Exponentiated Weibull-Exponential Distribution with Applications

Transcription

1 Avlbl t Appl Appl Mth ISSN: Vol, Issu (Dcmb 07), pp Applctos d Appld Mthmtcs: A Ittol Joul (AAM) Epottd Wbull-Epotl Dstbuto wth Applctos M Elghy, M Shkl d BM Golm Kb 3 Abstct Gdut Studs d Sctfc Rsch Jddh Uvsty Jddh, Kgdom of Sud Ab m_lghy85@yhoocom Dptmt of Mthmtcs Mm Dd Collg Hlh, FL 330, USA mshkl@mdcdu 3 Dptmt of Mthmtcs d Sttstcs Flod Ittol Uvsty Mm, FL 3399, USA kbg@fudu Rcvd: Dcmb 3, 06; Accptd: Ju 9, 07 I ths tcl, w fou-pmt cotuous modl, clld th pottd Wbull potl dstbuto, s toducd bsd o pottd Wbull-G fmly (Hss d Elghy, 06) Th w modl cots som w dstbutos s wll s som fom dstbutos Vous mthmtcl popts of ths dstbuto studd Gl plct pssos fo th qutl fucto, pso of dstbuto d dsty fuctos, momts, gtg fucto, Réy d q tops, d od sttstcs obtd Th stmto of th modl pmts s dscussd usg mmum lklhood mthod Th pctcl mpotc of th w dstbuto s dmosttd though l dt st wh w comp t wth svl lftm dstbutos Kywods: Etopy; Epotl dstbuto; Epottd Wbull-G fmly of dstbutos; Momts; Od sttstcs; Mmum lklhood stmto MSC 00 No: 6E5, 6E99 70

2 AAM: It J, Vol, Issu (Dcmb 07) 7 Itoducto I th lst fw ys, w gtd fmls of cotuous dstbutos hv ttctd svl sttstcs to dvlop w modls Ths fmls obtd by toducg o o mo ddtol shp pmt(s) to th bsl dstbuto Som of th gtd fmls : th bt-g (Eug t l, 00; Jos, 004), gmm-g (typ ) ( Zogfos d Blksh, 009), Kumswmy-G (Kw-G; Codo d d Csto, 0), McDold-G (Mc-G; Ald t l, 0), gmm-g (typ ) (Rst c d Blksh, 0), tsfomdtsfom (T-X; Alzth t l, 03), Wbull-G (Bougugo t l (04), Kumswmy odd log-logstc by Alzdh t l (05), typ hlf-logstc fmly of dstbutos by Codo t l (06), Kumswmy Wbull-G by Hss d Elghy (06), ddtv Wbull-G by Hss d Sd (06) mog oths Ghy gtd fmly of dstbutos toducd by Elghy t l (06), Hss t l (07) toducd typ II hlf logstc G Hss d Elghy (06b) toducd w fmly clld pottd Wbull-gtd (EW-G) Th cumultv dstbuto fucto (cdf ) of pottd Wbull-gtd fmly s gv by G ( ) F( ) p [ ] ; 0 ;,, 0, G ( ) () wh, 0 th two shp pmts d 0 s th scl pmt Th cdf () povds wd fmly of cotuous dstbutos Th pobblty dsty fucto (pdf) cospodg to () s gv by wh 0,,, 0 G( ) [ ] G( ) ( G( )) g( ) G( ) f ( ) p [ ], G ( ) G ( ) I ths pp, w toduc w fou-pmt modl s compttv tso fo th Wbull dstbuto usg th EW-G dstbuto Th w modl tds som ct dstbutos d povds som w dstbutos Th st of th pp s outld s follows I Scto, w df th pottd Wbull potl ( EWE ) dstbuto d povd som spcl modls I Scto 3, w dv vy usful pstto fo th EWE dsty d dstbuto fuctos I th sm scto w dv, som gl mthmtcl popts of th poposd dstbuto Th mmum lklhood mthod s ppld to dv th stmts of th modl pmts Scto 4 Scto 5 gvs umcl mpl to pl how l dt st c b modld by EWE d ths pp ds wth som coclusos Scto 6 () Th Epottd Wbull Epotl Dstbuto I ths scto, th fou-pmt EWE dstbuto s obtd bsd o th EW-G fmly

3 7 B M Golm Kb t l Lt th dom vbl X follow th followg potl dstbuto wth scl pmt, g( ; ) ;, 0 0 Th cdf of potl dstbuto s gv by (3) G( ; ) (4) Substutg fom pdf (3) d cdf (4) to cdf (), th th cdf of pottd Wbull EWE,,,, tks th followg fom potl dstbuto, F( ; ) p( ( ) ) ;,,, 0, 0, (5) wh, (,,, ) s th st of pmts Istg th pdf (3) d cdf (4) to (), th th pdf of EWE tks th followg fom ( ; ) [ ] p[ { ( ) }] p( ( ) ) f (6) Th pdf (6) cots som w dstbutos s wll s som cut dstbutos Tbl lsts th spcl sub-modls of th EWE dstbuto (6) Tbl Spcl sub modls of th pottd Wbull potl dstbuto Modl Dstbuto fucto Rfcs Epottd Epotl F ( ) p( ( )) w potl (EEE) Epottd Rylgh potl ( ) p( ( ) ) F w (ERE) 3 Wbull potl (WE) 4 Epotl F ( ) p( ( ) ) Ogutud t - - F ( ) p( ( )) w l (05) potl (EE) 5 Rylgh Epotl (RE) - - F ( ) p( ( ) ) w

4 AAM: It J, Vol, Issu (Dcmb 07) 73 Th suvvl fucto, hzd t fucto, vsd-hzd t fucto d cumultv hzd t fucto of EWE, spctvly, gv by d R( ; ) p( ( ), [ ] p[ { ( ) }] p( ( ) h ( ; ), p( ( ) [ ] p[ { ( ) }] ( ; ), p( ( ) ) H( ; ) l R( ; ) l p( ) Plots of th cdf, pdf, suvvl fucto, hzd t fucto, d vsd hzd t fucto of EWE dstbuto fo som pmt vlus dsplyd Fgus,, 3, 4, d 5 spctvly Fgu Plots of th cdf of th EWE dstbuto fo som pmt vlus Fom Fgu t pps tht th poposd dstbuto s skwd to th ght Thus, t mght b usful to modl som lf tstg dt

5 74 B M Golm Kb t l Fgu Plots of th pdf of th EWE dstbuto fo som pmt vlus Fgu 3 Plots of th suvvl fucto of th EWE dstbuto fo som pmt vlus

6 AAM: It J, Vol, Issu (Dcmb 07) 75 Fgu 4 Plots of th hzd t of th EWE dstbuto fo som pmt vlus Fgu 5 Plots of th vsd hzd t fucto of th EWE dstbuto fo som pmt vlus 3 Sttstcl Popts I ths scto som sttstcl popts of th EWE dstbuto dscussd 3 Usful Epsos I ths subscto psttos of th pdf d cdf fo pottd Wbull potl dstbuto dvd

7 76 B M Golm Kb t l Usg th glzd boml thom, wh 0 s l o tg d z, z ( ) z 0 (7) Th, by pplyg th boml thom (7) (6), th dstbuto fucto of EWE dstbuto wh s l d postv bcoms f j j ( ) ( ) ( ) (8) j0 j By usg th pow ss fo th potl fucto, w obt Usg Equto (9), Equto (8) bcoms, ( j)[ ] k k k ( ) ( j) k0 k! [ ] k (9) ( ) ( j) f jk k k ( k) ( ) [ ], jk, 0 k! j Th bov quto c lso b wtt s, ( ) ( j) ( k ) m f j, k, m0 k! j m jk k k m( k) ( ) [ ] Usg th boml thom, th bov quto combto of potl dstbut,, wh j, k, m, f ( ) ( ) j, k, m,, j, k, m, 0 c b pssd s ft l (0) k jk k ( ) ( j) ( k ) m m k k! j m Now, w pss th cumultv dsty fucto s ft l combto of potl dstbuto Sc,

8 AAM: It J, Vol, Issu (Dcmb 07) 77 th, ( ) h h [ F( )] [ ], p( ) h ph [ F( )] ( ) p0 p Now, usg th pow ss fo th potl fucto th bov quto, w obt pq q h ( ) ( p) h q [ F( )] [ ], pq, 0 q! p whch c lso b wtt s, [ F( )] h ( ) ( p) h pq q p q q p! ( ), 0 q By usg th boml pso, th bov quto c b wtt s pq q h ( ) ( p) h q t qt [ F( )] [ ] p, q, t0 q! p t Th, wh, h [ F( )] p, q, t,, p, q, t, () p, q, t, 0 pq q ( ) ( p) h q t q t q! p t 3 Qutl d Md Th qutl fucto, sy Q( u) F ( u) of X c b obtd by vtg (5) s follows u Q( u) ( p( ( ) )) Aft som smplfctos, t ducs to Q( u) l (l( u ) ), ()

9 78 B M Golm Kb t l wh, u s ufom dom vbl o th ut tvl b dvd fom () by sttg 33 Momts u 05 0, Tht s, th md s gv by Md Qu ( ) l (l( 05 ) ) I ptcul, th md c Ths subscto povds th momt d momt gtg fucto of EWE dstbuto Momts mpott y sttstcl lyss, spclly pplctos It c b usd to study th most mpott ftus d chctstcs of dstbuto (g tdcy, dspso, skwss d kutoss) If X hs th pdf (6), th ts th momt c b obtd though th followg lto E f d (X ) ( ; ) (3) Substtutg (0) to (3) ylds: Lt y ( ) Th, ( l ) ( X ) j, k, m, l d 0 j, k, m, l ' E bcoms ( ), j, k, m, (4) j, k, m, 0 ( ) Wh k jk k ( ) ( j) ( k ) m m k j, k, m, k! j m d () s gmm fucto Bsd o th fst fou momts of th EWE dstbuto, th msus of skwss A () d kutoss k () of th EWE dstbuto c b obtd s d ( ) 3 ( ) ( ) ( ) A( ), ( ) ( )

10 AAM: It J, Vol, Issu (Dcmb 07) 79 ( ) 4 ( ) ( ) 6 ( ) ( ) 3 ( ) k( ) ( ) ( ) Glly, th momt gtg fucto of followg lto EWE dstbuto s obtd though th 34 Réy d q Etops t t ( ) M X t E X!! ( ) j, k, m, (5), j, k, m, 0 ( ) ( ) 0 Th topy of dom vbl X s msu of vto of uctty d hs b usd my flds such s physcs, gg d coomcs Accodg to Réy (96), th Réy topy s dfd by I ( X ) log f ( ; ) d, 0 d By pplyg th boml thom (7) th pdf (6), th pdf f ( ; ) follows wh f W ( ) ( ; ) j, k, m,, j, k, m, 0 c b pssd s W j, k, m, k jk ( ) ( j ) k k! k m k m j m Thfo, th Réy topy of EWE dstbuto s gv by d th, Th q- topy s dfd by I X W d ( ) ( ) log j, k, m,, j, k, m, 0 0 I Wj, k, m, j, k, m, 0 ( ) ( X) log

11 70 B M Golm Kb t l q Hq( X ) log f ( ; ) d, q 0 d q q Thfo, th q- topy of EWE dstbuto s gv by H q W q ( ) j, k, m, j, k, m, 0 (6) ( X) log Fo mo o dfft kds of tops w f ou ds to Ahsullh t l (04) mog oths 35 Od Sttstcs Lt X : X : X : b th od sttstcs of dom smpl of sz followg th pottd Wbull potl dstbuto, wth pmts, d Th th pdf of th kth od sttstc (Dvd, (98)), c b wtt s follows, f( ( )) k k k f X ( ( ) ( )) ( ( )), k k F k B( k, k ) v0 v v vk (7) wh, B (,) s th bt fucto By substtutg (0) d () (7), d plcg h wth v k, lds to f X( k ) k * ( k) t( k ) ( ( k) ), (8) B( k, k ) v0 j, k, 0 q, t, 0 wh k v v j, k, q, t, * Momts of od sttstcs s gv by: Substtutg (8) (9 ), lds to Th, ( k ) ( k ) ( k ) ( k ) E(X ) f ( ) d (9) k ( k) t ( k) ( k) ( k) B( k, k ) v j, k, 0 q, t, 0 E d * ( k ) (X ) 0 0

12 AAM: It J, Vol, Issu (Dcmb 07) 7 k * ( ) E(X ( k )) B( k, k ) v0 j, k, 0 q, t, 0 ( k) t 4 Mmum Lklhood Estmto Th mmum lklhood stmts (MLEs) of th ukow pmts fo th pottd Wbull potl dstbuto dtmd bsd o complt smpls Lt b obsvd vlus fom th log-lklhood fucto fo th vcto of pmts EWE dstbuto wth st of pmts (,,, ) c b pssd s l L( ) l l l l ( ) l ( ) l X,, X T Th totl Th lmts of th sco fucto U( ) ( U, U, U, U ) gv by U l, (0) ( ), () U d U l l ( ) l, () U ( ) ( ) ( ) ( ) ( ) ( ) (3)

13 7 B M Golm Kb t l Th, th mmum lklhood stmts of th pmts, α, β d λ obtd by solvg th mmum lklhood Equtos (0-3) d pplyg th Nwto-Rphso s tto mthod d usg th comput pckg such s Mpl o R o oth softw 5 Dt Alyss I ths scto, o l dt st lyzd to llustt th mt of EWE dstbuto comp to som sub-modls; mly, Wbull potl (WE), potl potl (EE), d Rylgh potl (RE) dstbutos W obt th MLE d th cospodg stdd os ( pthss) of th modl pmts To comp th pfomc of dfft dstbuto modls, w cosd ct lk mus of log-lklhood fucto Kolmogoov-Smov ( K S) sttstc, Akk fomto cto ( AIC ), th coct Akk fomto cto ( CAIC ), Bys fomto cto ( BIC ) d p-vlu Howv, th btt dstbuto cospods to th smll vlus of ct d bggst p-vlu Futhmo, w plot th hstogm fo ch dt st d th stmtd pdf of th EWE, WE, EE d RE modls Moov, th plots of mpcl cdf of th dt sts d stmtd pdf of EWE, WE, EE d RE modls dsplyd Fgus 5 d 5, spctvly l L, AIC, ( l L), BIC, CAIC,, HQIC K S Th dt hv b obtd fom Bjkdl (960) d pst th suvvl tms ( dys) of 7 gu pgs fctd wth vult tubcl bcll Th dt st s follows: 0, 033, 044, 056, 059, 07, 074, 077, 09, 093, 096,,, 0, 05, 07, 07, 08, 08, 08, 09,, 3, 5, 6,,,,, 4, 3, 34, 36, 39, 44, 46, 53, 59, 6, 63, 63, 68, 7, 7, 76, 83, 95, 96, 97, 0, 3, 5, 6,, 3, 3, 4, 45, 5, 53, 54, 54, 78, 93, 37, 34, 347, 36, 40, 43, 458, 555 Tbl 5 gvs MLEs of pmts d th stdd o (SE) of th EWE dstbuto Th vlus of th log-lklhood fuctos, AIC, CAIC, BIC, HQIC, K-S d p-vlu pstd Tbl 5 Tbl 5 Th MLEs d SE of th modl pmts MLEs S E Mod â ˆ ˆ ˆ SE( â ) SE( l ˆ ) SE( ˆ ) SE( ˆ ) EWE WE EE RE

14 AAM: It J, Vol, Issu (Dcmb 07) 73 Tbl 5 Th vlus of -LL, AIC, BIC, CAIC, HQIC, K-S d p-vlu fo th thd dt st Dstbut - o LL EWE 70 4 WE EE RE AIC CAIC BIC HQIC K-S p- vlu W fd tht th EWE dstbuto wth fou pmts povds btt ft th th spcl sub-modls It hs th smllst K-S, AIC, CAIC, BIC d HQIC vlus mog thos cosdd h Plots of th fttd dsts d th hstogm gv Fgus 5 d 5, spctvly Fgu 5 Estmtd cumultv dsts of th Fgu 5 Estmtd dsts of th modls modls fo dt st fo dt st 6 Cocluso I ths pp, w hv toducd w fou-pmt pottd Wbull potl dstbuto d studd ts dfft sttstcl popts It s otd tht th poposd EWE dstbuto hs svl dsbl popts Th EWE dstbuto covs som stg dstbutos d cots som w dstbutos Th pctcl mpotc of th w dstbuto ws dmosttd pplcto wh th EWE dstbuto povdd btt fttg

15 74 B M Golm Kb t l compso to svl oth fom lftm dstbutos Applctos showd tht th EWE modls c b usd std of oth kow dstbutos Ackowldgmt: Th uthos thkful to th Edto--Chf d th oymous fs fo th vlubl commts d suggstos, whch ctly mpovd th qulty d psttos of th pp gtly Autho, B M Golm Kb ddctd ths pp to ll who scfcd thmslvs dug th lgug voluto of Fbuy, 95 so tht w c spk Bgl (Bgl), my lgug Bgldsh ow REFERENCES Ahsullh, M, Kb, B M G, d Shkl, M (04) Noml d Studt s t Dstbutos d Th Applctos Atlts Pss, Ps, Fc Ald, C, Codo, GM, Otg, EMM, Sb, JM (0) Glzd bt gtd dstbutos Computtol Sttstcs d Dt Alyss 56: Alzdh, M, Emd, M, Doostpst, M, Codo, GM, Otg, EMM, Pscm, RR (05) Kumswmy odd log-logstc fmly of dstbutos: Popts d pplctos Hcttp Uvsty Bullt of Ntul Sccs d Egg Ss B: Mthmtcs d Sttstcs Fothcomg, vlbl t DOI: 0567/HJMS Alzth, A, L, C d Fmoy, F (03) A w mthod fo gtg fmls of cotuous dstbutos Mto, 7 (), Bjkdl, T (960) Acqusto of sstc Gu Ps fctd wth dfft doss of Vult Tubcl Bcll Amc Joul of Hyg, 7, Bougugo, M, Slv, RB, Codo, GM (04) Th Wbull G fmly of pobblty dstbutos Joul of Dt Scc,, Codo GM d d Csto M (0) A w fmly of glzd dstbutos Joul of Sttstcl Computto d Smulto, 8: Codo, G M, Alzdh, M d Dz, P R (06) Th typ I hlf-logstc fmly of dstbutos Joul of Sttstcl Computto d Smulto, 86 (4), Dvd, HA (98) Od sttstcs Scod dto, Wly, Nw Yok Elghy, M, Hss, AS d Rshd, M (06) Ghy Gtd Fmly of Dstbutos wth Applcto, Mthmtcl Thoy d Modlg,6,-5 Eug, N, L C, Fmoy, F (00) Bt-oml dstbuto d ts pplctos, Commucto Sttstcs Thoy Mthods, 3, Gwood, JA, Ldwh, JM, d NC Mtls, (979) Pobblty wghtd momts: Dftos d ltos of pmts of svl dstbutos pssbl vs fom Wt Rsoucs Rsch, 5, Gupt, RC (975) O chctzto of dstbutos by codtol pcttos Commuctos Sttstcs- Thoy d Mthods, 4, Hss, A S d Elghy, M, (06 ) Kumswmy Wbull-gtd fmly of dstbutos wth pplctos Advcs d Applctos Sttstcs, 48, 05-39

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