Studies on Properties and Estimation Problems for Modified Extension of Exponential Distribution

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1 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5 Studes o Propertes ad Estmato Problems for odfed Exteso of Expoetal Dstrbuto.A. El-Damcese athematcs Departmet Faculty of Scece Tata Uversty Tata Egypt ABSTRACT The preset paper cosders modfed exteso of the expoetal dstrbuto wth three parameters. The ma propertes of ths ew dstrbuto s studed wth specal emphass o ts meda mode ad momets fucto ad some characterstcs related to relablty studes. For odfed- exteso expoetal dstrbuto (EXED) have bee obtaed the Bayes Estmators of scale ad shape parameters usg Ldley's approxmato (L-approxmato) uder squared error loss fucto. But through ths approxmato techque t s ot possble to compute the terval estmates of the parameters. Therefore Gbbs samplg method s developed to geerate sample from the posteror dstrbuto. O the bass of geerated posteror sample the Bayes estmate of the ukow parameters s computed ad costructed 95 % hghest posteror desty credble tervals. A ote Carlo smulato study s carred out to compare the performace of Bayes estmators wth the correspodg classcal estmators terms of ther smulated rsk. A real data set has bee cosdered for llustratve purpose of the study. Keywords odfed- exteso expoetal dstrbuto (EXED) axmum lkelhood estmator Bayes estmator squared error loss fucto Ldley s approxmato method ad Gbbs samplg method. INTRODUCTION I the feld of lfetme modellg expoetal dstrbuto (ED) has greater mportace to study the relablty characterstcs of ay lfetme pheomeo. The popularty of ths model has bee dscussed by several authors. Although t became most popular due to ts costat falure rate patter but may practcal stuato ths dstrbuto s ot suted to study the pheomeo where falure rate s ot costat. I recet years several ew classes of models were troduced based o modfcato of expoetal dstrbuto. For example Gupta ad Kudu (999) ad Gupta ad Kudu () troduced a exteso of the expoetal dstrbuto typcally called the geeralzed expoetal (GE) dstrbuto. Therefore t s sad that the radom varable x follows the GE dstrbuto f ts desty fucto s gve by g x; e x ( e x ) () where x > > ad > wth otato s used X~GE( ) for a radom varable wth such dstrbuto. ore recetly Nadarajah ad Haghgh () troduced aother exteso of the expoetal model so that a radom varable X follows the Nadarajah ad Haghgh s expoetal dstrbuto (NHE) f ts desty fucto s gve by g x; + x e +x Da. A. Ramada athematcs Departmet Faculty of Scece asoura Uversty asoura Egypt () where x > > ad > wth otato s used X~NHE. Sajay et al. () explaed the classcal ad Bayesa estmato of ukow parameters ad relablty characterstcs exteso of expoetal dstrbuto. Both dstrbutos have the expoetal dstrbuto (E) wth scale parameter as a specal case whe that s g x; g x; e x () where x > ad > wth the otato X~E. Other extesos of the expoetal model the survval aalyss cotext are cosdered the arshall ad Olk s (7) book. The ma object of ths paper s to preset yet aother exteso for the expoetal dstrbuto that ca be used as a alteratve to the oes metoed above. Some propertes are dscussed for ths ew dstrbuto. The classcal ad Bayesa estmato of the ukow parameters ad relablty characterstcs of a ew exteso of expoetal dstrbuto s developed. It s observed that the LEs of the ukow parameters caot be obtaed ce closed form as expected ad they have to obta by solvg two olear equatos smultaeously. It s remarkable that most of the Bayesa ferece procedures have bee developed wth the usual squared-error loss fucto whch s symmetrcal ad assocates equal mportace to the losses due to overestmato ad uderestmato of equal magtude. However such a restrcto may be mpractcal most stuatos of practcal mportace. For example the estmato of relablty ad falure rate fuctos a overestmato s usually much more serous tha a uderestmato. I ths case the use of symmetrcal loss fucto mght be approprate as also emphaszed by Basu ad Ebrahm (99). Further the Bayesa ferece of the ukow parameters s cosdered uder the assumpto that both parameters have depedet gamma prors. It s observed that the Bayes estmators have ot bee obtaed explct form. Therefore Ldley s approxmato method s used. Ufortuately by usg Ldley s approxmato method t s ot possble to costruct the hghest posteror desty (HPD) credble tervals. Therefore ote Carlo arkov Cha method (Gbbs samplg procedure) s used to costruct the 95% HPD credble tervals for the parameters ad estmates are also coded o the bass of CC samples. ote Carlo smulatos are coducted to compare the performaces of the classcal estmators wth correspodg Bayes estmators obtaed uder squared error loss fucto both formatve ad o-formatve set-up for complete sample. Further cofdece tervals s costructed 95% approxmate ad hghest posteror desty (HPD) credble tervals for the parameters.

2 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5. DENSITY AND PROPERTIES A radom varable X s dstrbuted accordg to the modfed exteded expoetal dstrbuto (ExED) wth parameters ad f ts desty fucto ad the cumulatve dstrbuto fucto of ths ew famly of dstrbuto ca be gve as h t f(t) R(t) + t + t + t ; t. (7) f x + x + x + x e +x+x () where x ad wth the otato X~ExED. ad Fg : Desty plot for varous choce for ad F x e +x+x (5) where x ad. The modfed exteded expoetal dstrbuto (ExED) ca be a useful characterzato of lfe tme data aalyss. The relablty fucto (R) of the modfed exteded expoetal dstrbuto (ExED) s deoted by R(t) also kow as the survvor fucto ad s defed as R t e +t+t ; t Fg : Relablty plot for varous choce for ad Oe of the characterstc relablty aalyss s the hazard rate fucto (HRF) defed by (6) Fg : Falure Rate for varous choce for ad It s mportat to ote that the uts for h t s the probablty of falure per ut of tme dstace or cycles. These falure rates are defed wth dfferet choces of parameters. The cumulatve hazard fucto of the modfed exteded expoetal dstrbuto s deoted by H t ad s defed as t H t h x dt + x + x + x dx t + t + t.. STATISTICAL ANALYSIS. eda ad mode It s observed as expected that the mea of ExED( ) caot be obtaed explct forms. It ca be obtaed as fte seres expaso so geeral dfferet momets of ExED( ). Also caot get the quatle x q of ExED( ) a closed form by usg the equato F X x q ; q.thus by usg Equato (5) fd that x q + x q l q < q <. (9) The meda m X of ExED( ) ca be obtaed from (9) whe q.5 as follows x.5 + x.5 l.5. () oreover the mode of ExED( ) ca be obtaed as a soluto of the followg olear equato. d dx f X x; d dx + x + x + x e +x+x. (8) (). omet The r th momets of the ExED s deoted by μ r ad t s gve by μ r m r r m () e mr mr ( ) rm Γ m +

3 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5 The mea ad varace of ExED are ad E(x) Var x m m m + m m m Γ m + () e m m ( () m + ) Γ m + () m m + ) Γ. m + e m m () e m m ( (). CLASSICAL ESTIATION I ths secto the maxmum lkelhood estmates (LEs) of the parameters have bee obtaed relablty fucto ad hazard fucto for the cosdered model. Let us suppose that uts are put o a test wth correspodg lfe tmes beg detcally dstrbuted wth probablty desty fucto () ad cumulatve dstrbuto fucto (5). The the lkelhood fucto ca be wrtte as L x\ f ; e ((++x ) ) + l L l + l L + l L l + + l x x (5) (6) (7) (8) l L + l + x x l + + (9) axmum lkelhood estmates ca be obtaed by solvg the above two equatos smultaeously but these equatos caot be expressed explct form. Therefore Nolear maxmzato techque ( bult commad R software) has bee used to compute the LEs of the parameters. Further let are the LEs of ad respectvely. Therefore usg varace property of LEs the Bayes estmators of relablty fucto R ad hazard fucto h for ay specfed tme t are gve by followg equatos. ad R t e +t+t () h t + t + t + t. (). Asymptotc tervals for the parameters I ths subsecto the Fsher formato matrs obtaed to compute 95% asymptotc cofdece tervals for the parameters based o maxmum lkelhood estmators (LEs). The Fsher formato matrx ca be obtaed by usg loglkelhood fucto (6). Thus we have where l L l L l L I( ) + l L l L l L l L l L l L l L l L l L x + + x x + l () () () (5)

4 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5 l L l L x + l l L + x + x l. (6) (7) (8) All the above dervatves are evaluated at ( ). The above matrx ca be verted to obta the estmate of the asymptotc varace-covarace matrx of the LEs ad dagoal elemets of I ( ) provdes asymptotc varace of ad respectvely. The above approach s used to derve the ( γ)% cofdece tervals of the parameters as the followg forms ± Z γ Var ± Z γ Var ad ± Z γ Var. (9) 5. BAYESIAN ESTIATION OF THE PARAETERS I ths secto the expresso posteror dstrbutos have bee derved for the cosdered model. Let X (x x x x ) be a radom sample of sze observed from () ad the the lkelhood fucto s gve as (5). So ths model s a good alteratve of the several expoetated famly ad reduces expoetal famly for a ad. Sce for ths dstrbuto ot a sgle cojugate pror s kow tll date. Therefore we cosder depedet gamma prors for shape.e. ~ gamma(a b) as well as scale parameter.e ~ gamma c d ad ~ gamma(g f). Therefore the jot pror of ( ) s gve as π a c g ebd f () where abcdg ad f are the hyper parameters. Therefore the jot posteror dstrbuto ca wrtte as P X +a +c g e bd f e + +x + x + + x. () Uder squared error loss fucto (SELF) the Bayes estmate s the posteror mea of the dstrbuto. Therefore the Bayes estmate of ( ) Relablty fucto R(t)ad Hazard fucto h(t)ca be expressed followg equatos. K +a +c g e bd f e + d d d + +x K +a +c g e bd f e + x + + x d d d K +a +c g e bd f e R t + x + + x d d d K +a +c g e ad () + +x () + +x bd f +t+t () +x e + + x + + x d d d (5) h t K +a +c g where K e + t + t + t + +x ebd f + d d d. +a +c g e bd f e (6) + +x + x + + x d d d. (7) From the above t s easy to observed that the aalytcal soluto of the Bayes estmators are ot possble. Therefore the Ldley s approxmato methods ad arkov Cha ote Carlo method have bee used to obta the approxmate solutos of the above Eqs. ( 6). 5. Ldley s approxmato It may be oted here that the posteror dstrbuto of ( ) takes a rato form that volves a tegrato the deomator ad caot be reduced to a closed form. Hece the evaluato of the posteror expectato for obtag the Bayes estmator of ad wll be tedous. Amog the varous methods suggested to approxmate the rato of tegrals of the above form perhaps the smplest oe s Ldley's (98) approxmato method whch approaches the rato of the tegrals as a whole ad produces a sgle umercal result. ay authors have used ths approxmato for obtag the Bayes estmators for some lfetme dstrbutos; see amog others Howlader ad Hossa () ad Jahee (5).

5 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5 Thus the use of Ldley's (98) approxmato s proposed for obtag the Bayes estmator of ad by cosderg the fucto I(x) defed as follows; I x E u u el +G d e L +G d (8) where u s a fucto of ad oly L s log of lkelhood G s log jot pror of ad Accordg to Ldley (98) f L estmates of the parameters are avalable ad s suffcetly large the the above rato of the tegral ca be approxmated as: I x u + u a + u a + u a + a + a 5 + A u ς +u ς +u ς +B u ς +u ς +u ς +C u ς +u ς +u ς (9) where a ρ ς + ρ ς + ρ ς a u ς + u ς + u ς a 5 (u ς + u ς + u ς ) A ς L + ς L + ς L +ς L + ς L + ς L B ς L + ς L + ς L +ς L + ς L + ς L C ς L + ς L + ς L +ς L + ς L + ς L ad subscrpts o the rght-had sdes refer to respectvely ad let θ θ ad θ ρ ρ u θ u θ θ θ θ u j u θ θ θ θ θ j j L j L θ θ θ θ θ j j L jk L θ θ θ j k. θ θ j θ k ad ς j s the j th elemet of the verse of the matrx L j all evaluated at the LE of parameters. For the pror dstrbuto () we have ρ l π a l + c l + g l b + d + f ad the we get ρ a b ρ c d ρ g f Also the values of L j ca be obtaed as follows for j L + x + x l + + x L L + l L L L L L L l x + + x + x + + x x + + x. ad the values of L jk for j k L + x + x l + + x L L L L L L L L L L L L l + + l + l + + l + + l + l L L L L L L + l 5

6 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5 L + L L L L L L L After substtuto the Eqs. (-6) reduces lke Ldleys tegral therefore for the Bayes estmates of the parameter If u the ef BS + ab ς + Aς +Bς +Cς. ς + cd ς + ad smlarly the Bayes estmate for uder SELF s If u the BS + a b ς + c d + e f ς + Aς +Bς +Cς. ad smlarly the Bayes estmate for uder SELF s If u the BS + a b ς + c d + e f ς + Aς +Bς +Cς. () ς () ς () Further the Bayes estmates of the relablty fucto ad hazard fucto uder SELF are gve by Relablty: If u e +t+t the the correspodg dervatves are u e +t+t + t + t l + t + t u e +t+t + t + t l + t + t + t + t u te +t+t + t + t l + t + t + t + t u t e +t+t + t + t l + t + t + t + t u te +t+t + t + t u e +t+t t + t + t t + t + t u t +t+t e + t + t + + t + t u t e +t+t + t + t u e +t+t t + t + t t + t + t remag L ad (a a a a a 5 ) terms are same as above. Therefore relablty estmate s; R BS t e +t+t + u a + u a + u a + a + a 5 + A u ς +u ς +u ς +B u ς +u ς +u ς +C u ς +u ς +u ς. () Hazard: I the case of hazard fucto If u + t + t + t the the correspodg dervatves are u + t + t + t + l + t + t u + t + t + t l + t + t + l + t + t u + t + t + l + t + t + t + t + t + t + ( ) l + t + t u t + t + t + l + t + t + t + t + t + t + ( ) l + t + t u + t + t + ( )t + t + t + t u t + t + t + t + t + t + t u t + t + t + t + t + t + t u t + t + t + ( )t + t + t + t u t + t + t + t + t + t + t remag L ad (a a a a a 5 ) terms are same as above. Therefore relablty estmate s; h BS t + t + t + t + u a + u a + u a + a + a 5 + A u ς +u ς +u ς +B u ς +u ς +u ς +C u ς +u ς +u ς. () 5. arkov cha mote carlo method I ths subsecto Gbbs samplg procedure s dscussed to geerate sample from posteror dstrbuto. For more detals about arkov Cha ote Carlo ethod (CC) see Smth ad Roberts (99) Hastgs (97) ad Sgh et al. (). Che ad Shao () developed a ote Carlo method for usg mportace samplg to compute HPD (hghest probablty desty) tervals for the parameters of 6

7 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5 terest or ay fucto of them. Thus utlzg the cocept of etropols Hastgs (-H) uder Gbbs samplg procedure geerate sample from the posteror desty fucto () uder the assumpto that parameters ad have depedet gamma desty fucto wth hyper have depedet gamma desty fucto wth hyper parameters (a b) (c d) ad (g f) respectvely. To mplemet ths techque we cosder full codtoal posteror destes of ad as; parameters (a b) (c d) ad (g f) respectvely. To mplemet ths techque we cosder full codtoal posteror destes of ad as; π X +a e b e π X +c e d e π X g e f e + +x + + +x + + +x (5) (6) + x + + x. (7) -H uder Gbbs samplg algorthm cosst the followg steps: Step : Geerate ad from (5 ) (6) ad (7) respectvely. Step : Obta the posteror sample. by repeatg step tmes. Step : The Bayes estmates of the parameters.e. Relablty fucto R(t) ad Hazard fucto h(t) wth respect to the SELF are gve as; ad s C E π \X s C E π \X s C E π \X R(t) s C E π R t \ X h t s C E π h t \ X k k k k k k k k e + k t+ k t k k k + k t + k t + k t k (8) (9) (5) (5) (5) respectvely. Step : After extractg the posteror samples we ca easly costruct the 95% HPD credble tervals for ad. Therefore for ths purpose order N as < < < N N as < < < N ad N as < < < N. The ϑ % credble tervals of ad are N ϑ + Nϑ N ad () N ϑ + Nϑ N. N ϑ +.. Nϑ N Here x deotes the greatest teger less tha or equal to X. The the HPD credble terval whch has the shortest legth. 6. REAL DATA ANALYSIS I ths secto a real data set s studed to llustrate how the proposed methodology ca be appled real lfe pheomeo. To check the valdty of proposed model Akake formato crtero (AIC) ad Bayesa formato crtero (BIC) have bee dscussed see Table. Further we have also provded emprcal cumulatve dstrbuto fucto (ECDF) plot ad theoretcal cumulatve dstrbuto fucto (CDF) plots for maxmum lkelhood estmator (LE) as well as Bayes estmator of the parameters see fgure of ECDF. After all t s observed that proposed model works qute well. The cosdered data are the falure tmes of the ar codtog system of a ar-plae take from of sze see Lhart ad Zucch (986). I ths case the four dstrbutos amely expoetal expoetated expoetal gamma ad Webull have bee ftted. Both estmato procedures have bee take to accout for the cosdered real data set. The cosdered methodology ca be llustrated as follows; AIC l L X θ k BIC l L X θ k l() where L X θ s the lkelhood fucto k s the umber of parameters assocated wth model. Table : Table shows the values of varous adaptve measures for dfferet models regardg fttg of the cosdered real data odel log L AIC BIC ED(θ) EED( ) Gamma( ) Webull( ) ExED( ) ExED( ) I classcal set-up the maxmum lkelhood estmates (LEs) of relablty fucto ad hazard fucto (R(t) h(t)) are calculated as (..8.) ( ) respectvely. The 95% asymptotc cofdece tervals of ad based o fsher formato matrx are obtaed as ( 75.) (.5) ad ( 9.695) respectvely. 7. CONCLUSION Ths paper troduces a comprehesve accout of mathematcal propertes of the ew dstrbuto. The scaleexpoetal dstrbuto ca be see as a partcular case of the 7

8 Iteratoal Joural of Computer Applcatos ( ) Volume 5 No. September 5 ew model. It s show that the dstrbuto fucto hazard fucto ad momet fucto ca be obtaed closed form. We have cosdered the classcal ad Bayesa estmato of ukow parameters ad relablty characterstcs modfed exteso of expoetal dstrbuto. From the smulato we ca obtas that the Bayes estmates wth o-formatve pror behave lke the maxmum lkelhood estmates but for formatve pror the Bayes estmates behave much better tha the maxmum lkelhood estmates. 8. ACKNOWLEDGENTS The authors thak the referees ad edtor for ther valuable commets ad suggestos for mprovg the paper. Our thaks to the experts who have cotrbuted towards developmet of the template. 9. REFERENCES [] Basu A.P. ad Ebrahm N. (99). "Bayesa approach to lfe testg ad relablty estmato usg asymmetrc loss fucto". Joural of Statstcal Plag ad Iferece [] Che.H. Shao Q.. ad Ibrahm J.G. (). "ote Carlo methods Bayesa computato". Sprger-Verlag New York. [] Gupta R. D. ad Kudu D. (999). "Geeralzed expoetal dstrbutos". Australa ad New Zealad Joural of Statstcs.() [] Gupta R. D. ad Kudu D. (). "Expoetated expoetal famly: A alteratve to Gamma ad Webull dstrbuto". Bometrcal Joural. () 7. [5] Hastgs W.K. (97). "ote Carlo samplg methods usg arkov chas ad ther applcatos". Bometrka. 57() [6] Howlader H. A. ad HossaA. (). "Bayesa survval estmato of Pareto dstrbuto of the secod kd based o falure-cesored data". Computatoal Statstcs ad Data Aalyss [7] Jahee Z. F. (5). "O record statstcs from a mxture of two expoetal dstrbutos". Joural of Statstcal Computato ad Smulato.75() pp. -. [8] Ldley D.V. (98). "Approxmate Bayesa method". Trabajos de Estad. 7. [9] Lhart H. ad Zucch W. (986). "odel selecto". Wley New York. [] arshall A. W. ad Olk I. (7). "Lfe Dstrbutos: Structure of Noparametrc". Semparametrc ad Parametrc Famles. [] Nadarajah S. ad Haghgh F. (). "A exteso of the expoetal dstrbuto". Statstcs: A Joural of Theoretcal ad Appled Statstcs. 5(6) [] Sajay K.S. Umesh S. ad Abhmayu S. Y. (). "Relablty estmato ad predcto for exteso of expoetal dstrbuto usg formatve ad oformatve prors". Iteratoal Joural of System Assurace Egeerg ad aagemet. [] Sgh S.K. Sgh U. ad Sharma V.K. (). "Bayesa predcto of future observatos from verse Webull dstrbuto based o type-ii hybrd cesored sample". Iteratoal Joural of Advaced Statstcs ad Probablty.. [] Smth A.F.. ad Roberts G.O. (99). "Bayesa computato va the Gbbs sampler ad related arkov cha ote Carlo methods". Joural of the Royal Statstcal Socety: Seres B (Statstcal ethodology). 55(). IJCA T : 8