Semi-Parametric Method to Estimate the Time-to- Failure Distribution and its Percentiles for Simple Linear Degradation Model

Transcription

1 Joural o Modr Applid Saisical Mods Volum 6 Issu Aricl Smi-Paramric Mod o Esima Tim-o- Failur isriuio ad is Prcils or Simpl Liar gradaio Modl Laila Nai Ba ak Yarmouk Uivrsiy, Irid, Jorda, la00_ma@yaoo.com Moammd Al-Ha Eram Yarmouk Uivrsiy, Irid, Jorda, m_assa@omail.com Omar Eidous Yarmouk Uivrsiy, Irid, Jorda, omarm@yu.du.o Follow is ad addiioal works a: p://digialcommos.way.du/masm Par o Applid Saisics Commos, Social ad Bavioral Scics Commos, ad Saisical Tory Commos Rcommdd Ciaio ak, L. N. B., Eram, M. A.-H., & Eidous, O. (07). Smi-Paramric Mod o Esima Tim-o-Failur isriuio ad is Prcils or Simpl Liar gradaio Modl. Joural o Modr Applid Saisical Mods, 6(), doi: 0.37/ masm/ Tis Rgular Aricl is roug o you or r ad op accss y Op Accss Jourals a igialcommos@waysa. I as accpd or iclusio i Joural o Modr Applid Saisical Mods y a auorizd dior o igialcommos@waysa.

2 Joural o Modr Applid Saisical Mods Novmr 07, Vol. 6, No., doi: 0.37/masm/ Copyrig 07 JMASM, Ic. ISSN Smi-Paramric Mod o Esima Tim-o-Failur isriuio ad is Prcils or Simpl Liar gradaio Modl Laila Nai Ba ak Yarmouk Uivrsiy Irid, Jorda Moammd Al-Ha Eram Yarmouk Uivrsiy Irid, Jorda Omar Eidous Yarmouk Uivrsiy Irid, Jorda Mos rliailiy sudis oaid rliailiy iormaio y usig dgradaio masurms ovr im, wic coais usul daa aou produc rliailiy. Paramric mods lik maximum likliood (ML) simaor ad ordiary las squar (OLS) simaor ar usd widly o sima im-o-ailur disriuio ad is prcils. I is aricl, w sima im-o-ailur disriuio ad is prcils y usig a smi-paramric simaor a assums paramric ucio o av a alormal disriuio or a xpoial disriuio. T prormac o smi-paramric simaor is compard via simulaio sudy wi ML ad OLS simaors y usig ma squar rror ad lg o 95% oosrap coidc irval as asis criria o compariso. A applicaio o ral daa is giv. I gral, i r ar assumpios o radom c paramr, ML simaor is s; orwis krl smiparamric simaor wi al-ormal disriuio is s. Kywords: gradaio modl, smi-paramric simaor, maximum likliood simaor, ordiary las squar simaor Iroducio Mkr ad Escoar (998) did rliailiy o a ui as proailiy a a ui will prorm is idd ucio uil a spciid poi o im udr courd us codiios. Tr ar may proposd applicaios o masur rliailiy o ay produc. O o s applicaios is simaio o imo-ailur disriuio ad is prcils. I simaio, radiioal li ss ar Ms. Laila Nai Ba ak is a gradua sud i parm o Saisics. r a: la00_ma@yaoo.com. Pro. Moammd Al-Ha Eram is a Prossor o Saisics. m a: m_assa@omail.com. Pro. Omar Eidous is a Prossor o Saisics. m a: omarm@yu.du.o. 3

3 AKHN ET AL o o mos ici way o oai rliailiy iormaio caus w ailur im daa ar osrvd y d o s; i is diicul o us radiioal rliailiy aalysis a rcords oly ailur im daa o aalyz li im daa. Tus, i is possil o g ailur daa y dgradaio masurms ovr im wic may coai usul daa aou produc rliailiy. gradaio is usually masurd as a ucio o im T. L () do acual squc or pa o dgradaio o a paricular ui ovr im or ac sampl ui a will osrvd, ad l do criical lvl or dgradaio pa wr ailur as occurrd. T ocus is o liar dgradaio modl or simaig 00r prcil o im-o-ailur disriuio. Gr s ak ad Kordoskiĭ (966/969) discussd dgradaio prolm rom a girig poi o viw. Ty prsd Brsi disriuio, wic dscris im-o-ailur disriuio or a simpl liar modl wi radom ircp ad radom slop. Amsr ad Hoopr (983) proposd a simpl dgradaio modl or sigl, mulipl, ad sp-srss li ss. Ty xplai ow o us is modl o sima cral dcy o im-o-ailur disriuio. Lu, Mkr, ad Escoar (996) compard dgradaio aalysis ad radiioal ailur im aalysis i rms o asympoic icicy. Ty dmosrad a dgradaio aalysis givs mor prcisio a radiioal ailur im aalysis i gral. Al-Ha Eram, Eidous, ad Kmail (009) proposd oparamric classical krl mod o sima im-o-ailur disriuio ad is prcils or simpl liar dgradaio modl. Ty compard prormac o is mod wi xisig paramric mods lik ML ad OLS. Ty gav im-o-ailur disriuio asd o classical krl mod (y assumig Gaussia krl), wic is i FT Ф i wr Φ is disriuio ucio o sadard ormal disriuio. Ty compu adwid usig ormula (Silvrma, 986) 5 i 0.9A () 33

4 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL wr A = mi{s, IQR/.34}, S is sampl sadard dviaio, ad IQR is sampl ir-quaril rag. T krl ucio K is ak o Gaussia ucio u K u, u () ad smooig paramr o oparamric simaor is compud usig ormula i (). Modl ad Tim-o-Failur isriuio Cosidr ollowig simpl liar dgradaio modl o sima im-oailur disriuio: y (3) i i i i wr y i is osrvd dgradaio masurm o i ui a im i, β i is radom c paramr ( slop o liar dgradaio modl or ui i), i is ailur im or dgradaio modl, ad ε i is radom rror rm, wr N 0,. ε i ar iid wi I gral, im-o-ailur disriuio ca wri as a ucio o dgradaio modl paramrs. T ailur im T is did as im w acual pa () crosss criical dgradaio lvl, i.. T is soluio o By cosidrig simpl liar dgradaio modl (3), T T disriuio ucio o im-o-ailur is 34

5 AKHN ET AL F T PT P P G, 0 (4) wr β is a radom c paramr ad G(.) is disriuio ucio o β. L 00r prcil o im-o-ailur disriuio dod y r. To id r, w d o solv wi rspc o r. Tis givs r FT r G r r G r (5) I is clar a, or a ixd valu o, disriuio o T ad 00r prcil dpd o disriuio o β, radom c paramr. I som simpl cass, a closd-orm xprssio or F() could oaid, u or mos pracical pa modls, i is cssary o valua F() usig umrical mods. For xampl, cosidr liar dgradaio modl (3) wi radom c disriud as N(μ, σ ). From quaio (4), ad, rom quaio (5), F Ф r T r r Ф r 35

6 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL wr, Φ(z) is sadard ormal cumulaiv disriuio ucio. As aor xampl, i β ~ xp(θ), ad T F r l r For aov wo xampls, paramrs μ, σ, ad θ ca simad usig ML or OLS mods, or v y ay good saisical mod. Esimaig Prcils o Tim-o-Failur isriuio Usig Smi Paramric siy Mod I is a simaor o θ, w ca cosruc ollowig smi-paramric simaor o (x): x; x Xi SP x K X i;, x i (6) Tis simaor is smi-paramric caus i comis a oparamric simaor, classical krl simaor, ad a paramric simaor, (x, θ). I is work, wo vrsios o SP x ar cosidrd ad sudid. T irs o assums (x, θ) ollows a al-ormal disriuio ad or assums (x, θ) ollows a xpoial disriuio. I dgradaio modl is a simpl liar as i (3), ad i β, β,, β is a radom sampl rom ukow pd (g β ()), w proposd i is scio smi-paramric simaor or im-o-ailur disriuio ad is prcils. T Hal-Normal isriuio T smi-paramric simaor o g β () a dpds o al-ormal disriuio is 36

7 AKHN ET AL 37 SPH v g K v wr v wi i i Takig krl ucio K(u) o a Gaussia ucio, SPH g

8 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL 38 T simaio o disriuio ucio o im-o-ailur is SPH F G g T SPH d d d P r o Q wr Q is a radom varial disriud as N, T

9 AKHN ET AL F T Φ (7) wr Φ(.) is a sadard ormal disriuio. To sima 00r prcils (dod y r SPH F T SPH r SPH r umrically or r SPH. ), w sould solv T Expoial isriuio T smi-paramric simaor o g β () asd o xpoial disriuio is wr v g K v SPE v wi i i By akig K(u) o a Gaussia ucio, w oai g SPE 39

10 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL 330 T simaio o disriuio ucio o im-o-ailur y usig SPE ĝ is SPE SPE F G g ro P T d d d S wr S is a radom varial disriud as

11 AKHN ET AL N, Tror F T Φ (8) wr Φ(.) is a sadard ormal disriuio. To sima 00r prcils (dod y F T SPH r SPH r umrically wi rspc o. r SPH r SPH ) w sould solv Esimaig Prcils o Tim-o-Failur isriuio Usig Maximum Likliood (ML) Esimaor Mod Cosidr simpl liar dgradaio modl y, i,, ;,, m i i i i wr y i is osrvd dgradaio masurm o i ui a im i, β i is radom c paramr ( slop o liar dgradaio modl or ui i), i is so ailur or dgradaio modl, ad ε i is radom rror rm, wr N 0,. By usig ormula o im-o-ailur disriuio ε i ar iid wi i (4), w will cosruc ML simaor o r or ollowig disriuios: T Hal-Normal isriuio I β i ~ al ormal(σ ), im-o-ailur disriuio is 0 FT d (9) 33

12 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL By Liiz igral rul, ad y diriaig o sids o (9) wi rspc o, w oai pd o im-o-ailur disriuio, wic is T ; (0) Now, o id ML simaor o σ, l,,, a radom sampl rom (0); aural logarim o likliood ucio o σ is T i l L ;,,, l ; i l i i i i l l l i i i i Now, y diriaig wi rspc o σ, w oai By solvig d d l L ;,,, 4 i i d d w oai ML simaor o σ, wic is l L ;,,, 0 () MLH i i T ML simaor o r (dod y MLH quaio wi rspc o MLH : ) is oaid y solvig ollowig 33

13 AKHN ET AL r MLH MLH d () 0 MLH T Expoial isriuio I β i ~ xp(α), im-o-ailur disriuio is T F (3) By diriaig o sids o (3) wi rspc o, w oai pd o imo-ailur disriuio, wic is T ; (4) To id ML simaor o α l,,, a radom sampl rom (4); aural logarim o likliood ucio o α is Tror By solvig l L ;,,, l T i; i l i i i i l l l i i i i d l L ;,,, d i i 333

14 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL d l L ;,,, 0 d w oai ML simaor o α, wic is giv y MLE (5) i i T ML simaor o r (dod y MLE ) is MLE l l MLE r r i i (6) Esimaig Prcils o Tim-o-Failur isriuio Usig OLS Esimaor Mod By cosidrig sam dgradaio modl a was sudid i prvious scio, ad y lig β, β,, β a radom sampl o siz rom proailiy dsiy ucio g β (; μ) ad disriuio ucio G β (; μ), OLS simaor o μ (dod y ) will oaid as ollows: OLS Q m yi E yi i m i y i i E i (7) wr E(β i ) is a ucio o μ. By miimizig (7) wi rspc o μ w g OLS simaor o μ. T OLS simaor or im-o-ailur disriuio is FOLS G rols (8) 334

15 AKHN ET AL wr is OLS simaor o r a is giv y solvig r OLS r G rols By usig ormula o im-o-ailur disriuio (8) w oai OLS simaor or ollowig disriuios: T Hal-Normal isriuio I β i ~ al ormal(σ ), OLS simaor o σ (dod y oaid y miimizig OLSH ) is Q m yi E yi i wr y E i i Tror Q m yi i i ad d Q m y i i i d i Equaig aov drivaiv o zro, w g 335

16 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL m i y i i i 0 Now, solvig las quaio wi rspc o σ, w oai m y i i OLSH i m i i (9) ad r OLSH is giv y solvig ollowig quaio wi rspc o r OLSH : r G rolsh (0) T Expoial isriuio I β i ~ xp(α), OLS simaor o α (dod y miimizig OLSE ) is oaid y Q m yi E yi i wr E(y i ) = α i. Tus Q m y i i i ad d Q d m i By quaig aov drivaiv o zro, w g y i i i 336

17 AKHN ET AL m i y i i i 0 T OLS simaor o α is oaid y solvig las quaio wi rspc o α. Tus, OLSE m y i i i m i i () T simaor o r is giv y r OLSE rolse l OLSE l r r m i i m i y i i () Simulaio Sudy ad Rsuls Cosidr prormac o our simaors o r. T adwid or ac simaor is compud y usig ormula (). T ias (B), ma squar rror (MSE), ad lg o 95% oosrap coidc irval usig oosrap prcil mod (Ero & Tisirai, 993; Raci & MacKio, 007) o ac simaor ar compud rom daa o siz a is simulad rom slcd disriuios, al ormal(σ ) or xp(α). L β, β, β a radom sampl o siz grad rom o o aov disriuios. To compu B ad MSE or ac o our simaors o r, id xac valu o r. Cas I β i ~ al ormal(σ ), y usig quaio (9), disriuio ucio o im-o-ailur is F T d 0 337

18 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL ad, asd o quaio (5), xac valu o r is r H r wr H - (.) is ivrs disriuio ucio o al-ormal disriuio. Cas I β i ~ xp(α), asd o quaio (3), F T ad, asd o quaio (5), xac valu o r is r l r I simulaio, iiial valus ar, Sampl siz = 0, 40, or 60, r = 0., 0., or 0.3, ad σ or α = 0. T criical lvl o dgradaio = 0 or ac sampl o siz. T umr o iraios o compu B ad MSE is N = 000. T umr o oosrap iraios ar M = 000. Simulaio rsuls ar prsd i Tals

19 AKHN ET AL Tal. B, MSE, ad lg o 95% oosrap coidc irval o sima r rom al ormal(0) wi = 0 SPH simaor SPE simaor ML simaor OLS simaor Lg o 95% oosrap CI r r B MSE B MSE B MSE B MSE SPH SPE ML OLS Tal. B, MSE ad lg o 95% oosrap coidc irval o sima r rom al ormal(0) wi = 40 SPH simaor SPE simaor ML simaor OLS simaor Lg o 95% oosrap CI r r B MSE B MSE B MSE B MSE SPH SPE ML OLS Tal 3. B, MSE ad lg o 95% oosrap coidc irval o sima r rom al ormal(0) wi = 60 SPH simaor SPE simaor ML simaor OLS simaor Lg o 95% oosrap CI r r B MSE B MSE B MSE B MSE SPH SPE ML OLS

20 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL Tal 4. B, MSE ad lg o 95% oosrap coidc irval o sima r rom xp(0) wi = 0 SPH simaor SPE simaor ML simaor OLS simaor Lg o 95% oosrap CI r r B MSE B MSE B MSE B MSE SPH SPE ML OLS Tal 5. B, MSE ad lg o 95% oosrap coidc irval o sima r rom xp(0) wi = 40 SPH simaor SPE simaor ML simaor OLS simaor Lg o 95% oosrap CI r r B MSE B MSE B MSE B MSE SPH SPE ML OLS Tal 6. B, MSE ad lg o 95% oosrap coidc irval o sima r rom xp(0) wi = 60 SPH simaor SPE simaor ML simaor OLS simaor Lg o 95% oosrap CI r r B MSE B MSE B MSE B MSE SPH SPE ML OLS

21 AKHN ET AL From Tals -6, ollowig coclusios may mad: T MSE or ac simaor dcras as icrass. T MSE or ac simaor icras as r icrass. T lg o 95% oosrap coidc irval is dcras as icrass. By comparig MSE o our simaors, ML ad OLS simaors av smalls valus o MSE ad smi paramric al ormal simaor clos o m. ML simaor as smalls valu o MSE ad sors lg o a 95% oosrap coidc irval or ac disriuio ad dir sampl siz, so ML simaor as s prormac. Esimaig 0.5 Usig aa wi Misspciid siy I is scio, w will sudy ad compar prormac o smi-paramric mod, OLS, ad ML simaors or r w disriuio o radom c is o cos corrcly. To prorm is compariso, gra radom c rom Wiull(, 5) ad assum is grad sampl is rom al ormal(5) or xp(5). T ru valu o 0.5 is W wr, W - (.) is ivrs disriuio ucio o Wiull(, 5) disriuio. Udr is misspciicaio, sima 0.5 usig our simaors. Tal 7. Esimaig 0.5 or a sampl rom a Wiull(, 5) disriuio a is misspciid as a al ormal(5) disriuio Esimaor Bias (B) Ma Squar Error (MSE) 0.5SPH SPE ML OLS

22 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL Tal 8. Esimaig 0.5 or a sampl rom a Wiull(, 5) disriuio a is misspciid as a xp(5) disriuio Esimaor Bias (B) Ma Squar Error (MSE) 0.5SPH SPE ML OLS I is simulaio, iiial valus ar Sampl siz = 60, σ or α = 5, ad r = 0.5. T criical lvl o dgradaio = 0. T umr o iraios o compu B ad MSE is N = 000. Simulaio rsuls ar prsd i Tals 7 ad 8. From Tals 7 ad 8 w coclud ollowig: T ML ad OLS simaors prorm poorly w radom c disriuio is misspciid. T smi-paramric al-ormal simaor as s prormac. Ral aa Applicaio T lasr dgradaio daa givs prc icras i lasr opraig curr or GaAs lasrs sd a 80 C wic is prsd i Tal c.7 o Mkr ad Escoar (998, p. 64). I is aricl, ailur is assumd o occurr a criical dgradaio lvl = 5. Figur sows prc icras i opraig curr or GaAs lasrs sd a 80 C. aa Aalysis Cosidr daa o sima prcils o im-o-ailur disriuio or simaors wic av discussd prviously (smi-paramric simaors (SPH & SPE), maximum likliood simaor (ML), ad ordiary las squar simaor (OLS)). Ts simaors will compard y compuig ma squar rror (MSE) ad lg o 95% oosrap coidc irval o prcils (r = 0.5) o im-o-ailur disriuio. 34

23 AKHN ET AL Figur. Prc icras i lasr opraig curr or GaAs lasrs sd a 80 C Tal 9. Failur im ad slop or ac ui i Ui i i β i To compu MSE, ru valu o r ad valus o β, β, β, slops o liar modl (3), mus kow. From lasr dgradaio daa, scal ims o dgradaio masurms y dividig ac im y 50. To g ailur im, us liar irpolaio ad, y iig simpl liar rgrssio 343

24 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL w i ad y i, oai simaio o β, β, β. Tal 9 coais imo-ailur i ad valus o slop sima i. Esimaig 50 Prcil o Tim-o-Failur isriuio Udr assumpio a radom c paramr is disriud as al ormal(σ ) or xp(α), w sima 50 prcil o im-o-ailur disriuio, 0.5, or smi-paramric simaors usig ormulas (7) ad (8), OLS simaor, ad ML simaor as ollows: To sima 0.5 usig smi-paramric simaors:. Tak a radom sampl, wi rplacm, o siz 5 rom slops i Tal 9.. pdig o is sampl, oai 0.5 SPH ad 0.5 SPE y solvig F T SPH 0.5 SPH 0.5 F T SPE 0.5 SPE 0.5, rspcivly. ad To sima 0.5 usig ML simaor:. Tak a radom sampl, wi rplacm, o siz 5 rom ailur ims i i Tal 9.. pdig o is sampl, oai 0.5 ML usig () or (6) accordig o assumd disriuio. Tal 0. T rsuls o ral daa udr assumpio βi ~ al ormal(σ ) Esimaor Bias MSE Lg o 95% oosrap CI Smi paramric al ormal Smi paramric xpoial ML OLS

25 AKHN ET AL Tal. T rsuls o ral daa udr assumpio βi ~ xp(σ ) Esimaor Bias MSE Lg o 95% oosrap CI Smi paramric al ormal Smi paramric xpoial ML OLS To sima 0.5 usig OLS simaor:. Tak a radom sampl, wi rplacm, o siz 5 o vcors rom daa, wr ac vcor cosiss o y i ad im or i =,,, 5, =,,, 6.. pdig o is sampl, oai 0.5 OLS usig (0) or () accordig o assumd disriuio. From Tals 0 ad, ollowig may cocludd: T simaio o mdia o im-o-ailur disriuio usig smi-paramric xpoial simaor as smalls MSE valu ad smalls 95% coidc irval lg. T prormac o smi-paramric al-ormal simaor ad smi-paramric xpoial simaor ar comparal. T ML ad OLS simaors prorm poorly compard o smiparamric simaors. Coclusios W disriuio o radom c is assumd o kow, ML ad OLS simaors o r prorm r a smi-paramric simaors i rms o MSE ad lg o 95% oosrap coidc irval. Orwis, smi-paramric simaors prorm s. Rrcs Al-Ha Eram, M., Eidous, O., & Kmail, G. (009). Esimaig prcils o im-o-ailur disriuio oaid rom a liar dgradaio 345

26 SEMI-PARAMETRIC ESTIMATION FOR LINEAR EGRAATION MOEL modl usig krl dsiy mod. Commuicaio i Saisics Simulaio ad Compuaio, 38(9), 8-8. doi: 0.080/ Amsr, S. J., & Hoopr, J. H. (983, Augus). Acclrad li ss wi masurd dgradaio daa ad grow modls. Papr prsd a Amrica Saisical Associaio Aual Mig, Toroo, Caada. Ero, B., & Tisirai, R. (993). A iroducio o oosrap. Nw York, NY: Capma ad Hall. Gr s ak, I. B., & Kordoskiĭ, K. B. (969). Modls o ailur (Scripa Tcica Ic., Tras.). Nw York, NY: Sprigr-Vrlag. (Origial work pulisd 966) Lu, C. J., Mkr, W. Q., & Escoar, L. A. (996). A compariso o disriuio ad ailur im aalysis mods or simaig a im-o-ailur disriuio. Saisica Siica, 6(3), Availal rom p:// Mkr, Q. W., & Escoar, L. A. (998). Saisical mod or rliailiy daa. Nw York, NY: Jo Wily ad Sos, Ic. Raci, J. S. & MacKio, J. G. (007). Irc via krl smooig o oosrap P valus. Compuaioal Saisics & aa Aalysis, 5(), doi: 0.06/.csda Silvrma, B. W. (986). siy simaio or saisics ad daa aalysis. Lodo, UK: Capma ad Hall. 346