momet.48847 0.00226 2.02469 0.277068 0.835294 0.44974 MLE.450706 0.02652.465996 0.098385 0.369279.389329 50 Momet.486772 0.002559 2.02393 0.280288 0.839375 0.436563 MLE.428652 0.02249.479297 0.6298 0.42548.263559 Momet.484908 0.0025 2.08496 0.26979 0.83897 0.437078
Comparg Dfferet Estmators of three Parameters for Trasmuted 525 MLE.92978 0.040379.705662 0.2688 0.368094 2.742997 00 Momet.973206 0.00826 2.02343 0.002347 0.52494 2.78 MLE.898322 0.025738.72495 0.267703 0.35805 2.799576 50 Momet.9842 0.00922 2.07832 0.00444 0.522892 2.890 MLE.902926 0.034388.685888 0.277782 0.359036 2.760898 Momet.980908 0.00869 2.024545 0.0079 0.52248 2.830 MLE 2.378908 0.068307.900365 0.600425 0.774849 3.059782 00 Momet 2.470776 0.00545 2.09433 0.23278 0.83887 2.76806 MLE 2.385457 0.066937.8564 0.60525 0.74752 3.7936 50 Momet 2.476763 0.002922 2.02674 0.225673 0.836975 2.765736 MLE 2.4956 0.033727.986879 0.368897 0.80434 2.93893 Momet 2.47297 0.003367 2.0832 0.232856 0.838828 2.759542
526 Dr. Saad Ahmed Abdulrahma MLE.55055 0.024024.5788 0.087765 0.4373.88452 00 Momet.4863 0.003064.525887 0.00945 0.04253.9487 BEST Mom Mom MLE MLE.543378 0.06455.5320 0.063799 0.5428.8506 50 Momet.487496 0.002448.52755 0.002903 0.04482.947472 BEST Mom Mom MLE MLE.49042 0.07287.550058 0.4637 0.252493.637832 Momet.47644 0.00402.522963 0.0065 0.0477.946669 BEST Mom Mom MLE MLE.9905 0.0290.64688 0.300492 0.27682 3.034896 00 Momet.96580 0.03273 2.03039 0.00497 0.524533 2.7722 MLE.9066 0.032289.706694 0.262349 0.34944 2.80987 50 Momet.983545 0.003284 2.02676 0.002204 0.52287 2.8972 MLE.922759 0.02298.7448 0.2203 0.350738 2.789928 Momet.96096 0.00965 2.039576 0.0643 0.524009 2.78594
Comparg Dfferet Estmators of three Parameters for Trasmuted 527 MLE 2.33505 0.45036.70079.024004 0.525276 4.028033 00 Momet 2.48006 0.00674 2.520366 0.003448 0.837385 2.764377 MLE 2.264643 0.34589.57284.4498 0.499034 4.02253 50 Momet 2.437227 0.09755 2.540256 0.005097 0.84583 2.750579 MLE 2.243644 0.78377.53607.292867 0.58390 3.764464 Momet 2.469335 0.003766 2.52844 0.002424 0.837829 2.762872 00 0 33.3% 22.2% 50 0 33.3% 22.2% 0 44.4% 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
528 Dr. Saad Ahmed Abdulrahma REFERENCES [] A. M. Sarha ad M. Zad. Modfed Webull dstrbuto, Appled Sceces, Vol., 2009, pp. 23-36. 2009. [2] C.-C. Lu. A Comparso betwee the Webull ad Logormal Models used to Aalyze Relablty Data. PhD thess Uversty of Nottgham, UK. 997. [3] Carrasco M, Ortega EM, Cordero GM. A geeralzed modfed Webull dstrbuto for lfetme modelg. Computatoal Statstcs ad Data Aalyss 2008;53(2):450-62. [4] Cordero GM, Ortega EM, Nadarajah S. The Kumaraswamy Webull dstrbuto wth applcato to falure data. joural of the Frakl Isttute 200:347:399-429. [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):575-583, 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, 40-407. 2009. [7] H. Pham ad C.-D. La. O Recet Geeralzatos of the Webull Dstrbuto. IEEE Trasactos o Relablty, 56(3):454-458, 2007. [8] Iferece for Webull Dstrbuto Stat 498B Idustral Statstcs, Frtz Scholz, May 22, 2008. [9] Joural of appled sceces research, 203 q(0): 5553-556, 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;:23-36. [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, 363-37.