PERFORMANCE RATING OF TRANSMUTED NADARAJAH AND HAGHIGHI EXPONENTIAL DISTRIBUTION: AN ANALYTICAL APPROACH
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1 Joua of Sascs: Advacs Thoy ad Appcaos Voum 7, Numb, 7, Pags 37-5 Avaab a hp://scfcadvacs.co. DOI: hp://d.do.og/.864/jsaa_787 PERFORMANCE RATING OF TRANSMUTED NADARAJAH AND HAGHIGHI EXPONENTIAL DISTRIBUTION: AN ANALYTICAL APPROACH A. A. OGUNDE, A. G. IBRAHEEM ad A. T. AUDU Dpam of Mahmacs ad Sascs Fda Poychc Ado-Ek Fda Schoo of Sascs Ibada Nga -ma: dbz95@yahoo.com admoagayu@gma.com audu@yma.com Absac Ths wok vsgas h poa usfuss of h asmud Nadaajah Haghgh poa dsbuo fo modg fm daa. Ths dsbuo ca b obad by usg h quadac ak asmuao map schm. Vaous sucua pops of h asmud Nadaajah Haghgh poa mod w vsgad cudg mom gag fuco, od sascs, mom, ma whch ps h avag f spa of a sysm, vaac, smao of h paams usg mamum khood ad h poa usfuss of h asmud Nadaajah Haghgh poa mod was show by mas of Kva 373/poy daa. Mahmacs Subjc Cassfcao: 65-XX, 7EXX. Kywods ad phass: quadac ak asmuao map, moms, Nadaajah Haghgh poa dsbuo, mamum khood smao, ods sascs. Rcvd May 6, 7; Rvsd May 7, 7 7 Scfc Advacs Pubshs
2 38 A. A. OGUNDE a.. Ioduco I hs pap, w oduc h asmud Nadaajah Haghgh poa dsbuo whch sms fom h foowg da; Shaw & Bucky [] cosdd compos maps of h foowg wo foms: samp asmuao maps ( STMs, y G ( F ( ad ak asmuao maps ( RTMs, G ( F ν ( w, wh F ad G a cumuav dsbuo fucos (CDFs. Gchs [9] fs o STMs ad RTMs as Q- asfomao ad P-asfomao, spcvy. Shaw & Bucky [] focusd o h RTM, whch uss as a oo fo h cosuco of w fams of o-gaussa dsbuos. Thy usd o modua a gv bas dsbuo fo h puposs of modfyg h moms, pacua h skw ad kuoss. A aaco of h appoach s ha f h CDF ad vs CDF (o qua fuco (QF a acab fo h bas dsbuo, h s a good chac fo h asmud dsbuo o b so. Accodg o hs appoach, a adom vaab X s sad o hav a asmud dsbuo f s cumuav dsbuo fuco (cdf sasfs h foowg aoshp: G ( ( F ( F (, ; ( g( f ( [( F ( ], ( wh F ( s h cdf of h bas mod, g ( ad f ( a h cospodg pobaby dsy fucos (pdf assocad wh G ( ad F (, spcvy. Ths pap vsgas h sasca pops of h asmud Nadaajah Haghgh poa dsbuo. Aya ad Tsokos [3] sudd h asmud Wbu dsbuo o aayz wo fm daa ss. Tasmud Loma dsbuo (Ashou ad Ehwy [], asmud poad Gamma dsbuo (Hussa [], asmud vs Raygh dsbuo (Ahmad a. [], asmud Pao dsbuo (Movc ad Puka [8], asmud
3 PERFORMANCE RATING OF TRANSMUTED NADARAJAH 39 modfd vs Wbu dsbuo (Ebaa [6], asmud addv Wbu dsbuo (Ebaa ad Aya [7], asmud compmay Wbu gomc dsbuo (Affy a. [], asmud vs poa (Oguud ad Adjumo [9]. Kha ad Kg [3, 4] poposd h asmud modfd Wbu ad h asmud gazd vs Wbu dsbuos ad dscussd sucua pops wh appcao o aby daa. Kha a. [] poposd h asmud vs Wbu dsbuo ad dscussd vaous sucua pops wh appcao o aby daa. Mo cy, Kha a. [5, 6, 7] sudd h asmud gazd poa, asmud Wbu, ad asmud gazd Gompz dsbuos aso, Owooko a. [8] amd h pfomac ag of h asmud poa dsbuo by usg QRTM chqu whch ds h bas mods fo modg fm daa. Abdu-Mom ad Sham [4, 5] amd h sasca pops of h asmud Gompz dsbuo ad aso h poad Nadaajah Haghgh poa dsbuo. A adom vaab X s sad o hav a Nadaajah ad Haghgh poa dsbuo (NHED wh shap paam ad h sca paam, f s pobaby dsy fuco ( pdf s h fom: ( f ;,, >, (, >. (3 Th cumuav dsbuo fuco (CDF ad suvva fuco (SF a: ( F, >, (, >, (4 ad ( S, >, (, >. (5
4 4 A. A. OGUNDE a.. Tasmud Nadaajah ad Haghgh Epoa Dsbuo Usg ( ad (4, w ca df h CDF of TNHED as foows: ( ;,, ( ( G, >, (, ad >. (6 Th pdf of TNHED s gv as: ( ;,, ( g, >, (, ad >. (7 W ca g h pdf fo h asmud poa (TED, Nadaajah ad Haghgh poa (NHED, ad poa (ED dsbuos by akg,, ad, spcvy. Th suvva (aby fuco S ( ad h hazad a fuco h ( fo TNHED w obad as foows: ( ;,, ( ( S, >, (, ad >, (8 ad h( ( ( ( (, >, (, ad >, (9 3. Qua Fuco, Mda ad Mod of h Tasmud Nadaajah Haghgh Epoa Dsbuo Th qua fuco q of h TNHED dsbuo ca b obad as h vs of Equao (6
5 PERFORMANCE RATING OF TRANSMUTED NADARAJAH 4 ( ( 4 q, ( ad h mda ca b obad as ( ( 4.5. ( Th ow qua ad upp qua ca aso b dvd fom Equao ( wh q.5 ad q.75, spcvy. Th mod of TNHED s h oo of h foowg quao: { }. ( ( ( Th gaph of cumuav dsy, pobaby dsy, ad h hazad fucos w pod bow fo vaous vaus of a, b, ad b. Fgu. Po of h cumuav dsy fuco fo TNHED.
6 4 A. A. OGUNDE a. Fgu. Po of h dsy fuco fo TNHED fo vaous vaus of h paams.
7 PERFORMANCE RATING OF TRANSMUTED NADARAJAH 43 Fgu 3. Po of h hazad fuco fo TNHED fo vaous vau of h paams. Fom h dagam abov Fgu dpc ha h cdf of TNHED covgs o o as ad ha shows ha h pdf of TNHED s a pop pdf. Fgu dmosas h fby of h dsbuo modg fm daa. Fgu 3 gvs a ca pcu of h fac ha h dsbuo s vsb boh o h gh ad o h f ad ca b usd o mod fm daa v hos ha possss h bahub-shap fau a.
8 44 A. A. OGUNDE a. 4. Tadoa Moms fo TNHED H w oba E(, hc w ca oba h ma ad h vaac fo TNHED as foows: ( ( ( ( ( E d. (3 Usg h asfomao, ( y, hs mps ha y ( ad d ( y dy, h Equao (3 w asfom o y y ( y ( dy. (4 ( E( Th abov quao ca b spd o I ( y dy; (5 ( ( y y I ( y dy; (6 ( y I3 ( y dy. (7 Epadg ( y usg h Tayo s ss w yd ( y y. (8
9 PERFORMANCE RATING OF TRANSMUTED NADARAJAH 45 Th padg Equaos (5, (6 ad (7 usg Equao (8, w oba ;, 4 y dy y I (9 ;, 5 y dy y I (., 6 y dy y I ( Thfo combg Equaos (9, (, ad (, w hav E,., ( To oba h ma of TNHED, w s Equao (, h w hav E,., (3 Th w hav,
10 A. A. OGUNDE a. 46 E,., (4 Aso sg Equao ( X E X E,., Fom abov, w hav X E,., X E *O smpfcao w hav X E,,.,, Th h vaac of TNHED s gv as
11 PERFORMANCE RATING OF TRANSMUTED NADARAJAH 47 X,, Va,,.,, (5 5. Mamum Lkhood Esmaos (MLE I hs sco, w cosd mamum khood smaos (MLE of TNHED. L,,, b a adom samp of sz fom TNHED, h h og-khood fuco Θ fo T Θ,, ca b w as [ ] Θ og og og og. og (6 Th oma quao bcom d d ; (7 ; og d d (8
12 48 A. A. OGUNDE a. ( d d (. (9 Th MLE of,, ad ca b oba by sovg h Equaos (7, (8, d d d ad (9, usg,, ad. d d d 6. Appcao of TNHED I hs sco, w us a a daa s o show ha h TNHED ca b a b mod ha o basd o h gazd poa dsbuo (GED, asmud poa dsbuo (TED, dd poa dsbuo (EED, Nadaajah ad Haghgh poa dsbuo (NHED, ad poa dsbuo (ED. W cosd a daa s of h f of fagu facu of Kva 373/poy ha a subjc o cosa pssu a h 9% sss v u a had fad, so w hav comp daa wh h ac ms of fau. Ths daa a:.5,.886,.89,.5,.33,.345,.4763,.565,.567,.6566,.6748,.675,.6753,.7696,.8375,.839,.845,.8645,.885,.93,.9,.9836,.483,.596,.773,.733,.57,.766,.985,.3,.353,.355,.4595,.488,.578,.5733,.783,.763,.746,.763,.7746,.875,.8375,.853,.888,.8878,.888,.936,.9558,.48,.48,.93,.93,.33,.,.46,.878,.33,.347,.353,.495,.56,.99, 3.56, 3.678, 3.445, , , , 3.943, 4.873, 5.45, , 5.595, 6.554, I od o compa h dsbuos, w cosd som oh co k LL, AIC (Akak fomao co, AICC (Akak fomao co cocd, ad BIC (Baysa fomao co fo h a daa s. Th bs dsbuo cospods o ow LL, AIC, AICC, ad BIC, wh p( p AIC P LL, AICC AIC, ad BIC p og(, p wh p s h umb of paams h sasca mod, s h
13 PERFORMANCE RATING OF TRANSMUTED NADARAJAH 49 samp sz ad s h mamzd vau of h khood fuco fo h smad mod. Tab shows h poaoy daa aayss of h daa. Tab shows paam MLE fo ach o of h fd dsbuos. Tab 3 shows h vaus of LL, AIC, AICC, ad BIC vaus. Tab. Summay of daa o fagu facu of Kva 373/poy a 9% sss v M Low mda Upp Ma Ma. Vaac Skwss Kuoss Rag qua qua Tab. Esmad paams of h TNHED, TED, GED, EED, NHED, ad ED Paams smas Mods TNHED TED GED EED NHED ED Tab 3. Ca fo compaso Mods LL AIC AICC BIC TNHED TED GED EED NHED ED
14 5 A. A. OGUNDE a. 7. Cocuso I w b obsvd fom h vaus of h paams smas ad aso h vaus of h co fo compaso, h mod ha coas h mmum fomao oss whch cospods o mmum, AIC, AICC, ad BIC s cosdd o b h bs mod h cass of mods cosdd. A appcao of h asmud Nadaajah ad Haghgh poa dsbuo o a daa shows ha h w dsbuo povds a b f ha h gazd poa dsbuo, asmud poa dsbuo, poad poa dsbuo, Nadaajah Haghgh poa dsbuo, ad poa dsbuo. Rfcs [] A. Z. Affy, Z. M. Nofa ad N. S. Bu, Tasmud compmay Wbu gomc dsbuo, Pak. J. Sa. Opao Rs. (4 (4, [] A. Ahmad, S. P. Ahmad ad A. Ahmd, Tasmud vs Raygh dsbuo: A gazao of h vs Raygh dsbuo, Mah. Thoy Mod 4(7 (4, 9-98; S. K. Ashou ad M. A. Ehwy, Tasmud Loma dsbuo, Am. J. App. Mah. Sa. (6 (3b, -7. [3] G. R. Aya ad C. P. Tsokos, Tasmud Wbu dsbuo: A gazao of h Wbu pobaby dsbuo, Euopa Joua of Pu Appd Mahmacs 4 (, 89-. [4] B. Abdu-Mom ad M. Sham, Epoad Nadaajah Haghgh poa dsbuo, Iaoa Joua of Mahmaca Aayss ad Appcaos (5 (5, [5] B. Abdu-Mom ad M. Sham, Tasmud Gompz dsbuo, Compuaoa ad Appd Mahmacs (3 (5, [6] I. Ebaa, Tasmud modfd vs Wbu dsbuo: A gazao of h modfd vs Wbu pobaby dsbuo, I. J. Mah. Ach. 4(8 (3, 7-9. [7] I. Ebaa ad G. Aya, O h asmud addv Wbu dsbuo, Ausaa Joua of Sascs 4( (3, 7-3. [8] Eahoo A. Owooko, Pum E. Oguud ad Adbowa O. Adjumo, Pfomac ag of asmud poa dsbuo: A aayca appoach, Spg Op Joua 4 (5, 88. DOI.86/s
15 PERFORMANCE RATING OF TRANSMUTED NADARAJAH 5 [9] Wa Gchs, Sasca Modg wh Qua Fucos, CRC Pss Ic.,. [] M. A. Hussa, Tasmud poad gamma dsbuo: A gazao of h poad gamma pobaby dsbuo, App. Mah. Sc. 8(7 (4, [] M. S. Kha ad R. Kg, Tasmud modfd Wbu dsbuo: A gazao of h modfd Wbu pobaby dsbuo, Euopa Joua of Pu ad Appd Mahmacs 6 (3, [] M. S. Kha, R. Kg ad I. Hudso, Chaacscs of h asmud vs Wbu dsbuo, ANZIAM J. 55 (EMAC3 (4, C97-C7. [3] M. S. Kha ad R. Kg, Tasmud modfd Wbu dsbuo: A gazao of h modfd Wbu pobaby dsbuo, Euopa Joua of Pu ad Appd Mahmacs 6( (3a, [4] M. S. Kha ad R. Kg, Tasmud gazd vs Wbu dsbuo, Joua of Appd Sasca Sccs (3 (3b, 5-3. [5] M. S. Kha, R. Kg ad I. Hudso, Tasmud gazd poa dsbuo: A gazao of h poa dsbuo wh appcaos o suvva daa, Commucaos Sascs: Smuao ad Compuao, o appa, 5a. [6] M. S. Kha, R. Kg ad I. Hudso, Tasmud Wbu dsbuo Pops ad Esmao. Commucaos Sascs: Thoy ad Mhods 46( (7, [7] M. S. Kha, R. Kg ad I. Hudso, Tasmud Gazd Gompz dsbuo: Pops ad Esmao, Paksa Joua of Sascs 3(3, (6, 6-8 [8] F. Movc ad I. Puka, Tasmud Pao dsbuo, Pobaby Sasca Foum 7 (4, -. [9] P. E. Oguud ad A. O. Adjumo, Th asmud vs poa dsbuo, Iaoa Joua Advac Sascs ad Pobaby 3( (5, -7. [] W. T. Shaw ad I. R. Bucky, Th achmy of pobaby dsbuos: Byod Gam-Cha pasos ad a skw kuoc oma dsbuo fom a ak asmuao map, Rsach Rpo, 7. [] W. T. Shaw ad I. R. Bucky, Th achmy of pobaby dsbuos: Byod Gam-Cha pasos, ad a skw-kuoc-oma dsbuo fom a ak asmuao map, 9. axv pp axv: g
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