The Beta Inverted Exponential Distribution: Properties and Applications

Transcription

1 Volum, Issu 5, ISSN (Ol): Th Bta Ivrtd Epotal Dstrbuto: Proprts ad Applcatos Bhupdra Sgh Dpartmt of Statstcs, Ch. Chara Sgh Uvrsty, Mrut, Ida Emal: Rtu Gol Dpartmt of Statstcs, Ch. Chara Sgh Uvrsty, Mrut, Ida Emal: Abstract I ths study, w propos a thr-paramtr bta vrtd potal dstrbuto whch cotas gralzd vrtd potal ad vrtd potal dstrbutos as spcal sub modls. Ths dstrbuto ca b usd ffctvly th aalyss of lftm data sc t accommodats omootoc, umodal ad vrs bathtub-shapd hazard fuctos. W drv th No-ctral momts, vrs momts, momt gratg fucto, vrs momt gratg fucto, mod, ad also am th dstrbutoal proprts of ordr statstcs. Th mamum lklhood stmats wth thr stadard rrors ad th asymptotc cofdc trvals ar obtad. Bays stmats alog wth postror stadard rrors ad hghst postror dsty trvals of th paramtrs ar also computd. Markov Cha Mot Carlo tchqus ar mployd to smulat compl postror dsts of th paramtrs. Two ral data sts ar usd to llustrat th comptcy of th proposd dstrbuto. Kywords Bta Ivrtd Epotal Dstrbuto, Mamum Lklhood Estmat, Bays Estmat, Asymptotc Cofdc Itrval, Hghst Postror Dsty Itrval, Gbbs Samplr, Mtropols Algorthm. I. INTRODUCTION I th past yars, may gralzd uvarat cotuous dstrbutos hav b troducd th statstcal ltratur. Th gralzato of ay dstrbuto s mportat ordr to mak ts shap mor flbl to captur th dvrsty prst th obsrv data. O of th promt classs of grator that s usd to gralz th wll kow dstrbutos s th bta grator class. Itally, [3] usd bta grator to gralz ormal dstrbuto, ad calld t as bta-ormal dstrbuto. Th class of bta-gralzd (BG) dstrbuto s dfd as G() a- b- F G() = w (- w) dw (.) Whr G() s th dstrbuto fucto (df) of a radom varabl (rv) X wth probablty dsty fucto (pdf) g(). For a cotuous rv X, th pdf for th bta-gralzd dstrbuto (.) s a- b- f G() = [G()] [- G()] g() (.) Whr a > ad b > ar th two w addtoal a- paramtrs ad b- B a,b = w - w dw s th bta fucto. Th class of bta-gralzd dstrbutos has Copyrght 5 IJASM, All rght rsrvd 3 maly two applcatos () th ablty of fttg skwd data that grally ot b fttd proprly by stg dstrbutos [5] () t s a class of gralzato of th dstrbuto of ordr statstcs for th rv X wth cdf G(). Thraftr, may w BG dstrbutos hav b appard th ltratur. Ths clud bta-gumbl [9], bta-frcht [7], bta-potal [8], bta- Wbull [6], bta-parto [], bta Ivrs Wbull [], Bta-Brbaum-Saudrs [7], bta gralzd Wbull [5] ad bta-cauchy [4]. I ths papr, w troduc bta vrtd potal (BIE) dstrbuto by takg G() to b th dstrbuto fucto of vrtd potal (IE) dstrbuto. Ths gralzato cluds vrtd potal [] ad gralzd vrtd potal [] dstrbutos as spcal cass. Th proposd BIE dstrbuto has b appld to modl fatgu tm of 66-T6 alumum coupos data [] ad brakg strss of carbo fbrs data []. It s obsrvd that th BIE dstrbuto ft both th data wll as compard to othr cosdrd modls. Th rst of th papr s outld as follows: I scto, th BIE dstrbuto s troducd. I scto 3 ad 4, w drv altratv forms of th cdf, th pdf of BIE dstrbuto ad strss-strgth rlablty fucto rspctvly. Hr, w prss th pdf of BIE dstrbuto as a ft wghtd lr combato of th pdf of IE dstrbuto. Th scto 5 ad 6 dals wth th drvato of momts, vrs momts, momt gratg fucto ad vrs momt gratg fucto of th BIE dstrbuto. I scto 7 ad 8, mod ad th dstrbuto of ordr statstc ar obtad. Mamum lklhood stmats (MLE), th lmts of Fshr formato matr ad Baysa stmats usg MCMC tchqus ar computd scto 9 ad. A smulato study s coductd scto. Th applcatos of th proposd BIE dstrbuto ar llustratv scto by fttg two ral data sts. Th papr s cocludd scto 3. II. BETA INVERTED EXPONENTIAL DISTRIBUTION Hr, w dvlop thr-paramtr BIE dstrbuto by takg G() to b th dstrbuto fucto of vrtd potal (IE) dstrbuto. Th IE dstrbuto was troducd by [] ad has b usd as a lftm modl by [3] dtal. Th cdf ad pdf of IE dstrbuto wth scal paramtr ar gv by -θ/ G IE()= ; >,θ > (.)

2 Volum, Issu 5, ISSN (Ol): θ IE -θ/ g ()= ;,θ > (.) Usg (.) (.), th gral form of th cdf of BIE dstrbuto ca b wrtt as -θ/ a- b- F()= w (- w) dw ; >,(a,b,θ)> Th BIE dsty fucto from (.) s f() = G() a- - G() b- g() B a,b (.3) Whr g() = dg()/d s th dsty of th basl dstrbuto. Thus, w hav θ -a θ/ -θ/ f()= - b- ; >,(a,b,θ)> (.4) Th survval ad hazard fuctos of th BIE dstrbuto dpds o th complt bta fucto rato ad ar gv by R()= - F() ad -θ/ ( ) -θ/ a- b- = - w (- w) dw = - I (a,b) θ -a θ/ -θ/ h()= - I -θ/ ( ) (a,b) - b- (.5) Spcal Cass: () For a=, w gt gralzd vrtd potal (GIE) dstrbuto. () For b=, w hav IE dstrbuto wth scal paramtrs a ad. () For a=b=, (.4) rducs to th pdf of IE dstrbuto wth scal paramtr. Th smulato from BIE dstrbuto s straght forward. Lt V follows Bta dstrbuto wth paramtrs a - ad b, th X = F (V)= -θ/logv follows BIE dstrbuto wth paramtrs a, b ad. Th possbl shaps of dsty ad hazard rat fuctos of BIE dstrbuto ar dpctd Fg. ad for som slctd valus of th paramtrs a, b ad. Th hazard rat fucto of BIE dstrbuto appars to b upsd dow bathtub shap. III. ALTERNATIVE FORMS OF CUMULATIVE DISTRIBUTION AND DENSITY FUNCTIONS W ca wrt by, F = I (a,b) (3.) G() y - a- b- Whr I (a,b)= w (- w) dw dots th y complt bta fucto rato.. th cdf of th bta Fg.. Probablty dsty fucto of BIE Fg.. Hazard rat fucto of BIE dstrbuto wth paramtrs a ad b. For gral a ad b, w ca prss (3.) trms of th wll-kow hyprgomtrc fucto [8] dfd by F α,β,γ; = [] = γ [] [] α β (3.) [ι] Whr α = α α+... α+ι - dots th ascdg factoral. Thus, w hav G a F() = F a,- b,a+;g() a (3.3) Furthr, for b postv ral o-tgr ad z <, w hav j b- (-) Γ(b ) j (- z) = z Γ(b - j) Γ(j +) Whr j= (.) s th gamma fucto. (3.4) Now, usg (3.) ad (3.4), F() ca b wrtt as F()= w G () (3.5) j θ(a+ j) j= j (-) Γb Hr w = ar costats Γ(b - j)γ(j +) (a+ j) j Copyrght 5 IJASM, All rght rsrvd 33

3 Volum, Issu 5, ISSN (Ol): such that j j= w = ad G θ(a+ j) () s th cdf of IE dstrbuto wth scal paramtr θ(a+ j). Th pdf of BIE dstrbuto s thus bcoms f()= w g () (3.6) j θ(a+ j) j= Th BIE survval fucto has th followg paso S()= - F()= w j S θ(a+ j) () (3.7) j= -θ(a+ j)/ Whr S θ(a+ j) ()= - s th survval fucto of th IE dstrbuto wth paramtr θ(a+ j). IV. STRESS-STRENGTH RELIABILITY For a strss-strgth systm modl, th systm oprats tll ts strgth cds th strss coutrd durg ts oprato. Thus, th strss-strgth rlablty ca b dfd as th probablty that th systm s strog ough to ovrcom th strss appld o t. Lt X ad X dot th strgth ad strss varabls, both ar dpdtly ad dtcally dstrbutd (d) as BIE wth paramtrs a, b ad θ. Th, w hav R = P X < X = [ f( )d ]f( )d - - = F( )f( )d = F().f()d ;sc X ad X ard. (4.) - Substtutg (.4) ad (3.5) to (4.), w obta θ -(aθ/) b- -(θ/) - a+ j θ/ R = w j(a,b) - d j= -(θ/) By makg th trasformato - = u, R taks th form a- b- a+ j R = w j(a,b) (- u) u - u du j= Fally, w hav R = w j(a,b) - B(a,a+b+ j) (4.) j= V. MOMENTS OF THE BIE DISTRIBUTION Momts ar grally usd to study varous charactrstcs of a dstrbuto (.. ctral tdcs, dsprso, skwss ad kurtoss). Hr, w drv th r th o-ctral ad vrs momts about zro of th BIE dstrbuto say ' μ r ad ' μ -r. Copyrght 5 IJASM, All rght rsrvd Th No-Ctral Momts Th r th o-ctral momt of th BIE dstrbuto ca b wrtt as ' r μ r = E X = r f()d Usg (3.6), w gt j ' θ - b r -θa+ j / r B a,b j= b - j j Now lttg Z= θ X, w obta r j ' θ - b - -r μ r = - r a+ j B a,b j= b - j j μ =. d (5..) 5. Ivrs Momts Th r th vrs momts about zro of th BIE ' dstrbuto μ-r s gv by ' μ -r = E r = f()d r Whr θ a+ j -(a+ j)θ/ τ r(j)= d r So, = r + a+ j θ -r j ' θ r + (-) b μ -r = a+ j B a,b j= b - j j r - r+ VI. GENERATING FUNCTION (5..) I ths scto, w drvd th momt gratg fucto ad th vrs momt gratg fucto of th BIE dstrbuto say MX t ad IM X t. 6. Momt Gratg Fucto M X t = t f()d j θ - b t -θa+ j / = d B a,b j= b - j j j k θ - b t / -θ a+ j = d B a,b j= b - j j k= k Now usg quato (5..), o gts, ' t k M X t = μk k= k (6..)

4 Volum, Issu 5, ISSN (Ol): Ivrs Momt Gratg Fucto Th vrs momt gratg fucto of th BIE dstrbuto s gv by, IM X (t)= E t / θ t / -aθ/ -θ/ = - d O makg trasformato θ =z, w hav t z/θ -az -z b- IM X (t) = - dz j t z/θ b (-) -z b - j j a+ j = dz B a,b j= j b - IM X (t) = B a,b j= b - j j t -(a+ j) θ VII. MODE b- (6..) Mod of BIE dstrbuto ca b obtad as a soluto of log(f()) =, whch gvs -θ/ -θ/ - aθ - -(b -)θ (- ) = (7.) For gv valus of a, b ad θ, th quato (7.) ca b solvd umrcally to gt mod. VIII. ORDER STATISTICS Th dsty of th th ordr statstcs X : say f :, a radom sampl of sz from th BIE dstrbuto, s gv by (for =,.,) f - : = f F - F - B, - + Whr, f() ad F() ar gv (.4) ad (3.3) rspctvly. Thrfor, th pdf of ordr statstcs bcoms, b(-+)- θ G() - G() f :()= - - B, - + a b - F a,- b,a +;G() F b,- a,b+;- G() IX. MAXIMUM LIKELIHOOD ESTIMATION Lt X, X,., X b a radom sampl of sz from th dsty of BIE (.4). Th log-lklhood fucto ca b wrtt as - = -θ/ b- - θa/ L()= θ - = = (9.) Takg log o both sds, w gt Copyrght 5 IJASM, All rght rsrvd 35 l = logl()= -log- log +logθ - θa + = = -θ / (b -) log - = Now, th MLEs of a, b ad θ ca b obtad by solvg th followg o-lar quatos: l θ = - ψ(a)-ψ(a+b) - a = l = - ψ(b)-ψ(a+b) + b log - -θ/ = ad -θ/ l = - a +(b -) θ θ -θ/ = = - Whr ψ(.) s th dgamma fucto. aˆ - a Th asymptotc dstrbuto of ˆ b - b, whch ˆ θ - θ - follows N 3(,I ), ca b usd to costruct th asymptotc cofdc trvals for th paramtrs a, b ad θ. Whr, I s th obsrvd Fshr formato matr ad s gv by aa ab aθ I= - ba bb bθ θa θb θθ a= ˆ a,b= ˆ b,θ=θ ˆ Th lmts of I ar l ' ' aa = - ψ a -ψ a+b a l ' ' bb = - ψ b -ψ a+b b l - = -(b -) θ θ - -θ/ = -θ/ l a =- a = l d ab = ψ a+b a b db -θ/ l b = b -θ / = - X. BAYESIAN ESTIMATION THROUGH MCMC TECHNIQUES For mplmtg Baysa stmato procdur, w assum dpdt gamma prors for th paramtrs θ,a

5 Volum, Issu 5, ISSN (Ol): ad b as Gamma(υ,η ), Gamma(υ,η ) ad Gamma(υ 3,η 3) wth rspctv dsts as: η υ η- -υθ ω (θ)= θ Γ(η ) ;θ >,(υ,η ) > (.) η υ η- -υa ω (a)= a Γ(η ) ;a >,(υ,η ) > (.) η υ 3 3 η3- -υ3b Γ(η 3 ) ω (b)= b ;b >,(υ,η ) > (.3) Usg lklhood fucto (9.) ad pror dstrbutos (.) (.3), th jot postror dstrbuto of θ, a ad b ad th data s gv by g(θ,a,b ) )= L( ),a,b)ω (θ)ω (a)ω 3(b) (.4) For drawg frcs o th paramtrs θ, a ad b, o d to valuat th jot postror dstrbutos of th paramtrs gv th sampl obsrvatos. Howvr, du to th multdmsoal complty, t s vry dffcult to valuat thm aalytcally. To ovrcom ths dffculty, w us MCMC mthod such as Gbbs samplr proposd by [6], whch grats obsrvatos from th codtoal postror dstrbuto of ach of th paramtrs usg th currt valus of th gv paramtrs. Th full codtoal postror dstrbutos of θ,a ad b ar as follows: - a +ν θ -θ b- +η - = π (θ,a,b) θ - = (.5) - θ +ν a η - π = (a,θ,b) a (.6) B a,b -θ b- η3- -ν3 b π 3(b,θ,a) - b B a,b = (.7) Gbbs algorthm. Grat θ from π (θ ),a,b) as gv (.5).. Grat a from th dsty π (a,θ,b) as gv (.6). 3. Grat b from th dsty π 3(b,θ,a)as gv (.7). 4. Rpat stps -3, M tms ad rcord th squc of Ω =(θ,a,b) aftr dscardg N bur- tratos to lmat th ffcts of th startg valus. (Ω,Ω,...,Ω ). N+ N+ M 5. Bays stmat of Ω say fucto s th * Ω = M - N M j=n+ * Ω udr squard rror loss Ω j 6. Th postror varac of s * * j M V Ω = Ω - Ω M - N j=n+ * * * 7. Lt Ω N+ < Ω N+ <...< Ω M dot rspctvly th ordrd valus of * * * Ω N+,Ω N+,...,Ω M. Th, followg [9], w obta (-γ )% hghst postror dsty (HPD) trvals for Ω. Not that th samplg stps -3 s do usg Mtropols-Hastgs algorthm [4, ] to grat Ω =(θ,a,b). XI. A SIMULATION STUDY Hr, w hav proposd a smulato study for drawg frcs o th paramtrs of th BIE dstrbuto. Assumg θ = 3, a =.3, b = ad θ = 5, a =.5, b= 4, w smulatd two sts of obsrvatos wth sampl szs =3, 5, 8 ad from th BIE dstrbuto (.4). As w hav s scto 9 that th MLEs caot b obtad closd forms, thrfor, w us malk() fucto of R-softwar to obta thm. Th MLEs of θ,a ad b alog wth thr stadard rrors (SE) hav b obtad ad ar lstd Tabls ad. Th 95% asymptotc cofdc trvals for θ,a ad b ar also costructd ad rportd Tabls ad. I Baysa stup, Gbbs samplg algorthm s usd to obta Bays stmats (BE) of th paramtrs wth thr postror stadard rrors (PSE) ad HPD crdbl trvals. Usg Gbbs algorthm, w gratd ralzatos of th Markov cha of θ,a ad b from th codtoal postror dstrbutos gv (.5), (.6) ad (.7) rspctvly. To ullfy th autocorrlato btw th succssv draws of th paramtrs, w oly rgstr vry th gratd valus of draws. Th rsultg MCMC rus, postror dstrbutos ad autocorrlato fuctos of θ,a ad b ar plottd Fg. 3-5, Fg. 6-8 ad Fg. 9- rspctvly. Th MCMC chas of th gratd draws of th paramtrs ar show to b wll mg. Bays stmats of θ,a ad b alog wth thr PSEs ad HPD crdbl trvals hav b summarzd Tabls ad. From th smulato rsults, t has b obsrvd that: I comparso to th MLEs, Bays stmats prform bttr rspct of th stmato rrors. Th wdths of HPD trvals ar comparatvly smallr tha thos of asymptotc cofdc trvals. SEs as wll as PSEs of th stmators td to dcras as sampl sz crass. Th sam trd s obsrvd wth wdths of th trvals. Copyrght 5 IJASM, All rght rsrvd 36

6 Volum, Issu 5, ISSN (Ol): Tabl-: MLEs (SE), 95% cofdc trvals, Bays stmats (PSE), ad 95% HPD trvals Paramtr MLE (SE) CI {wdth} BE (PSE) HPD trval {wdth} = 3.8 (.5) [, 6.756] {6.756}.347 (.45) [.534, 3.45] {.7} 3 a= (.99) [,.53] {.53}.4 (.47) [.33,.38] {.75} b= (.35) [, 7.576] {7.576}.997 (.) [.8,.98] {.386} = 3.78 (6.65) [, 4.6] {4.6}.484 (.368) [.844, 3.] {.377} 5 a= (.888) [, 4.87] {4.87}. (.38) [.38,.86] {.48} b=.8 (.93) [, 3.954] {3.954}.959 (.94) [.789,.5] {.363} = 3.5 (.46) [, 6.43] {6.43}.37 (.79) [.867,.934] {.67} 8 a= (.377) [,.94] {.94}. (.7) [.5,.6] {.9} b=.8 (.6) [.8, 3.6] {.396}.948 (.96) [.748,.] {.37} = (.537) [, 9.76] {9.76}.474 (.63) [.934,.964] {.3} a=.3.5 (.86) [,.3] {.3}.3 (.5) [.5,.5] {.} b=.648 (.44) [.837,.46] {.63}.876 (.93) [.695,.53] {.358} Tabl-: MLEs (SE), 95% cofdc trvals, Bays stmats (PSE), ad 95% HPD trvals Paramtr MLE (SE) CI {wdth} BE (PSE) HPD trval {wdth} = 5 5.6(.86) [, 5.566] {5.566} 4.354(.554) [3.7, 5.39] {.} 3 a =.5.486(.89) [,.6] {.6}.3(.6) [.88,.4] {.34} b=4 5.8 (8.795) [, 3.6] {3.6} (.43) [3.688, 4.38] {.55} = (4.7) [,.76] {.76} (.436) [3.46, 5.57] {.695} 5 a =.5.85 (.4) [, 3.5] {3.5}.98 (.47) [.9,.388] {.79} b= (.57) [, 9.485] {9.485} (.4) [3.68, 4.4] {.54} = (3.6) [, 9.8] {9.8} (.38) [3.796, 5.68] {.47} 8 a =.5.84 (.96) [,.78] {.78}.99 (.38) [.33,.379] {.46} b= (.69) [.5, 5.346] {4.94} (.38) [3.6, 4.8] {.56} = (.89) [.435, 9.8] {8.583} 4.6 (.334) [3.985, 5.46] {.6} a = (.377) [,.386] {.386}.3 (.33) [.4,.37] {.3} b= (.54) [.97, 6.83] {4.96} (.44) [3.578, 4.55] {.577} XII. REAL DATA ANALYSIS Wbull (BW) ad bta potal (BE), ad ar lstd I ths scto, w llustrat th applcablty of th Tabl-3. Th Akak formato crtro (AIC), proposd BIE modl to two ral data sts. W compar th ft of th BIE modl comparso to th othr Baysa formato crtro (BIC) ad Kolmogorov- Smrov (K-S) statstc wth corrspodg P-valu ar cosdrd modls amly, gralzd vrtd usd to compar th ft of th caddat dstrbutos. Th potal (GIE), vrtd potal (IE), vrtd rqurd umrcal calculatos hav b prformd Raylgh (IR), bta usg th programs dvlopd R softwar. Tabl 3: Modls dsts for comparso Modl f() Paramtrs Gralzd vrtd potal (GIE) aθ -θ/ -θ/ a- (a,θ)> (- ) Ivrtd potal (IE) θ -θ/ θ > Ivrtd Raylgh (IR) θ -θ/ θ > 3 Bta Wbull (BW) λ λ λθ λ λ- -b(θ) (- -(θ) ) a- (a,b,θ,λ)> Bta potal (BE) θ -bθ (- -θ ) a- (a,b,θ)> Copyrght 5 IJASM, All rght rsrvd 37

7 Volum, Issu 5, ISSN (Ol): Data st : Th followg data st wr usd by [] ad corrspod to th fatgu tm of 66-T6 alumum coupos cut paralll to th drcto of rollg ad oscllatd at 8 cycls pr scod (cps) , 9, 96, 97, 99,, 3, 4, 4, 5, 7, 8, 8, 8, 9, 9,,, 3, 4, 4, 4, 6, 9,,,,,, 3, 4, 4, 4, 4, 4, 8, 8, 9, 9, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 33, 34, 34, 34, 34, 34, 36, 36, 37, 38, 38, 38, 39, 39, 4, 4, 4, 4, 4, 4, 4, 4, 44, 44, 45, 46, 48, 48, 49, 5, 5, 5, 55, 56, 57, 57, 57, 57, 58, 59, 6, 63, 63, 64, 66, 66, 68, 7, 74, 96, Tabl 4 dsplays th MLEs wth corrspodg stadard rrors of th modls paramtrs alog wth log-lklhood valu, AIC, BIC, K-S statstc ad P-valu. Basd o th valus of logl ad K-S statstcs, w obsrvd that th proposd BIE modl s th bst fttd modl for ths data st as comparso to th othr cosdrd modls. Howvr, th valus of AIC ad BIC of GIE modl ar margally smallr tha thos of BIE modl. Th dsty ad survval plots of th cosdrd modls fttd to data ar dsplayd Fg (a) ad Fg (b). From ths plots, t s amply clar that th proposd BIE modl s supror to th othr dstrbutos trms of modl fttg. Data st : Th followg data cotas obsrvatos o brakg strss of carbo fbrs ( Gba) ad s tak from [] , 3., 4.4, 3.8, 3.75,.96, 3.39, 3.3, 3.5,.8,.4,.76, 3.9,.59,.7, 3.5,.84,.6,.57,.89,.74, 3.7,.4, 3.9,.43,.53,.8, 3.3,.35,.77, 3.68, 4.9,.57,.,.7,.7,.39,.79,.8,.88,.73,.87, 3.9,.87,.95,.67, 4.,.85,.55,.7,.97, 3.68,.8,., 5.8,.69, 3.68, 4.7,.3,.8,.5,.47, 3., 3.5,.97,.93, 3.33,.56,.59,.83,.36,.84, 5.56,.,.48,.5,.48,.3,.6,.5, 3.6, 3.,.69, 4.9, 3.39, 3.,.55, 3.56,.38,.9,.98,.59,.73,.7,.8, 4.38,.85,.8,., For ths data st, th MLEs wth corrspodg stadard rrors of th modls paramtrs alog wth log-lklhood valu, AIC, BIC, K-S statstc ad P-valu ar gv Tabl 5. Aga, BIE modl turs out to b th bst fttd modl as t has hghst P-valu ad lowst K-S statstc valu amog thos of othr modls. BE dstrbuto s also fttg th data wll wth lowst valus of AIC ad BIC. Th stmatd dsty ad survval fucto plots ar show Fg 3(a) ad Fg 3(b). From ths plots, t ca b s that th stmatd dsty fucto ad survval fucto of BIE modl ar closly followd th pattr of th hstogram ad mprcal survval fucto of ths data st rspctvly. XIII. CONCLUSION I ths artcl, w propos a w bta gratd dstrbuto calld as bta vrtd potal dstrbuto. Ths dstrbuto ca b usd to ft th lftm data havg upsd-dow bathtub-shapd hazard fuctos. Th statstcal proprts cludg momts, vrs momts, MGF, vrs MGF ad mod ar dscussd. A smulato study s coductd to judg th prformacs of th MLEs ad Bays stmats. For llustratv purpos, th aalyss of two ral data sts s prstd. ACKNOWLEDGEMENT Th authors gratfully ackowldg th commts ad suggstos from two aoymous rvwrs that rally hlpd us to mprov th papr. Tabl 4: MLE (SE), K-S Statstcs, P-valu, logl, AIC ad BIC of th Modls Fttd to th data Modl MLE (SE) K-S (P-valu) logl AIC BIC a b BIE (.4) (4.68) (.357) (.758) GIE (4.53) (.7) (.78) IE (.966) (.-6) IR (.965) (8.83) BW 36. (.35) 8.5 (.367).344 (.4).697 (.49).847 (.4638) BE (.486).79 (.35).7 (.) -.74 (.67) Copyrght 5 IJASM, All rght rsrvd 38

8 Volum, Issu 5, ISSN (Ol): Tabl 5: MLE (SE), K-S Statstcs, P-valu, logl, AIC ad BIC of th Modls Fttd to th data Modls MLE (SE) K-S statstcs (P-valu) logl AIC BIC a b BIE (.39) (.9) (6.6) (.43) GIE (.67) (.88) (.64) IE ( (.4) ) IR (.) (.37) BW (.96) (.356) (.89) (.3) (.3) BE (.467) (.763).77 (.9) -.94 (.346) Copyrght 5 IJASM, All rght rsrvd 39

9 Volum, Issu 5, ISSN (Ol): Fg.. For alumum coupos fatgu tm data: (a) Estmatd dsty plot (b) Estmatd survval plot Fg.3. For tsl strgth data: (a) Estmatd dsty plot (b) Estmatd survval plot REFERENCES [] A. Akst, F. Famoy ad C. L Th bta-parto dstrbuto. Statstcs, 8, 4, pp [] A.Z. Kllr ad A. R. Kamath, Rlablty aalyss of CNC Mach Tools. Rlb. Egg., 98, vol. 3, pp [3] C. T. L, B. S. Dura ad T. O. Lws, Ivrtd gamma as lf dstrbuto. Mcrolctro Rlab., 989, 9(4), pp [4] E. Alshawarbh, F. Famoy ad C. L, Bta-Cauchy Dstrbuto: Som Proprts ad Applcatos. Joural of Statstcal Thory ad Applcatos, 3, vol., No. 4, pp [5] F. Castllars, L. C. Motgro ad G. M. Cordro, Th Bta Log-ormal Dstrbuto. Joural of Statstcal Computato ad Smulato, 3, vol. 83, No., pp [6] F. Famoy, C. L ad O. Olumolad. Th bta-wbull dstrbuto. Joural of Statstcal Thory ad Applcatos, 5, 4(), pp- 36. [7] G. M. Cordro ad A. J. Lmot, Th bta-brbaum- Saudrs dstrbuto: A mprovd dstrbuto for fatgu lf modlg. Computatoal Statstcs ad Data Aalyss,, 55, pp [8] I. S. Gradshty ad I. M. Ryzhk, Tabl of tgrals, srs, ad products. Acadmc Prss, Nw York,. [9] M. H. Ch ad Q. M. Shao, Mot Carlo stmato of Baysa crdbl ad HPD ts trvals. J Comput Graph Stat, 999, 6, pp [] M. S. Kha, Th bta vrs Wbull dstrbuto. Itratoal Trasactos Mathmatcal Sccs ad Computr,, 3, pp [] M. S. Kha, Thortcal Aalyss of Ivrs Gralzd Epotal Modls. I procdg of th 9 Itratoal Cofrc o Mach Larg ad Computg (ICMLC 9), Prth, Australa,,, pp.8 4. [] M.D. Nchols ad W.J. Padgtt, A bootstrap cotrol chart for Wbull prctls, Qualty ad Rlablty Egrg Itratoal, 6,, pp Copyrght 5 IJASM, All rght rsrvd 4

10 Volum, Issu 5, ISSN (Ol): [3] N. Eug, C. L ad F. Famoy, Bta-ormal dstrbuto ad applcatos. Commu Stat Thory Mthods,, 3, pp [4] N. Mtropols ad S. Ulam, Th Mot Carlo mthod. J. Amr. Statst. Assoc., 949, 44, pp [5] N. Sgla, K. Ja, ad S. K. Sharma, Th Bta Gralzd Wbull dstrbuto: Proprts ad applcatos. Rlablty Egrg & Systm Safty,, vol., pp.5 5. [6] S. Gma ad D. Gma, Stochastc rlaato, Gbbs dstrbutos ad th Baysa rstorato of mags. IEEE Trasactos o Pattr Aalyss ad Mach Itllgc, 984, vol. 6, pp [7] S. Nadarajah ad A. K. Gupta, Th bta Frcht dstrbuto. Far East J Thor Stat, 4, 4, pp-5-4. [8] S. Nadarajah ad S. Kotz, Th bta potal dstrbuto. Rlablty Egrg ad Systm Safty, 5, 9, pp [9] S. Nadarajah ad S. Kotz, Th bta Gumbl dstrbuto. Math Prob Eg, 4,, pp [] W. K. Hastgs, Mot Carlo samplg mthods usg Markov Chas ad thr applcatos. Bomtrka, 97, 57, pp [] Z. W. Brbaum, ad S. C. Saudrs, Estmato for a famly of lf dstrbutos wth applcatos to fatgu. Joural of Appld Probablty, 969, 6, pp AUTHORS PROFILE Dr. Bhupdra Sgh H was bor o 7 th Sptmbr, 97 Ghazabad, Uttr Pradsh. H has compltd hs M.Sc, M.Phl ad Ph.D Statstcs from th Dpartmt of Statstcs, Ch. Chara Sgh Uvrsty, Mrut, Ida. At prst, h s workg as Assocat Profssor th Dpartmt of Statstcs, Ch. Chara Sgh Uvrsty, Mrut, Ida. Hs rsarch ara cluds Ifrtal Rlablty thory, Baysa Statstcs ad Qualty Cotrol. Ms. Rtu Gol Sh was bor o 6 th Sptmbr 99 Mrut. Sh obtad hr M.Sc dgr Statstcs from th Dpartmt of Statstcs, Ch. Chara Sgh Uvrsty, Ida. Sh was th Uvrsty toppr M.Sc Statstcs amato of th Ch. Chara Sgh Uvrsty, Mrut ad was awardd Gold Mdal. Sh s currtly prusg M.Phl Statstcs from th sam Uvrsty. Copyrght 5 IJASM, All rght rsrvd 4