THE WEIBULL LENGTH BIASED DISTRIBUTION -PROPERTIES AND ESTIMATION-

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1 THE WEIBULL LENGTH BIASED DISTIBUTION -POPETIES AND ESTIMATION- By S. A. Shaba Nama Ahme Bourssa I.S.S. aro Uversy I.N.P.S. Algers Uversy ABSTAT The legh-base verso of he Webull srbuo kow as Webull leghbase WLB srbuo s cosere s show ha s umoal hroughou examg s shape. Oher properes of he srbuo were sue such as he momes a he hazar rae fuco. I s show ha he hazar fuco s upse bahub shape for values of he shape parameer ha are less ha uy a creasg oherwse. Bayesa a o Bayesa esmao problems are also cosere. a umercal example s rouce for llusrao. Keywors: Legh-base Hazar rae fuco elably fuco Webull srbuo Mome meho Maxmum lkelhoo esmaes Bayesa esmaes. MS 000: Prmary: 60E05 secoary: 6E5 6F0 a 6F5. - INTODUTION The cocep of weghe srbuo ca be race o Fsher hs paper The suy of effec of mehos of ascerame upo esmao of frequeces 94; whle hs of legh-base samplg was rouce by ox 96 see Pall 00. These wo coceps f varous applcaos bomecal area such as famly hsory a sease survval a ermeae eves a laecy pero of AIDS ue o bloo rasfuso Gupa a Akma 995. The suy of huma famles a wllfe populaos was he subjec of a arcle evelope by Pall a ao 978. Pall e al. 986 presee a ls of he mos commo forms of he wegh fuco useful scefc a sascal leraure as well as some basc heorems for weghe srbuos a sze-base; as specal case hey arrve a he cocluso ha he legh-base verso of some mxure

2 of scree srbuos arses as a mxure of he legh-base verso of hese srbuos. Gupa a Trpah 990 sue he error mae f a orary srbuo s use sea of he legh-base verso hs error was ame as ype III error or moelg error by ao Gupa a Trpah 990. They erve a geeral form for hs error whe he raom varable uer suy follows a we class of scree srbuos kow as mofe power seres srbuo. The resuls were apple o suy he famly hsory a seases usg he Posso a he geeralze Posso srbuos. Gupa a Trpah 996 sue he weghe verso of he bvarae hree-parameer logarhmc seres srbuo whch has applcaos may fels such as: ecology socal a behavoral sceces a speces abuace sues. They frs erve he weghe verso of hs srbuo a gave a explc form for he probably mass fuco a he probably geerag fuco he legh-base case. Much work was oe o characerze relaoshps bewee orgal srbuos a her legh base versos. A able for some basc srbuos a her legh base forms s gve by Pall a ao 978 such as Logormal Gamma Pareo Bea srbuo. Kharee 989 presee a useful resul by gvg a relaoshp bewee he orgal raom varable X a s legh-base verso Y whe X s eher Iverse Gaussa or Gamma srbuo. He showe ha he legh-base raom varable Y ca be wre as a lear combao of he orgal raom varable X a a ch-square raom varable Z a versely he orgal raom varable ca be characerze hrough hs relaoshp. elaoshps he coex of relably were reae by several auhors such as Pall e al. 986 Ja e al. 989 Gupa a Krma 990 a recely by Oulye a George 00; I hese works he survval fuco he falure rae a he mea resual lfe fuco of he legh-base srbuo were expresse relao wh hese of he orgal srbuo. Webull srbuo plays a mpora role lfe esg a relably sues. I was rouce by he Swesh sces Wallo Webull 95 Kapur a Saxea 00. If X s a raom varable havg he Webull srbuo s pf akes he form: g x x exp x x 0 > 0 > 0 -

3 Where: s he shape parameer a scale parameer. The Webull srbuo s very flexble a hs s ue o s ably moelg boh creasg > a ecreasg < falure raes. The Webull srbuo has he followg rh mome: r r r E X r - I hs pa per he legh-base verso of he webulll srbuo s cosere whch has may applcaos foresry a lfe esg. Le T be a o egave raom varable T s sa o have he Webull legh-base srbuo wll be abbrevae as WLB f s esy fuco s gve by: f e > 0 > 0 - The esy - ca be obae by combg he efo of he legh- base srbuo gve by: g f -4 E a he esy of he orgal srbuo -. Ths srbuo ca be explae as follows: Suppose ha he lfemes of a gve sample of ems s Webull a ha he ems oes have he same chace of beg selece bu each oe s selece accorg o s legh or lfe legh he he resulg srbuo s o Webull bu Webull legh-base. I ca be oe ha - s a geeralze gamma as efe by Sacy 96 wh parameers η k.the WLB srbuo clues he gamma srbuo as specal case. The relably fuco of he WLB srbuo s gve by: γ The umeraor represes he complee gamma fuco efe as: -5 x a γ a x e -6 0 The rh mome assocae wh - s:

4 E T r r r r -7 The same resul ca be obae by usg he momes of he orgal srbuo Webull va he followg relaoshp: r r r E X r E T f. g -8 E X E X 0 0 The res of he paper s orgaze as follows: he shape of he WLB srbuo a s hazar rae fuco are sue secos wo a hree respecvely. Bayesa a o Bayesa esmao problems are cosere seco four. A umercal example s gve seco fve o llusrae he above mehos. - THE SHAPE: The shape of he esy fuco gve - ca be clarfe by suyg hs fuco efe over he posve real le [0 ] a he behavor of s ervave as follows: -- Lms a ervaves of he fuco We have: - lm f lm exp 0 0 bu: lm 0 a lmexp 0 resuls ha: 0 lm f lm f If we pu lm exp z he above lm ca be rewre as: - 4

5 lm bu : f lm lm z exp z lm f 0 z z exp z from hs follows ha 0 - Secoly we suy he ervave sce he fuco f a s logarhm are maxmze a he same po a orer o smplfy calculaos we wll ake he ervave of he logarhm of he fuco f gve by l f l l l l - Takg he ervave of hs fuco wh respec o yels: l f Equag hs ervave o zero gves: The he ervave s equal zero a 0 egave for values of ha excee 0 a posve oherwse. To verfy f he po 0 0 s a maxmum or mmum he seco ervave of f wh respec o s erve whch s equal o: l f -5 Ths quay s egave for all values of. From he above resuls he fuco f creases akes s maxmum a 0 he ecreases aga. The followg fgure llusraes some of he possble shapes of he esy f for specfe values of. The scale parameer was ake o be uy sce oes fluece he shape of hs fuco. 5

6 Fgure : The esy fuco of he WLB srbuo f The shape of he srbuo ca be sue more eals usg he followg wo coeffces. -- oeffce of skewess: Deoe by Sk he coeffce of skewess eables us o kow f he srbuo uer suy s symmerc or o s efe by: µ Sk -6 σ Where: µ s he hr mome abou he mea a σ s he saar evao of he srbuo. The skewess s zero for symmercal srbuos posve for skewe rgh srbuos a egave f he srbuo s skewe o he lef Frak a Alhoe 994 hs meas ha he sg of he coeffce caes he reco of he skew. From equao - for r a equao -. eplacg hem equao -6 gves: SK

7 I ca be oe from -7 ha he coeffce of skewess of he WLB srbuo oes epe o he scale parameer a s fuco of he shape parameer oly he we ca wre as Sk. Numercal vesgao of Sk caes ha he WLB srbuo s symmerc for.448 -a hs po he mea s equal o he mea - posvely skewe wh a al o he rgh for values <.448 a egavely skewe wh a al o he lef for >.448. I pracce more aeo s gve o he frs case.e. for < oeffce of kuross: Deoe by Kur he coeffce of kuross measures he flaess of he op a s efe by: µ Kur 4-8 σ 4 The kuross s equal zero for he ormal srbuo posve for he more all a slm curves ha he ormal oe he eghborhoo of he moe hs case he srbuo s sa o be lepokurc. I s egave for playkurc srbuos.e. flaer ha he ormal srbuo. Usg he momes from - a he varace from - he coeffce of kuross for he WLBD s gve by: Kur Kur s also a fuco o he parameer oly a ca be wre as Kur Kur he Kuross s posve for values of he shape parameer ha are.64 or > he he WLB srbuo s curve s lepokurc h s playcurc for.64 < sce he coeffce s egave for hs case. I s ear zero he eghborhoo of each of he wo pos. 7

8 -HAZAD ATE FUNTION The hazar rae fuco s efe by he rao / F e h γ f akes he form: - I orer o suy he behavor of hs hazar rae we apply resuls of Glaser 980 gve he form of lemma -. Lemma -: Le T be a couous raom varable wh wce ffereable esy f fuco f. Defe he quay η where f eoe he frs ervave f of he esy fuco wh respec o. suppose ha he frs ervave of η -ame η - exss. Glaser gave he followg resuls for more eals see Glaser If η < 0 for all >0 he he hazar rae s mooocally ecreasg falure rae DF. - If η > 0 for all >0 he he hazar rae s mooocally creasg falure rae IF. - If here exss 0 such ha > 0 η for all 0 < < 0 ; 0 0 η a η < 0 for all > 0. I ao o ha lm f 0 ; he he hazar rae s upse ow 0 bahub shape UBT. 4- If here exss 0 such ha η < 0 for all 0<< 0 ; 0 0 all > 0. Ag o ha η a η > 0 lm f. cosequeces ha he hazar rae s 0 bahub shape BT. For WLB srbuo we beg by compug he quay η ; by frs akg he ervave of he esy fuco gve - wh respec o whch s gve by: f f f - Dvg boh ses of he equao - by he measure f we oba: η akg s ervave wh respec o yels: η - Accorg o he values of he shape parameer : for 8

9 - For < s easly see ha he hr par of he lemma follows; where 0 s soluo of. I resuls ha he hazar rae s UBT η shape. - For η whch s srcly posve fuco for all values of. I resuls from he lemma 4- ha h s IF hs case also he WLBD reuces o gamma srbuo wh parameer k wh a creasg hazar rae. - For > η > 0 for all he he hazar rae s mooocally creasg IF; hs agrees wh he heorem gve Gupa a Krma 990 whch cae ha he legh-base verso preserves he IF propery of he orgal raom varable. These resuls ca be summarze hrough he followg heorem: Theorem -: le T be a o egave raom varable havg he Webull legh-base srbuo; he s hazar rae h s IF for values of he shape parameer ha are greaer or equal oe a UBT oherwse meas for 0 < <. The shapes of he hazar rae of he WLB srbuo for specal values of he shape parameer are llusrae fgure he scale parameer was ake o be he uy sce oes fluece he shape of he hazar ra 9

10 Fgure : The Hazar rae of he WLB srbuo for gve values of h ESTIMATION I hs seco esmaes of he wo parameers of he WLB srbuo a he relably fuco are obae by he meho of mome maxmum lkelhoo meho a Bayesa a approxmae Bayesa meho -usg Lley s expaso- assumg epee o-formave pror for each parameer. 4-- Mome esmaes Ths meho follows by equag he populao momes from -7 o he sample momes hs yels he followg sysem of wo equaos: * * * * * * * * Solvg hs sysem wll yel * * 4--a 4--b he mome s esmaes of a respecvely. These esmaes are geerally use as al values for he maxmum lkelhoo meho whe o close form exss for he MLE a he ormal equaos ees o be solve eravely. eplacg hese esmaes he formula of he relably -5 yels: 0

11 * * γ * * 4- * Ths esmae ca be calle he mome esmae of he relably fuco. 4-- Maxmum lkelhoo esmaes MLE: Suppose ha a sample was raw from - he he logarhm of he lkelhoo s gve by: l l l l l 4- Dffereag 4- wh respec o a urs a equag he ervaves o zero we ge he followg ormal equaos: a l l l 0 Ψ 4-4-b Where he Ps-fuco Ψ a s efe as he ervave of he logarhm of he gamma fuco wh respec o a. see he Habook of Mahemacal Fucos 970 page 59 a Ψ a l a a>0 4-5 a a The Ps-fuco s kow as gamma fuco whch ca be approxmae by: Ψ a ~ l a a a 0a 5a The sysem 4-4-a 4-4-b ca be reuce o oly oe equao by exracg from he frs equao: 4-7 A replace he seco oe.e. equao 4-4-b we oba:

12 0 l l l l l Ψ χ 4-8 Ths olear equao oes seem o have a close form soluo a mus be solve eravely o oba he esmae of he shape parameer whch wll be replace 4-7 o ge he MLE of he scale parameer. Or he sysem of he wo equaos ca be solve smulaeously. The asympoc expece varace-covarace marx of he esmaes ca be obae by verg he formao marx see he Appex wh elemes ha are egaves of he expece values of he seco ervaves of he lkelhoo fuco wh respec o he parameers a : l E I 4-9-a Ψ l l E I I 4-9-b Ψ l ξ l E I Ψ Ψ l 4-9-c Where s he ervave of he gamma fuco ca be approxmae by akg he ervave of 4-6 a Ψ. a z ξ s ema s zea fuco see Grashey a yzhk 965 pages:07-07 s efe by: 0 0 exp exp z z a a z a z ξ 4-0 By replacg z a a by her values or formula we ge: 0 0 exp exp ξ 4- The he varace-covarace marx of he esmaes of he parameers ca be obae by verg he formao marx as follows:

13 I I Var ov Var 4- I I ov Var A observe varace-covarace marx ca be obae also by replacg he MLE he formao marx a verg whou akg expecaos Specal cases: If oe of he wo parameers s kow we have he followg resuls: - If he shape parameer s kow he MLE of he scale parameer s gve by: 4--a Wh varace Var 4--b - If he scale parameer s kow he MLE of he shape parameer ca be obae by solvg equao 4-4-b afer replacg by a s varace s obae by verg 4-9-c a replacg by oo Maxmum lkelhoo esmae of he relably fuco The relably fuco ca be regare as a parameer a ees o be esmae. Usg he varace propery of he maxmum lkelhoo meho he MLE of he relably ca be obae by replacg a he maxmum lkelhoo esmaes of a he formula -5 s gve by γ 4-4 Usg Taylor expaso of orer oe abou he parameer esmaes of we ca wre:

14 By akg he expecao of he above formula a from he properes of he MLE resuls ha s asympocally ubase esmae of wh varace: V V V OV 4-5 Where he varaces a covaraces of he maxmum lkelhoo esmaes of he parameers were gve he marx Bayes esmaes: Suppose ha a lle formao s avalable abou he parameers a he he approprae pror for hs case assumg epeece s: π 4-6 : beg he sg of proporoaly. Usg Bayes heorem whch combes he lkelhoo fuco wh he pror gve 4-6 we oba he followg jo poseror: π / T exp l 4-7 Usg a square error loss fuco he Bayes esmaes of ay fuco of parameers s s poseror expecao gve by: E Ω u η / 4-8 u η π η / η Ω π η / η Where η s a parameer whch s hs case η or s equal o oe of he wo parameers f he oher oe s kow a Ω s he parameer space. Whe he aalycal meho s o aracable we refer o umercal egrao o oba he Bayes esmaes Bayes esmae of he scale parameer : Pug u η u 4-8 a usg he poseror 4-7 we ge he Bayes esmae of he scale parameer by a rao of wo egrals hs meas: ~ π / T 00 E / T 4-9-a 00 π / T Where s he ormalzg cosa; s gve by: 4

15 0 l l exp 4-9-b A 0 l l exp 4-9-c The varace of he Bayes esmae of scale parameer ca be obae by applyg he followg formula: / / ~ T E T E Var 5-9- The seco poseror mome ca be obae by seg η u u 4-8 a usg he poseror 4-7 hs yel: / / / T T T E π π 4-9-e 0 l l exp 4-9-f No close form soluos have bee fou for he egrals 4-9-b 4-9-c a 4-9-f whe boh parameers are ukow a umercal egrao s ecessary o evaluae hem. If we suppose ha he shape parameer s kow he egrals 4-9-a a 4-9-b wll am close form soluos he Bayes esmae of he scale parameer a s varace are ecal o hose of he maxmum lkelhoo meho gve by 4-- a 4--b The shape parameer : Seg η u u 4-8 a usg he poseror 4-7 we ge he Bayes esmae of he shape parameer by a rao of wo egrals hs meas: / / / ~ T T T E π π 4-0-a The eomaor was gve by equao 4-9-b gve a he umeraor s gve by: 5

16 0 4 l l exp 4-0-b A / / / T T T E π π 4-0-c 0 5 l l exp 4-0- The varace of he Bayes esmae of he shape parameer s he gve by: 4 5 / / ~ T E T E Var 4-0-e Whe he scale parameer s kow he egrals 4-9-b 4-0-b 4-0- o have a close form expresso a wll reuce o: 0 l exp 4--a 0 4 l exp 4--b 0 5 l exp 4--c The boh cases kow or ukow umercal egrao s ecessary o evaluae hese egrals The relably fuco: Pug u u η 4-8 a usg he poseror 4-7 we ge he Bayes esmae of he shape parameer by a rao of wo egrals hs meas: 6

17 ~ / π / T 0 0 E T Wh: 0 0 π / T γ 6 A s varace ca be obae from: exp l 4--a 4--b Var ~ E / T E / T 4--c Where: π / T 00 E / T 00 π / T γ exp l e No close form soluos exs for he egrals 4--a a 4--b eve f oe of he wo parameers s kow hese egrals ca be compue va umercal egrao Approxmae Bayes esmaes: Whe he egrals occurrg Bayesa aalyss o am close form soluo we refer o umercal egrao o f a soluo as was suggese he precee seco. Lley 980 gave a alerave meho o approxmae he egrals ha occur Bayesa sascs. The form of rao of egrals cosere by Lley 980 s as gve bellow: w η exp l η v η exp l η η η 4- Where: η η η L ηm s he parameer l η s he logarhm of he lkelhoo fuco a w. v. are arbrary fucos of η. Le v η π η he pror esy of he parameer η w η u η π η a ρ η l π η he rao 4-5 wll be he 7

18 poseror expecao of he fuco u η uer square error loss fuco a we wre: u η exp l η ρ η η E u η / 4-4 exp l η ρ η η Ths rao s equvale o he rao gve 4-8; The basc ea o evaluae s o expa o Taylor seres he fucos volve abou he maxmum lkelhoo η of η hs lea o he followg formula where he frs erm ome s O : E u η / ~ u uj u ρ j σ j ljk ul σ j σ 4-5 kl j j k Where each suffx eoes ffereao oce wh respec o he varable havg ha suffx; hs meas: l η u η u η ρ η l jk uj u ρ ec. σ j s he j eleme η η η η η η η j k j of he varace covarace-marx wh elemes ha are he verse of egaves of he seco ervaves of he log lkelhoo wh respec o he parameers. All he quaes 4-5 are o be evaluae a he MLE of a he summao ru over all suffxes from oe o m he mesoaly of. Lley 980 gave he oe parameer a woparameer verso of 4-5 as follows: For he oe- parameer case: 4 E u η/ ~ u η u uρ σ luσ 4-6-a Where: u η u η ρ l l u u ρ l σ l l η η η η η A for he wo-parameer case: E u η / ~ u σ u u ρ σ u u ρ σ u u ρ σ u u ρ u u σ σ l0 σ l uσ σ u σ σ σ l u σσ σ uσσ l0 u σσ uσ 4-6-b All he quaes 4-6-a a 4-6-b are o be evaluae a he MLE of he parameer η. All he eee resuls o ge he approxmae Bayes esmaes are gve he appex. 8

19 4-4-- The scale parameer Pug u η u 4-4 we ge he Bayes esmae of he scale parameer. Takg he ervaves of he fuco uη whch respec o each parameer ur yel: u u u u 0 u. eplacg hese ervaves wh he above resuls evaluae a he MLE of he parameers 4-6-b gves: ~ 4-7-a By he same way we pu u η u whch has ervaves u u u u u 0 o ge s poseror seco mome. E / 4-7-b If we suppose ha he shape parameer s kow he approxmae Bayes esmae of he scale parameer a s varace are gve by: ~ ~ 4-7-a ~ ~ Var Var Var 4-7-b Ths meas ha he MLE he Bayes a approxmae Bayes esmaes are ecal a have he same varace The shape parameer Seg u η u whch has ervaves u u 0 u u u 4-4 gves he Bayes esmae of he shape parameer. Usg he ervaves of he fuco uη a he resuls gve he appex. eplacg hem 4-6-b gves: ~ 4-8-a a we pu also u η orer o oba he varace he ervaves of he fuco "u" hs case are: u u u u u 0. Ths gves: E / 4-8-b If we suppose ha he scale parameer s kow we ge: 9

20 4 A D B A D 4-8-c A D B A D 4-8- Wh: l Ψ Ψ A 6l 6 Ψ Ψ B l 4 D l 4-8-e The relably fuco: To f he esmae of he relably fuco we se u u η 4-4 he ervaves of he relably fuco are gve he appex such ha u u u u u hs yel: u ~ 4-9-a To ge he seco poseror mome of he relably fuco we pu he frs a he seco ervaves of hs fuco are: u u u η u u u u a u we ge: / E 4-9-b - If we suppose ha he shape parameer s kow we ge: f 4-9-c f f 4-9- Where s he WLB esy evaluae a he MLE of he ukow parameers. f - If he scale parameer s assume o be kow we ge 0

21 A B A D D 4-9-e 4 A B A D D 4-9-f 4 5-NUMEIAL EXAMPLE To llusrae he above formulas a mehos he followg aa were ake from Gupa a Akma 998 hey represe mllo of revoluos o falure for ball beargs fague es: These aa have bee prevously fe assumg Webull logormal Iverse Gaussa a legh-base verse Gaussa. The Kolmogorov-Smrov es oes rejec ha hs aa come from a WLB srbuo. Some properes of he sample were compue such as he meat 7. 4 he varace VT.44 0 he Skeweess Sk.008 a kuross Kur 0.96 ; from he values of hese wo las coeffces he srbuo of hs aa s posve skewe rgh a lepokurc. The parameers of he sample were esmae umercally sce here was o close form for hem excep whe s kow he sysem 4--a a 4--b was solve umercally a yels he mome esmaes * *.56 a whch were use as al values for he ormal equaos 4-4-a a 4-4-b o oba he maxmum lkelhoo esmaes. Bayes a approxmae Bayes esmaes were also obae a he resuls are gve he followg able: Table MLE Bayes a approxmae Bayes esmaes of he parameers a her varaces. Parameers Esmaes MLE BE ABE

22 *.: Icaes he varace : MLE: maxmum lkelhoo esmaes :BE: Bayesa esmaes : ABE: Approxmae Bayes esmae. We observe ha he esmaes of he shape a he scale parameers are close a he Bayes oe have he smalles varace for he wo parameers. The relably fuco was evaluae for cera values of me by boh classcal a Bayesa mehos he resuls are gve able below: Table Mome MLE Bayes a approxmae Bayes esmaes of he relably fuco wh varaces Esmaes ~ * ~ Approxmae mome esmae MLE Bayes esmae Bayes esmae Tme * * From able we observe ha * ~ ~ a are sgushable whle preses a slgh fferece. The approxmae meho gves he smalles varaces for all values of "".

23 APPENDIX The followg resuls are useful o compue he approxmae Bayes esmaes A-- The ervaves of he log lkelhoo fuco: The log lkelhoo fuco of he WLB srbuo was gve equao 4- seco four. The seco ervaves of hs fuco evaluae a he MLE are gve by: l l 0 l l l l l0 l l Ψ Ψ A- Takg he egaves of hese quaes wll gve he observe formao marx whch wll be vere o f he observe varace-covarace marx wh elemes: l 0 l σ σ V A- l l 0 σ σ The hr ervaves of he log lkelhoo fuco evaluae a he MLE s are gve by: l l l l 0 l l l l 0 Ψ 6Ψ l 4 Ψ l 6 6l A- 4 A-- The ervaves of he logarhm of he pror fuco: The logarhm of he pror esy s gve by: ρ l π l l Dffereag hs fuco wh respec o each parameer ur we ge:

24 ρ ρ ρ ρ A-4 A-- The ervaves of he relably fuco: By ffereag he relably fuco gve -5 we ge wh respec o a ur we ge: A--- The frs ervaves: exp f l I f Ψ A-5 A---The seco ervaves: f Ψ l l f Ψ Ψ A-6 l l l Ψ Ψ I I f Where: x x e x x I 0 l x e x x I x 0 l 4

25 EFENES Abramowz M. a Segu I Habook of Mahemacal fucos wh formulas graphs a mahemacal ables. Dover Publcaos Ic.New york. Frak H. a Alhoe S Sascs coceps a applcaos. ambrge Uversy Press Grea Bra. Glaser.E Bahub a relae falure rae characerzao. J. Amer. Sascal Assoc Vol Gupa.. a Akma H.O O he relably sues of a weghe verse Gaussa moel. Joural of Sascal Plag a Iferece Gupa.. a Krma S. N. V. A The role of weghe srbuo sochasc moelg. ommu. Sas. Theory Meh Gupa.. a Trpah Effec of legh-base samplg o he moelg error. ommucao sascs Theory Meh Gupa.. a Trpah Weghe bvarae logarhmc seres srbuos. ommu. Sas. Theory Meh Grashey I.S. a yzhk I.M. 965: Table of egrals seres a proucs. Acaemc press Ic. New York a Loo. Ja K. SghH. a Baga I elaos for relably measures of weghe srbuos.. ommu. Sas. Theory Meh Kapur J.N. a Saxea H Mahemacal sascs 0h eo S. ha & ompay LTD New Delh Ia repr 00. Kharee haracerzao of Iverse-Gaussa a Gamma srbuos Through her legh-base srbuos. IEEE Trasacos o elably Lley D.V. 980.Approxmae Bayesa Mehos. Trabojos e Esasca Vol Olyee B.O. a George E.O.00. O sochasc Iequales a comparsos of relably measures for weghe srbuos. Mahemacal Problems Egeerg. Vol.8 -. Pall G.P. 00. Weghe srbuos. Ecyclopea of Evromecs Vol Joho Wley & Sos. 5

26 Pall G.P. a ao Weghe srbuos a sze-base samplg wh applcao o wllfe populaos a huma famles. Bomercs Pall G.P. a ao.. a aaparkh M.V O scree weghe srbuos a her use moel choce for observe aa.. ommu. Sas. Theory Meh Sacy E.W. 96: A geeralzao of he gamma srbuo A. Mah. Sa. Vol