On Estimation of Unknown Parameters of Exponential- Logarithmic Distribution by Censored Data

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1 saqartvlos mcrbata rovul akadms moamb, t 9, #2, 2015 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol 9, o 2, 2015 Mathmatcs O Estmato of Ukow Paramtrs of Epotal- Logarthmc Dstrbuto by Csord Data Al Pjya Dpartmt of Mathmatcs, Faculty of Eact ad Natural Sccs, Tbls Stat Uvrsty, Tbls (Prstd by Acadmy Mmbr Elzbar Nadaraya) ABSTRACT Th problm of stmato of paramtrs of Epotal-Logarthmc dstrbuto th cas of csord data s cosdrd W usd psudo mamum lklhood mthod ad costructd a procdur to solv ths problm Thorm of cosstcy s provd Smulato s usd to study th proprts of stmators drvd 2015 Bull Gorg Natl Acad Sc Ky words: Epotal-Logarthmc dstrbuto, psudo mamum lklhood stmators, cosstt stmators, partly csord data Th Epotal Logarthmc (EL) dstrbuto s a famly of lftm dstrbutos wth dcrasg falur rat, dfd o th trval [0, ), s [1] Ths dstrbuto s paramtrzd by two paramtrs: p (0,1) ad >0 (EL) dstrbuto s usd th study of lgths of orgasms, dvcs, matrals, tc, whch s of major mportac th bologcal ad grg sccs Th gv work studs th problm of paramtrs stmato of (EL) dstrbuto wth a spcfc obsrvato pattr Th obsrvatos ar assumd to group ad full o obsrvatos of dvdual ralzato valus ar possbl Applcato of a modfd a crta ss, mthod of mamum lklhood s suggstd ad th offrd procdur s show to lad to cosstt stmators Lt X b a radom varabl wth a dstrbuto fucto F( ) F(, ), whr s a ukow vctor paramtr a ft dmsoal Euclda spac q R Suppos s compact W hav to costruct a cosstt stmator basd o obsrvatos of a radom varabl X Th prmt s dsgd such a way that th actual umbr of ralzatos s ukow to us, w kow oly a part of thos ralzatos Lt fd pots 0 t1 t2 t b gv o th l R (cludg th cas wh th last pot taks a ft valu) Ths pots mak thr catgors of trvals: 0) Th trval ( t, t 1) blogs to zro catgory f w do ot kow thr th dvdual valus of th sampl, or th umbr of sampl valus of th radom varabl apparg ths trval 1) Th trval ( t, t 1) blogs to th frst catgory f dvdual sampl valus ar ukow, but w 2015 Bull Gorg Natl Acad Sc

2 34 Al Pjya kow th umbr of sampl valus of th radom varabl X ths trval As usual, w dot ths umbr by 2) Th trval ( t, t 1) blogs to th scod catgory f w kow dvdual sampl valu 1, 2,, Furthr w dot summato ad tgrato th trvals of th zro, frst ad scod catgors by,(1) ad (2), rspctvly (s [2]) W call ths typ of sampl partally groupd sampl wth csorg Evdtly, csord sampls of both typs as wll as trucatd sampls mak a spcal cas of th statd problm Absc of formato th zro catgory trval crats crta dffcults, whch w try to ovrcom basd o our kowldg of th dstrbuto typ of F(, ) ad th umbr of sampl mmbrs that do ot appar th zro catgory trval: (1)(2) Lt A ( t, t 1) b zro catgory trval W dot by m th umbr of sampl mmbrs apparg A Th, r m th total umbr of obsrvatos Not that m m apparg A If F ( ) r F r (, ) dots a mprcal dstrbuto fucto, th m F ( r t 1) F r ( t ) m s, rlatv frqucy of X ad by vrtu of Broull s law of larg umbrs t covrgs to p ( ) F( t 1, ) F( t, ) wth probablty 1 By summato of Equalts (1) ovr all zro catgory trvals w obta whch lads to F r ( t 1) F r ( t ) m, 1 F ( r t 1) F r ( t ) F r ( t 1) F r ( t ) m (2) 1 ( F r t 1) F r ( t ) W td to apply th mthod of psudo mamum lklhood Assum that th radom varabl X has a dstrbuto dsty wth rspct to Lbsgu masur f ( ) f (, ) Th th lklhood fucto has th followg form: j (1) j1 m 1 1, (3) L ( ; ) F( t ) F(t ) F( t ) F(t ) f ( ) whr m, ar dfd by Formulas (2) Fdg mamum pots of th fucto L( ; ) bcoms complcatd sc t s dffcult to study smoothss proprts of mprcal fuctos Thrfor, w cosdr a modfd lklhood fucto: F ( t ) F ( t ) 1 1 F ( t1 ) F ( t ) 1 1 (4) L ( ; ) F( t ) F(t ) F( t ) F(t ) f ( ) j (1) j1 (1) Bull Gorg Natl Acad Sc, vol 9, o 2, 2015

3 O Estmato of Ukow Paramtrs of Epotal-Logarthmc Dstrbuto 35 Lmma: Lt th followg codtos b satsfd: a) Th dstrbuto fucto F(, ) s cotuous both varabl ad has a cotuous drvatv F (, ) f (, ) ; b) Th fucto L ( ; ) has a absolut mamum Th s cosstt ad asymptotcally ffct stmator of th tru valu of th paramtr 0 Th proof rsults from th corrspodg thorms of [3 4] Estmato of th paramtrs p ad of (EL) Dstrbuto Lt X b a radom varabl dstrbutd wth rspct to (EL) dstrbuto, wth dsty ad dstrbuto fucto 1 (1 p) f (, p, ) l p 1 (1 p) l(1 (1 p) ) F( ) 1, l p p (0,1), 0, [0, ) Lt [1, ) b a zro catgory trval Howvr, w hav obsrvatos X1, X 2,, X th trval [0,1) W hav to stmat ad p paramtrs wth rspct to ths obsrvatos W apply psudo mamum lklhood stmats I ordr to costruct lklhood fuctos ot that f w dot by k th umbr of gral sampl mmbrs that appar [1, ), th k wll b th frqucy k of thm apparg th trval [1, ) Thrfor by Broull Kolmogorov thorm w hav F( ) F(1) k 1 F( ) F(1) Sc th probablty of k lmts of th sampl apparg th black hol s [F( ) F(1)] k, w ca wrt th psudo lklhood fucto as Hc, l(1 (1 p) ) l pl(1 (1 p) ) l(1 (1 p) ) 1 (1 p) L ( ) (5) l p 1 l p 1 (1 p) l(1 (1 p) ) l(1 (1 p) ) 1 l L l l( ) l(1 p) l p l(1 (1 p) ) l p l p l l (1 (1 p) ) 1 1 Computg l L w gt Bull Gorg Natl Acad Sc, vol 9, o 2, 2015

4 36 Al Pjya (1 p) (1 p) (1 (1 p) )(l p l(1 (1 p) )) 1 (1 p) 1 1 (1 p) l p l(1 (1 p) ) l 2 (6) 1 (1 p) )(l p l(1 (1 p) )) l p l L Smlarly, p l(1 (1 p) ) l l p p p p p p 2 (l l(1 (1 ) )) (l l(1 (1 ) ))l Studyg (6), wh 0 ad w gt l p l(1 (1 p) ) 1 (1 p) p (7) p l p 1 p 1 (1 p) 1 l L lm 0 ad l L lm 1 Th cotuous fucto l L chags ts sg ad hc thr sts a pot, such that l L ( ) 0 W com to th sam cocluso rlatd to paramtr p: l L lm p 0 p ad l L lm p p 2 1 Th cotuous fucto l L p chags ts sg ad hc thr sts a pot p, such that l L ( p p ) 0 p O th bass of abov coclusos ad takg to cosdrato Lmma w ca stat that th followg thorms of cosstcy ar tru: Thorm 1: Lt X1, X 2,, X b radom varabls of a sampl dstrbutd wth rspct to (EL) dstrbuto wth ukow paramtr ad kow p Th obsrvatos ar mad o th trval [0,1) ad [1, ) thr sampl mmbrs, or thr umbr ar rcordd Th th psudo mamum lklhood stmator for paramtr sts ad s a uqu root of th quato: Bull Gorg Natl Acad Sc, vol 9, o 2, 2015

5 O Estmato of Ukow Paramtrs of Epotal-Logarthmc Dstrbuto 37 ad th stmator s cosstt (1 p) (1 p) (1 (1 p) )(l p l(1 (1 p) )) 1 (1 p) 1 1 (1 p) l p l(1 (1 p) ) l 0, 2 1 (1 p) )(l p l(1 (1 p) )) l p Thorm 2: Lt X1, X 2,, X b radom varabls of a sampl dstrbutd wth rspct to (EL) dstrbuto wth ukow paramtr p ad kow Th obsrvatos ar mad o th trval [0,1) ad [1, ) thr sampl mmbrs or thr umbr ar rcordd Th th psudo mamum lklhood stmator for p paramtr sts ad s a uqu root of th quato: l(1 (1 p) ) l l p p p p p p 2 (l l(1 (1 ) )) (l l(1 (1 ) ))l l p l(1 (1 p) ) 1 (1 p) p ad th stmator s cosstt 0, p l p 1 p 1 (1 p) 1 Tabl Th rsult of th prmt β p =100 =250 =500 =1000 p= p= p= β= β= β= Smulato Th objctv of our smulato s to chck rsults rcvd prcdg scto ad to compar th actual paramtrs valus wth valus, whch wr gott va prmts I ths study, th sampl sz chos s = 100, 250, 500 ad 1000 W cosdr svral cass for both ad p paramtrs Estmato rsults ar show th tabl blow Frst colum shows stmatd paramtrs ad th scod colum w cosdr svral cass for paramtr stmatg Th rsults show th Tabl mply that usg ths mthod stmators hav thr rror dcrasg as th sampl sz crass Bull Gorg Natl Acad Sc, vol 9, o 2, 2015

6 38 Al Pjya matmatka qspocalur-logartmul gaawlbs ucob paramtrbs Sfasba czurrbul moacmbt a pja javasvls salobs Tblss salmwfo uvrstts zust da sabubsmtyvlo mcrbata fakultts matmatks mmartulba (warmodgla akadms wvrs adaraas mr) statas warmodgla qspocalur-logartmul gaawlbs ucob paramtrbs Sfasba czurrbul moacmbt REFERENCES 1 Tahmasb R, Rza S, (2008) Computatoal Statstcs ad Data Aalyss, 52(8): Kulldorff G, Cotrbutos to th thory of stmato from groupd ad partally groupd sampls ALMQVIST&WIKSELL, STOCKHOLM, GOTEBORG UPPSALA 3 Frguso TS (1996) A Cours larg sampl thory Chapma&Hall 4 Dllo JV, Lbao G (2010) Statstcal ad Computatoal Tradoffs Stochastc Compos Lklhood arxv: ,v1:29 p Rcvd July, 2015 Bull Gorg Natl Acad Sc, vol 9, o 2, 2015

Reliability of time dependent stress-strength system for various distributions

Reliability of time dependent stress-strength system for various distributions IOS Joural of Mathmatcs (IOS-JM ISSN: 78-578. Volum 3, Issu 6 (Sp-Oct., PP -7 www.osrjourals.org lablty of tm dpdt strss-strgth systm for varous dstrbutos N.Swath, T.S.Uma Mahswar,, Dpartmt of Mathmatcs,