A Topp-Leone Generator of Exponentiated Power. Lindley Distribution and Its Application

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1 Appled Mathematcal Sceces Vol o HIKARI Ltd A Topp-Leoe Geerator of Epoetated Power Ldley Dstrbuto ad Its Applcato Srapa Aryuyue Departmet of Mathematcs ad Computer Scece Faculty of Scece ad Techology Rajamagala Uversty of Techology Thayabur Phatum Tha 1110 Thalad Copyrght 018 Srapa Aryuyue. Ths artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orgal work s properly cted. Abstract A ew framework for geeratg lfetme dstrbutos s proposed whch s called the Topp-Leoe Epoetated Power Ldley (TL-EPL) dstrbuto. Submodels of the TL-EPL dstrbuto such as the Topp-Leoe Power Ldley Topp-Leoe Geeralzed Ldley ad Topp-Leoe Ldley are troduced. Some statstcal characterstcs of the dstrbutos are vestgated (.e. mea varace ad fuctos of survval hazard ad quatle). The mamum lkelhood estmato s used to estmate the parameters of each dstrbuto. Some real data sets are ftted order to llustrate the usefuless of the proposed dstrbuto. Keywords: Topp-Leoe epoetated power Ldley quatle hazard lfetme data 1 Itroducto The Epoetated Power Ldley (EPL) dstrbuto wth three parameters as proposed by Warahea-Lyaage ad Parara [18] wll provde may applcatos dfferet felds such as egeer bology ad others. All parameters of EPL dstrbuto deed to ths dstrbuto makes t more fleble descrbg dfferet types of real data tha ts sub-models.e. the power Ldley (PL) dstrbuto [5] the geeralzed Ldley (GL) dstrbuto [11] ad the Ldley (L) dstrbuto [9]. The EPL dstrbuto due to ts fleblty accommodatg dfferet forms of the hazard fucto seems to be a sutable dstrbuto that ca be appled to a varety of problems fttg survval data. Ashour ad Eltehwy

2 568 Srapa Aryuyue [] appled the EPL dstrbuto to real data about flood levels ad the actve repar tmes (h) for a arbore commucato trascever. The EPL dstrbuto provdes better ft tha the other dstrbutos (.e. PL GL L epoetal modfed Webull Webull dstrbutos). Hece the data pot from the EPL dstrbuto has better relatoshp ad hece ths dstrbuto s good model for lfe tme data. The EPL dstrbuto s specfed terms of the probablty desty fucto (pdf) cumulatve desty fucto (cdf) ad quatle fucto [18].e. respectvely 1 1 ( ) (1 ) 1 (1 )e e g 1 1 (1) G ( ) 1 (1 )e 1 () 1/ 1 1 1/ G( ) 1 ( 1)(1 )e Q u W u (3) where ad 0. Whe u s dstrbuted as the uform o the terval (01) ad W (.) the Lambert W fucto [3]. The probablty fuctos of the sub-models of the EPL dstrbuto are preseted. For 1 we have the pdf ad cdf of the PL dstrbuto wth postve parameters ad.e. g 1 ( ) (1 ) 1 e 1 ad G1 ( ) 1 (1 )e. 1 For 1 we have the pdf ad cdf of the GL dstrbuto wth postve parameters ad.e. respectvely 1 (1 ) g( ) 1 (1 )e e 1 1 G ( ) 1 (1 )e. 1 ad For 1 we have the pdf ad cdf of the L dstrbuto wth postve parameter.e. respectvely g3 ( ) (1 )e 1 1 ad G3 ( ) 1 e. 1

3 A Topp-Leoe geerator of epoetated power Ldley dstrbuto 569 Statstcal dstrbutos are used to model the lfe of a tem order to study ts mportat propertes. Proper dstrbuto may provde useful formato that result soud coclusos ad decsos. Whe there s a eed for more fleble dstrbutos may researchers are about to use the ew oe wth more geeralzato. Recetly applyg ew geerators for cotuous dstrbutos became more terestg [14]. Ths methodology ca mprove o the goodess of ft ad determe tal propertes. These features have bee establshed by the results of may geerators such as the beta dstrbuto ad Topp-Leoe (TL) dstrbuto. The TL dstrbuto s oe of the cotuous dstrbutos that s attractve as a geerator. Ths dstrbuto was proposed by Topp ad Leoe [16] for emprcal data wth J-shaped hstograms such as powered bad tool ad automatc calculatg mache falures. Let T be a radom varable whch s dstrbuted as the TL dstrbuto wth parameter whch s deoted by T ~ TL( ) the cdf ad pdf of T respectvely 1 1 F ( t) t ( t) t (1 t)( t) TL ad ftl( t) (4) 1 t ( t) where 0t 1 ad 0. Solvg for the quadratc equato of (see [7]) obtas the quatle fucto of TL dstrbuto.e. 1/ t t u 0 Q ( t) F ( u) 1 1 u (5) TL 1 1/ TL where u s dstrbuted as the uform o the terval (01). The TL dstrbutos as a geerator for cotuous dstrbutos are proposed such as two-sded geeralzed Topp ad Leoe dstrbuto [17] Topp-Leoe geeralzed epoetal dstrbuto [14] Topp-Leoe Gumbel dstrbuto [19] ad Geeralzed Topp-Leoe famly of dstrbutos [10]. Geeratg a ew famly of dstrbuto requres two prcpal compoets whch are a geerator ad a paret dstrbuto (see [1 7 14]). Ideed the pdf of a geerator s trasformed to a ew pdf through the cdf G() of a paret dstrbuto. The TL dstrbuto s a cotuous umodal dstrbuto wth a wde rage of applcatos relablty felds ad s used for modelg lfetme pheomea whch has a J-shaped desty fucto wth a bathtub-shaped hazard fucto [15]. I ths artcle s proposed a ew Topp-Leoe geerated famly dstrbuto where the paret dstrbuto s the EPL dstrbuto. Sub-models of the proposed dstrbuto are studed. Some statstcal characterstcs of the dstrbutos are vestgated. The mamum lkelhood estmato (MLE) s used to estmate the parameters of each dstrbuto. Some real data sets are preseted order to llustrate that the data fts by usg the proposed dstrbuto. A Topp-Leoe geerated famly dstrbuto Alzaatreh et al. [1] preseted a method for geeratg a ew famly of dstrbuto

4 570 Srapa Aryuyue wth the followg defto of a radom varable T [ c1 c] c1 c ad a radom varable X wth cdf G ( ). Let W[ G ( )] be a fucto of G () ad satsfy these codtos: a) W[ G( )] [ c1 c ] b) W[ G ( )] s dfferetable ad mootocally o-decreasg ad c) W[ G( )] a as ad W[ G ( )] b as. Let T be a radom varable of a geerator dstrbuto wth pdf rt () defed as [ c1 c ]. Let X be a cotuous radom varable wth cdf G ( ). Thus the cdf ad pdf of a ew famly of dstrbutos are gve respectvely as W[ G( )] d FTL-G ( t) r( t) dt ad f 0 TL-G ( t) r{w[ G( )]} W[ G( )]. d If a radom varable T s dstrbuted as the TL ad bouded o [01]. Let X be a cotuous radom varable wth the TL-G dstrbuto. The cdf ca wrtte as F ( ) G ( ) [ G ( )] (6) TL-G where 0 s a shape parameter. The assocated pdf s f g G G G (7) 1 1 TL-G ( ) ( )[1 ( )] ( ) [ ( )] where g( ) dg( ) d. I addto a TL radom varable wth fte support has the same bouds as the cdf G () of ay other radom varable. Therefore the relato of a radom varable X havg the TL dstrbuto s X G 1 ( T) where T ~ TL( ). Let Q () G be the quatle fucto of a paret dstrbuto by whch ca be smulated the TL-G radom varate from Q u (8) 1/ G (1 1 ) 1/ where 1 1 u s the quatle fucto of the TL dstrbuto (5). The results obtaed Secto 3 ca be a ew TL-G dstrbuto. The momet of the TL-G dstrbuto ca be computed from the probablty weghted momets order ( sr) of the paret dstrbuto. Let G () be the cdf of the paret dstrbuto the the ( s r) th probablty weghted momet of X (see [1 14]) wll be 1 s r s r s r G 0 sr E X G( X ) G( X ) g( )d Q ( u) u d. The momet of the TL-G dstrbuto s s E G ( X ) ( 1) ( ) s 1. 1 (9)

5 A Topp-Leoe geerator of epoetated power Ldley dstrbuto A ew TL-G dstrbuto I ths secto we troduce a ew dstrbuto whch s called the Topp-Leoe epoetal power Ldley (TL-EPL) dstrbuto. 3.1 The TL-EPL dstrbuto From the cdf (6) ad pdf (7) of the TL-G dstrbuto let g ( ) ad G () be the pdf ad cdf of the EPL dstrbuto (see [ 16]). Cosequetly a radom varable X of the TL-EPL dstrbuto X ~ TL-EPL( ) has the cdf ad pdf as follows; F TL-EPL ( ) 1 (1 )e 1 (1 )e 1 1 (10) 1 1 TL-EPL ( ) (1 ) 1 (1 )e e f (1 )e 1 (1 )e. 1 1 (11) We defe the hazard fucto of the TL-EPL dstrbuto as follows H TL-EPL 1 ( ) 1 (1 )e e 1 1 (1 )e (1 ) 1 (1 )e STL-EPL ( ) 1 1 where S ( ) TL-EPL s the survval fucto of the TL-EPL dstrbuto.e. STL-EPL ( ) 1 1 (1 )e 1 (1 )e. 1 1 The quatle fucto of the TL-EPL dstrbuto s obtaed by substtutg (3) for (8).e. 1/ 1 1 1/ 1/ 1 TL-EPL( ) 1 ( 1) 1 (1 1 ) e Q u W u (1) where u s dstrbuted as the uform o the terval (01) ad W () s the Lambert W fucto [3]. From the momet of the TL-G dstrbuto (9) ad the

6 57 Srapa Aryuyue quatle fucto of the EPL dstrbuto (3) we obta the mea ad varace of X.e. respectvely E TL-EPL ( X) ( 1) ( ) ad VTL-EPL( X ) ( 1) ( ) 1 ( 1) ( ) where / 1 1 1/ 1 1 ( 1)(1 )e d 1 W u u ad / 1 1 1/ 1 1 ( 1)(1 )e d. 1 1 W u u 0 Some pdf plots of the TL-EPL dstrbuto are show Fgure 1 ad Fgure. Fgure 1: Plots of the pdf of the TL-EPL dstrbuto wth (a) dfferet values of ad (b) dfferet values of Fgure : Plots of the pdf of the TL-EPL dstrbuto wth (a) dfferet values of ad (b) dfferet values of

7 A Topp-Leoe geerator of epoetated power Ldley dstrbuto 573 The pdf of the TL-EPL dstrbuto s umodal. It creases or decreases for varous values of the parameters gvg the shapes obtaed Fgure (a). I Fgure 1(b) for values of 1 ad 1 the pdf seems almost symmetrc. 3. Sub-models The EPL dstrbuto wth three parameters ad has three sub-models (see [ 16]). Thus the TL-EPL dstrbuto has three sub-models whch are preseted as follows The Topp-Leoe power Ldley dstrbuto For X ~ TL-EPL( ) whe 1 we obta the Topp-Leoe power Ldley (TL-PL) dstrbuto whch s deoted by X ~ TL-PL( ). The TL- PL dstrbuto has the cdf ad pdf.e. respectvely F ( ) 1 (1 ) e TL-PL ftl-pl ( ) (1 ) (1 ) 1 (1 ) e e (13) (14) 3.. The Topp-Leoe geeralzed Ldley dstrbuto For X ~ TL-EPL( ) whe 1 we obta the Topp-Leoe geeralzed Ldley (TL-GL) dstrbuto whch s deoted by X ~ TL-EL ( ). The cdf ad pdf of the TL-GL dstrbuto are F TL-GL ( ) 1 (1 )e 1 (1 )e ftl-gl ( ) (1 ) 1 (1 )e e (1 )e 1 (1 )e. 1 1 (15) (16) 3..3 The Topp-Leoe Ldley dstrbuto For X ~ TL-EPL( ) whe 1 we obta the Topp-Leoe Ldley (TL-L) dstrbuto whch s deoted by X ~ TL-L( ). The TL-L dstrbuto has the cdf ad pdf (17) ad (18) respectvely F ( ) 1 (1 ) e TL-PL 1 (17)

8 574 Srapa Aryuyue 1 1 ftl-pl( ) (1 ) (1 ) 1 (1 ) e e (18) 3.4 Mamum lkelhood estmato I ths secto we descrbe the MLE procedure to obta the estmated value of the parameters of the TL-EPL based o the radom sample ( 1... ) of sze. Let X for 1... be depedet ad detcally dstrbuted. The log-lkelhood fucto of X ~ TL-EPL( ) o the observed sample s log L( ) ( ) gve by ( ) log(1 ) ( 1) log log log log log log log( 1) ( 1) log 1 ( ; ) ( ; ) ( ; ) (19) ( 1) log log where ( ; ) 1 (1 ( 1))e. By dfferetatg ( ) the partal dervatves of ( ) wth respect to ad are gve by ( ) log ( ; ) log ( ; ) (0) 1 1 ( ) ( 1) log ( ; ) log1 ( ; ) ( 1) log ( ; ) (1) 1 1 ( ) log(1 ) log ( 1) log ( ; ) log 1 ( ; ) () 1 1 ( ) log ( ; ) log 1 ( ; ) 1 1 ( 1) log ( ; ). (3) 1

9 A Topp-Leoe geerator of epoetated power Ldley dstrbuto 575 The epresso of these dfferetal equatos (19)-(3) are ot the closed form. I ths study the parameter estmates of ˆ ˆ ˆ ad ˆ ca be obtaed by usg the umercal optmzato wth the lm fucto the R laguage [13] whch R code for the MLE of the TL-EPL dstrbuto are as follows: #The TL epoetated power Ldley (TL-EPL) dstrbuto <- c(... ) logl_tl_epl <- fucto(theta0) { beta <- (theta0[1]) lambda <-(theta0[]) omega <- (theta0[3]) alpha <- (theta0[4]) G <- (1-(1+(beta*^lambda)/(beta+1))*ep(-beta*^lambda))^omega g1 <- ((lambda*beta^*omega)/(beta+1))*(1+^lambda) *(^(lambda-1))*ep(-beta*^lambda) g <- (1-(1+(beta*^lambda)/(beta+1)) *ep(-beta*^lambda))^(omega-1) g <- g1*g logl <- -log(*alpha)-log(g)-log(1-g)-(alpha-1)*log(g)-(alpha-1) *log(-g) retur(sum(logl)) } theta0 <- c(...) Est<-lm(logL_TL_EPLtheta0) 3.5 Applcato study I ths secto we provde a data aalyss order to assess the goodess-of-ft of the TL-EPL model wth two real data sets. I addto the sub-models of the TL- EPL dstrbuto (.e. TL-PL TL-GL ad TL-L dstrbutos) ad the EPL dstrbuto ad ts sub-models (e.g. PL GL L dstrbuto) are cosdered. The parameter (s) of each dstrbuto are estmated by the MLE method. To verfy whch dstrbuto fts better wth real data sets the Komogorov-Smrov test (KS test) wll be employed. Other crtera cludg the Akake Iformato Crtero (AIC) ad Bayesa Iformato Crtero (BIC) are cosdered.e. AIC log L p ad BIC log( L) plog( ) where s the sample sze ad p s the umber of parameters of each dstrbuto. The frst dataset cossts of 0 observatos wth the respect to mamum flood level data to see how the ew model works practce. The data has bee obtaed from Dumoceau ad Atle (see [ 4]) as show Table 1. For the results of Table the KS test dcates that the TL-EPL dstrbuto s a strog compettor compared to other dstrbutos.

10 576 Srapa Aryuyue Table 1. Mamum flood levels data from Dumoceau ad Atle [4] Table. Values of parameter estmates ad Statstcal crtera cocerg mamum flood level data. Dstrbuto ˆ Parameter estmate values Statstcal values ˆ ˆ ˆ -Log AIC BIC KS p-value TL-EPL TL-PL TL-GL TL-L EPL PL GL L For a secod applcato we aalyze a real data set o the actve repar tmes (h) for a arbore commucato trascever. The data s gve Table 3 ad ts source s Jorgese (see [ 8]). For the KS test Table 4 the TL-EPL dstrbuto s the best dstrbuto correspodg to a hgh p-value of the KS test. Table 3. Actve repar tme (h) for a arbore commucato trascever [8] Table 4. Values of parameter estmates ad Statstcal crtera of the actve repar tme data. Dstrbuto ˆ Parameter estmate values Statstcal values ˆ ˆ ˆ -Log AIC BIC KS p-value TL-EPL TL-PL TL-GL TL-L EPL PL GL L

11 A Topp-Leoe geerator of epoetated power Ldley dstrbuto Cocluso The Topp-Leoe Epoetated Power Ldley (TL-EPL) dstrbuto s proposed whch has the Topp-Leoe Power Ldley Topp-Leoe Geeralzed Ldley ad Topp-Leoe Ldley are sub-model. Some statstcal characterstcs of the dstrbutos are vestgated. The mamum lkelhood estmato s used to estmate the parameters of each dstrbuto. We provde a data aalyss order to assess the goodess-of-ft of the TL-EPL model wth two real data sets. The results of KS test dcates that the TL-EPL dstrbuto s a strog compettor compared to other dstrbutos (.e. TL-PL TL-GL TL-L PL GL ad L dstrbutos). Ackowledgemets. We wsh to gratefully ackowledge the referee of ths paper who helped to clarfy ad mprove ts presetato. Refereces [1] A. Alzaatreh C. Lee ad F. Famoye A ew method for geeratg famles of cotuous dstrbutos METRON 71 (013) [] S.K. Ashour ad M.A. Eltehwy Epoetated power Ldley dstrbuto Joural of Advaced Research 6 (015) [3] R.M. Corless G.H. Goet D.E.G. Hare D.J. Jeffrey ad D.J. Kuth O the Lambert W fucto Adv. Comput. Math. 5 (1996) [4] R. Dumoceau ad C. Atle Dscrmato betwee the Log-Normal ad the Webull dstrbutos Techometrcs 15 (01) [5] M.E. Ghtay D.K. Al-Mutar N. Balakrsha ad LJ. Al-Eez Power Ldley dstrbuto ad assocated ferece Comput. Stat. Data Aal. 64 (013) [6] J.A Greewood J. Ladwehr N. Matalas ad J. Walls Probablty weghted momets: Defto ad relato to parameters of several dstrbutos epressble verse form Water Resources Research 15 (1979) [7] M.C. Joes Famles of dstrbutos arsg from dstrbutos of order statstcs. TEST 13 (004)

12 578 Srapa Aryuyue [8] B. Jorgese Statstcal Propertes of the Geeralzed Iverse Gaussa Dstrbuto New York: Sprger-Verlag [9] D.V. Ldley Fducal dstrbutos ad Bayes theorem JR Stat. Soc. Ser. A 0 (1958) [10] A. Mahdavl Geeralzed Topp-Leoe famly of dstrbutos J. Bostat. Epdemol. 3 (017) [11] S. Nadarajah H.S. Bakouch ad R.A. Tahmasb А Geeralzed Ldley dstrbuto Sakhya B 73 (011) [1] S. Nadarajah ad S. Kotz Momets of some J-shaped dstrbutos Joural of Appled Statstcs 30 (003) [13] R Core Team A Laguage ad evromet for Statstcal computg R Foudato for Statstcal Computg Vea Austra (016). [14] Y. Sagsat ad W. Bodhsuwa The Topp-Leoe geerator of dstrbutos: propertes ad fereces Sogklaakar J. Sc. Techol. 38 (016) [15] H. Sulta ad S.P. Ahmad Bayesa Aalyss of Topp-Leoe Dstrbuto uder dfferet loss fuctos ad dfferet prors J. Stat. Appl. Pro. Lett. 3 (016) [16] C.W. Topp ad F.C. Leoe A famly of J-shaped frequecy fuctos Joural of the Amerca Statstcal Assocato 50 (1995) [17] D. Vcara J.R.V Dorpb ad S. Kotzb Two-sded geeralzed Topp ad Leoe (TS-GTL) dstrbutos Joural of Appled Statstcs 35 (018) [18] G. Warahea-Lyaage ad M. Parara A Geeralzed power Ldley dstrbuto wth applcatos ASIAN Joural of Mathematcs ad Applcatos (014) Artcle ID ama

13 A Topp-Leoe geerator of epoetated power Ldley dstrbuto 579 [19] W. Bodhsuwa The Topp-Leoe Gumbel Dstrbuto 1th Iteratoal Coferece o Mathematcs Statstcs ad Ther Applcatos (ICMSA) Bada Aceh Idoesa. (016). Receved: Aprl ; Publshed: May 3 018