ESTIMATION AND TESTING

Transcription

1 CHAPTER ESTIMATION AND TESTING. Iroduco Modfcao o he maxmum lkelhood (ML mehod of emao cera drbuo o overcome erave oluo of ML equao for he parameer were uggeed by may auhor (for example Tku (967; Mehrora ad Nada (974; Pero ad Rooze (977; Cohe ad Whe (980; Tku e al. (986; Balakrha (990; Tku ad Sureh (99. They have obaed modfed maxmum lkelhood (MML emae of he parameer by makg lear approxmao o cera expreo( ML equao of parameer of ormal, log-ormal, logc, expoeal ad Raylegh drbuo. The modfcao or approxmao uggeed o ome erm he log lkelhood equao by auhor meoed he above le are for emao he repecve dee from complee or ceored ample. Roaah e al. (993a ad 993b codered he MML emao gamma drbuo wh ad whou pror relao bewee parameer, Kaam ad Srvaa Rao (993 Raylegh drbuo from lef ceored ample, Kaam ad Dharma Rao (993 half logc drbuo from complee ample, Kaam ad Srvaa Rao (00 exeded MML emao o log-logc drbuo, Kaam ad Sr Ram (003 legh baed vero of expoeal model, Roaah e al. (005 expoeaed log-logc drbuo. I all he above work, he geeral procedure log lkelhood equao of cale parameer or he locao ad cale parameer of he 0

2 correpodg dey fuco are codered ad are mplfed o olve for he correpodg ML emae. Whe he rucural form of he lkelhood equao doe o yeld a cloed form emaor, a poro of he equao or ome poro of he equao are defed, whch, whe approxmaed by admble lear form would gve cloed form emaor. The admble learao meoed above vare from auhor o auhor. For ace, Tku (967 uggeed leary by dervg he lope ad ercep of he approxmaed lear expreo by verg he drbuo fuco a arrow eghbourhood of populao quale. Balakrha (990 o he oher had chooe o ge he lear expreo by coderg Taylor ere expao of pecfc erm he log-lkelhood equao aroud he correpodg adard populao quale. I h chaper, we adop he wo approache o emae he cale parameer vere Raylegh drbuo from complee ad Type-II ceored ample ad comparo Seco.. The Relably emao from complee ad Type-II ceored ample ad comparo are preeed Seco.3. The gee ad he developme of he model wh eceary pu abou a ohomogeeou Poo proce o model ofware relably growh are preeed Seco.4.. ML ad Modfed ML emao Mo of he lfe drbuo ued relably ad urvval ude are characerzed by a moooe falure rae. However, he probably dey fuco (pdf of oe-parameer vere Raylegh drbuo (IRD wh parameer

3 ( exp( / 3 f for > 0, 0 (.. = 0 for 0 The cumulave drbuo fuco (cdf of a vere Raylegh drbuo F( exp( / for > 0, 0 (.. = 0 for 0 Voda (97 uded ome propere of he MLE of he cale parameer of vere Raylegh drbuo. The relably fuco ad hazard fuco of vere Raylegh drbuo are repecvely gve by R( exp( / >0, 0 (..3 ad h( exp( 3 exp( / / 3 [exp( / ] (..4 Mukherjee ad Sara (984 howed ha for gve, he drbuo IFR or DFR accordg a < or > Mukherjee ad Ma (997 dcued he percele emaor of cale parameer vere Raylegh drbuo. Le... be a doubly Type-II ceored ample r r (r lef mo obervao ad rgh mo obervao are ceored from a ample of ze from a vere Raylegh drbuo. The lkelhood fuco of he ample o emae, from he gve doubly Type-II ceored ample gve by

4 L [F( r r ] f ( [ F( ] r Takg logarhm o boh de ad droppg he parameer free coa of proporoaly, we ge he remag log-lkelhood fuco a log L = r log [ F( ] log f ( log[ F( ] r r The ML equao o ge MLE of log L 0 r r ( r r exp(. [ / F(z ] 0 r r ( r r. f (z. [ F(z ] 0 (..5 where z ad f(., F(. are repecvely he pdf ad cdf of vere Raylegh drbuo. for I ca be ee ha equao (..5 cao be olved aalycally becaue of he rucural form of oluo of equao (..5 he MLE of f (z F(z. The erave ay ˆ from doubly Type-II 3

5 ceored ample. Therefore, we approxmae he fuco f (z F(z by ug he modfed ML mehod uggeed by Tku (967 ad Balakrha (990. Cae ( :- If we ake r=0, =0 equao (..5, we have he complee ample. I h cae, he MLE of ay ˆ ˆ (..6 (/ Cae (:- If we ake =0 equao (..5, we have he lef ceored ample. I h cae, he ML emae of ay ˆ L ( r ˆ L (..7 r r r Cae (:- If we ake r=0 equao (..5, we have he rgh ceored ample. I h cae, he ML equao of (. f (z. [ F(z ] 0 (..8 The equao (..8 cao be olved aalycally for ML emaor of. Hece he ay ˆ R a erave oluo of equao (..8. 4

6 Modfed ML Emao I ca be ee ha o ge he ML emae of rgh or doubly Type-II ceored ample, we have o olve he correpodg ML equao eravely. Hece o overcome h dffculy, we coder he followg modfed maxmum lkelhood mehod : Modfed ML mehod-i (Tku, 967 Le G(z [ f (z F(z ] where f(. ad F(. are gve (.. ad (... I h mehod, we approxmae G(z equao (..5 by a lear fuco G (z z Now he ML equao of gve equao (..5 become r r ( r r. ( z 0 r r r. ( r.. 5

7 [( r r r r.. ] ˆ Therefore he Tku modfed ML emaor (TMMLE of ay ˆ (..9 (, To ge he TMML emaor ˆ of,,..., whch are obaed a follow:, we requre he coeffce Le p,,,... ad followg equao, z *, z ** be he oluo of he * pq G(z p ad G(z ** p p q where q p. The, we have G(z * z * G(z ** z ** 6

8 ad * G(z *.z where f (z * G (z * ad [ F(z * ] G (z ** f (z ** [ F(z ** ] For he vere Raylegh drbuo, he expreo z * ad z ** are * z l( p p q ad * z * l( p p q The value of,,(,,... for =5(0, 5, 0 are compued ad preeed Table... Ug Table.., we ca emae ay ˆ for a gve complee ad ceored (rgh ad lef ample. 7

9 Modfed ML mehod-ii (Balakrha, 990 I h mehod, we coder he Taylor ere expao of G(z aroud he quale of he populao ay gve by he oluo of he equao G(, =,,. up o he fr dervave. NowG ( (..0 where G (, G ( ad G ( f ( [ F( ] Here f(. ad F(. are he pdf ad cdf of a vere Raylegh drbuo ad hece G ( 3. [ f ( F( ] 3 f (. [ F( ] Ug (..0 equao (..5, we ge 3 [( r r r r. ] 8

10 Therefore, Balakrha MMLE (BMMLE of ay ˆ 3 ug modfed ML mehod-ii ˆ 3 3 The ercep- ad lope- for =,,., are compued for = 5(0, 5, 0 ad are preeed Table... The exac varace of ˆ ad ˆ 3 are o racable. Smlarly, ce he exac MLE a erave oluo, varace cao be obaed aalycally. Therefore, he mall ample varace of hee hree emaor ˆ, ˆ, ˆ 3 ca be obaed oly hrough Moe-Carlo mulao. Regardg he aympoc varace, we ca eablh ha all he hree emaor are aympocally equally effce a decrbed below. I he wo mehod decrbed above he bac prcple ha he fuco G(z f (z /[( F(z ] approxmaed by a lear fuco ome eghborhood of he populao. I ca be ee ha he coruco of he eghborhood over whch he above fuco leared deped o he ze of he ample alo. The larger he ze, he cloer he approxmao. Tha he exace of he approxmao become fer ad fer for large value of. Hece, he approxmae log- lkelhood equao ad he exac log-lkelhood equao dffer by lle quae for large. Therefore, he oluo of exac ad approxmae log-lkelhood equao ed o each oher a 9. Hece he exac ad modfed MLE are aympocally decal (Tku e al., 986. However, he ame cao be ad mall ample. A he ame me he mall ample varace of exac ad modfed MLE are o mahemacally

11 racable. We, herefore compared hee emae mall ample hrough Moe-Carlo mulao. Comparo of MLE ad MMLE of cale parameer The amplg drbuo of ˆ, ˆ, ˆ 3 are o mahemacally racable. Hece we have reored o Moe-Carlo mulao udy o ge he emprcal amplg characerc of he MLE ad MMLE - TMMLE, BMMLE of. We have compued he mulaed ba, varace ad mea quare error (MSE of ˆ, ˆ, ˆ 3 for 3000 ample of ze 5(0, 5, 0 geeraed from adard vere Raylegh drbuo. Thee are preeed Table... The effcece of MMLE are compued by akg he rao of mulaed varace of MLE o ha of mulaed varace of MMLE ad are preeed Table... I all he cae (lef, rgh ad doubly Type II ceored ample, he MMLE Tku MMLE (TMMLE ad Balakrha MMLE (BMMLE of he cale parameer vere Raylegh drbuo are more baed ha he exac MLE. However, he ba of MMLE are decreag a he ample ze creae. Bewee he MMLE, he BMMLE ug he Balakrha approach beer ha he Tku MMLE wh repec o ba, varace ad MSE for lef, rgh ad doubly Type-II ceored ample. BMMLE performg wh abou 80% effcecy whe he ample ze =5 ad performg wh abou 86% effcecy whe =0 ad 90% effcecy whe =5 or 0. Hece, we ca ugge BMMLE a alerave emaor o MLE, o emae he cale parameer vere Raylegh drbuo from Type-II ceored ample. However, complee ample, he MLE of form, here o eed of ug modfed ML mehod. he cloed 0

12 .3 Relably emao The relably fuco of IRD gve by equao (..3 R( exp( / >0, >0 (.3. Here, we ued he hree emaor of amely MLE of ( ˆ, TMMLE of ( ˆ ad BMMLE of ( ˆ 3 baed o complee ad Type-II ceored ample he above equao (.3. o emae R( a prefxed value of. Therefore, he relably emaor baed o ˆ, ˆ, ˆ 3 are repecvely Rˆ ( exp( ˆ / Rˆ ( exp( ˆ / Rˆ ( exp( ˆ / 3 3 Sce ˆ he MLE of, by he varace propery of MLE, Rˆ ( alo MLE of R( a a coequece, aympoc varace (Avar gve by R Avar ( Rˆ ( A var( ˆ (.3. where R e / Avar ( ˆ E log L

13 The aympoc varace of MLE ad MMLE are ame (Bhaacharya, 985. Thu all he hree relably emaor - Rˆ (, Rˆ (, Rˆ 3 ( of R( have he ame aympoc varace whch gve by equao (.3.. However, he ame may o be rue mall ample. A he ame me he aalycal varace of hee relably emaor - Rˆ (, Rˆ (, Rˆ 3 ( cao be mahemacally racable. Hece, we compare he relave performace of hee hree emaor mall ample hrough Moe-Carlo mulao baed o he emprcal amplg characerc ba, varace ad mea quare error. The mulao udy a follow: We have geeraed 3000 radom ample of ze =5(0,5,0 from a adard vere Raylegh drbuo. Each ample he ordered ad from h complee ample a doubly (cludg lef ad rgh ceored ample codered by deleg r malle ordered obervao ad large obervao wh all poble choce of r ad. From hee reula ample, we compue MLE - ˆ, TMMLE - ˆ ad BMMLE ˆ 3 ad o oba he emae of he relably fuco - Rˆ (, (=,,3 by ubug ˆ (=,,3 R(. I order o ge he relable ug hee emaed fuco, we have o prefx he value of a whch he relably o be compued. Here, we coder he value a =.0, , , a whch he populao rue relable are R(=0.5, 0.75, 0.90, 0.95 repecvely. The reul are preeed Table.3. o.3.4. Comparo of he relably emaor The mulaed amplg characerc ba, varace ad mea quare error (MSE of he relably emaor are ame he cae of

14 complee ample, ce he MLE ad MMLE of Raylegh drbuo. are ame vere I emag he relably fuco, ue of MLE- Rˆ ( ad TMMLE- Rˆ ( perform beer ha BMMLE- Rˆ 3 (. Wh repec o varace ad mea quare error (MSE rrepecve of he ample ze ad umber of obervao ceored (doubly ceored o boh de, he order of he preferece of he relably emaor Rˆ (, Rˆ (, Rˆ 3 (. Therefore, we ugge TMMLE beer alerave o MLE (whch a erave oluo, o emae he relably fuco. I ereg o oe ha he order of preferece relably emao dffere from ha of paramerc emao..4 Sofware Relably Growh Model a a o-homogeeou Poo proce The erm ofware relably may be defed a he probably of falure free fucog of a ofware raher ha he faul coaed. However we cao rk ou he fac ha ofware relably deped o he umber of faul alo. I h regard, heory of probably ad hece acal aaly have become eeal he developme of a model ha ca be ued o evaluae he relably of real world ofware yem. Wh h backdrop, we udy he modelg of ofware relably a a o homogeou Poo proce wh mea value fuco baed o log logc drbuo. Smlar aemp baed o Pareo drbuo made by Kaam ad Subba rao (009. Suppoe ha we are ereed obervg he occurrece of a repeaable eve over a perod of me. The uao releva here ca be 3

15 he umber of me a developed ofware fal a gve perod of eg/operaoal me. A falure do o occur a predcable way uch a falure proce ca be defed wh a radom coug proce geerally, defed a a cou of umber of eve ha have occurred pecfed erval of me. Le be deoed by N(, where ay o egave real umber. N( dcae he umber of radom occurrece he erval [0,]. A coug proce ad o be a Poo proce f he falure have aoary depede creme ad he umber of falure ay erval of legh ha a Poo drbuo wh mea λ gve by λ y e (λ P {N ( +-N( y}, y 0,,... (.4. y! Th mahemacal model dcae ha he chage N( from oe me perod o aoher me perod ay [, +] deped oly o he legh of he erval bu o o he exreme, + of he erval. λ called he falure rae ad alo falure ey. I he above equao E(N(,. If we hk of a Poo proce whoe mea deped o he arg me ad alo he legh of he erval uch a Poo proce ca be explaed by a equao a y (m( e m( P {N y}, y 0,,... (.4. y! I h equao, m( a pove value, o-decreag, couou fuco of, geerally edg o a fe lm a a, m( called he mea value fuco ad dervave wh repec o called he 4

16 ey fuco deoed by λ(. I follow ha m( E(N( λ( d 0 Equao (.4. called a o homogeeou Poo proce. If a ofware yem whe pu o ue fal wh probably F( before me, f a ad for he ukow eveual umber of falure ha lkely o experece, he he average umber of falure expeced o be expereced before me af(. Hece af( ake a he mea value fuco of a NHPP. I he heory of probably, F( ca be defed wh he cumulave drbuo fuco (CDF of a couou o -egave valued radom varable. Thu a NHPP deged o udy he falure proce of a ofware ca be coruced a a Poo proce wh mea value fuco baed o he cumulave drbuo fuco of a couou pove valued radom varable. Sce may drbuo are avalable acal cece, oe ca hk of a umber of NHPP model each baed o a CDF. The fr ad foremo of uch model due o Goel ad Okumoo (979 whch baed o he well-kow expoeal drbuo. Laer may uch model have bee uggeed ad uded by varou reearcher ha ca be foud Wood (996, Pham (000 ad Huag e al. (007 ad referece here. I h he, we coder NHPP wh he mea value fuco gve erm of he CDF of IRD a m(=a exp( / (.4.3 I ca be ee ha m( ed o a a, m( a pove valued o decreag fuco of. The correpodg falure ey fuco 5

17 (exp( / λ(= (.4.4 Specfc relao for he mea value fuco ad ey fuco he cae of our pree model are λ(= m({ m(} { m(} (.4.5 λ(={ m(} { m(} (.4.6 (.4.7 λ(= { m (} The relably he ofware yem wh he above modelg he probably of o falure he me erval [0,] ad gve by R( P{N( 0} e m( (.4.8 I geeral, he relably R x - he probably ha here are o falure he erval [, +x] gve by [m( x m(] R( x = P {N ( + x - N( 0} e (.4.9 Geerally, he expreo gve Equao (.4.9 called ofware relably baed o a NHPP ad he NHPP alo called ofware relably growh model (SRGM. If he mea value fuco compleely pecfed wh parameer we ca have he value of he ofware relably a ay me of our choce. If he parameer of he mea value fuco are o kow hey eed o be emaed by a ofware falure daa he form of falure cou whch ca be ued o ge 6

18 a emae of he ofware relably order o ae he ofware qualy. ML Emao ( Ierval doma daa Suppoe ha ofware falure daa are gve he form of (y, =,,,, where y he umber of falure oberved he erval [0,], =,,, wh Such a daa called falure cou daa. The log -lkelhood fuco o ge he parameer of he NHPP hall be of he form LLF ( y y log[m( m( ] m( (.4.0 Thu he maxmum of he LLF deermed by he equao m( m( m( 0 (y y m( m( (.4. where θ o be replaced by he wo parameer a ad ucceo. [m(] 0 a, a y (exp( / (.4. [m(] 0 7

19 exp( exp( y exp(.. (y y exp( exp( (.4.3 Ug (.4. ad (.4.3 (.4.0 ucceo ad mulaeouly olvg he equao for a gve ample daa, we ge MLE of a ad However, hee wo equao adm oly erave oluo. ( Tme doma daa σ. Aumg ha he daa are gve for he occurrece me of he falure or he me of ucceve falure,.e., he realzao of radom varable S j for j=,,3,. Gve ha he daa provde ucceve me of oberved falure S j for 0..., We ca cover hee daa o he me bewee falure X where X = S - S - for =,,. Gve he recorded daa o he me of falure, he lkelhood fuco of he followg form: (.4.5 LLF log (S m(s The MLE of ukow parameer,,..., by olvg he equao ca be obaed (S m(s 0 (S (.4.6 Where ( m( 8

20 I vere Raylegh drbuo, he MLE of parameer a ad σ are ˆ ˆ aad σ repecvely. We kow ha m(=a F(= a exp( / Coder m( ( = σ 3 a (exp( / = a f ( The maxmum of LLF wh parameer a he oluo of he followg equao. (S a m(s 0 (S a (.4.7 a (exp( / S (.4.8 The maxmum of LLF wh parameer obaed by olvg (S m(s 0 (S (.4.9 σ S S = (.4.0 The MLE of a ad ca be obaed ug ay ample daa. 9

21 Table-..: Coeffce of modfed MLE vere Raylegh drbuo I

22 Table-.. (Coued I

23 Table-..: Emprcal Samplg characerc of MLE, MMLE of Scale parameer from complee ad ceored ample vere Raylegh drbuo Ba Varace MSE Effcecy r MLE TMMLE BMMLE MLE TMMLE BMMLE MLE TMMLE BMMLE TMMLE BMMLE

24 Table-.. (Coued Ba Varace MSE Effcecy r MLE TMMLE BMMLE MLE TMMLE BMMLE MLE TMMLE BMMLE TMMLE BMMLE

25 Table-.. (Coued Ba Varace MSE Effcecy r MLE TMMLE BMMLE MLE TMMLE BMMLE MLE TMMLE BMMLE TMMLE BMMLE

26 Table-.. (Coued Ba Varace MSE Effcecy r MLE TMMLE BMMLE MLE TMMLE BMMLE MLE TMMLE BMMLE TMMLE BMMLE

27 Table-.. (Coued Ba Varace MSE Effcecy r MLE TMMLE BMMLE MLE TMMLE BMMLE MLE TMMLE BMMLE TMMLE BMMLE

28 Table-.. (Coued Ba Varace MSE Effcecy r MLE TMMLE BMMLE MLE TMMLE BMMLE MLE TMMLE BMMLE TMMLE BMMLE

29 Table-.3.:Emprcal Samplg characerc of Relably emae ug MLE, MMLE from complee ad ceored ample vere Raylegh drbuo whe =.0 [R(=0.50]. Ba Varace MSE Effcecy r Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ 3 (

30 Table-.3. (Coued Ba Varace MSE Effcecy r Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ 3 (

31 Table-.3. (Coued Ba Varace MSE Effcecy r Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ 3 (

32 Table-.3. (Coued Ba Varace MSE Effcecy r Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ 3 (

33 Table-.3. (Coued Ba Varace MSE Effcecy r Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ ( Rˆ 3 ( Rˆ ( Rˆ 3 (