ON ESTIMATION OF STRESS STRENGTH MODEL FOR GENERALIZED INVERTED EXPONENTIAL DISTRIBUTION

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1 Joural of Rlablt ad Statstcal Studs; ISSN Prt: , Ol: Vol. 6, Issu 3: ON ESTIMATION OF STRESS STRENGTH MODEL FOR GENERALIZED INVERTED EXPONENTIAL DISTRIBUTION Mohad A. Hussa Dpartt of Mathatcal Statstcs, Isttut of Statstcal Studs ad Rsarch ISSR, Caro Uvrst, Egpt. Eal: Rcvd Aprl 6, 3 Abstract I ths papr, th stato of RPrY < Y, wh X ad Y ar two gralzd vrtd potal dstrbutos wth dffrt paratrs s cosdrd. Th au lklhood stator MLE of R ad ts asptotc dstrbuto ar obtad. Eact ad asptotc cofdc trvals of R ar costructd usg both act ad asptotc dstrbutos. Assug that th coo scal paratr s kow, MLE, Bas stators ad cofdc trvals of R ar vstgatd. Bas stators ar basd o foratv ad oforatv prors of th ukow paratrs. Mot Carlo sulatos ar prford to copar ad to valdat th dffrt proposd stators. K Words: Gralzd Epotal Dstrbuto, Sst Rlablt, Strss-Strgth, Bas, Mau Lklhood.. Itroducto Th stato of sst rlablt statstcal applcatos s vr coo ad has uch attto ltratur. Th ost wdl approach usd for rlablt stato s th wll-kow strss-strgth odl. Ths odl s usd a applcatos of phscs ad grg such as strgth falur ad sst collaps. I strss-strgth odlg, RPrY < X s a asur of copot rlablt wh t s subctd to rado strss Y ad has strgth X. I ths cott, R ca b cosdrd as a asur of sst prforac ad aturall ars lctrcal ad lctroc ssts. Othr trprtato ca b that, th rlablt, R, of th sst s th probablt that th sst s strog ough to ovrco th strss posd o t. It a b tod that R s of gratr trst tha ust rlablt sc t provds a gral asur of th dffrc btw two populatos ad has applcatos a ara. For apl, f X s th rspos for a cotrol group, ad Y rfrs to a tratt group, R s a asur of th ffct of th tratt. I addto, t a b tod that R quals th ara udr th rcvr opratg charactrstc ROC curv for dagostc tst or boarkrs wth cotuous outco Babr, 975. Th ROC curv s wdl usd, bocal, dcal ad halth srvc rsarch, to valuat th ablt of dagostc tsts or boarkrs to dstgush btw two groups of subcts, usuall o-dsasd ad dsasd subcts. For or dtals, o ca b advsd to Kotz t. al., 3. Ma authors hav studd th strss-strgth paratr R. Gogo ad Borah dals wth th strss vs. strgth probl corporatg ult-copot for ssts vz. stadb rdudac th cas of Epotal, Gaa ad Ldl

2 56 Joural of Rlablt ad Statstcal Studs, Dc. 3, Vol. 6 dstrbutos. Sgh t. al., hav dvlopd a r-odlg of strss-strgth sst rlablt whr th hav dfd th probablt that th sst s capabl to wthstad th au opratd strss at ts u strgth wh both strss ad strgth varabls ar Wbull dstrbutd. Barbro 3 studd statstcal frc for th rlablt of strss-strgth odls wh strss ad strgth ar dpdt Posso rado varabls, whras, Al t. al. hav vstgatd th stato of PrX < Y, wh X ad Y b to dffrt dstrbuto fals. Wog has costructd a asptotc cofdc trval for PrY < X whr X ad Y ar two dpdt gralzd Parto rado varabls wth a coo scal paratrs. Furthror, Rubo ad Stl, studd Basa stato of th strss-strgth odl th cas wh th argal dstrbutos of X ad Y ar dpdt dpdt rado varabls that b to classs of dstrbutos obtad b skwg scal turs of oral dstrbutos ad wh th varabl. I ths papr, stato of th sst rlablt, R, wh X ad Y ar dpdt but ot dtcall dstrbutd gralzd vrtd potal dstrbuto GIED varabls s cosdrd. Th GIED dstrbuto has th followg cuulatv dstrbuto fucto cdf ad probablt dst fucto pdf for X > : F [, >, wth f [, >, whr > s th scal paratr ad > s th shap paratr Abouaoh ad Alshgt, 9. Th rst of th papr s orgazd as follows. I scto, th sst rlablt s drvd ad scto 3, th au lklhood stato of R s dscussd. I scto 4, asptotc cofdc trval of R s obtad whl scto 5 s dvotd to th Basa stato of R. Nurcal solutos ad prforac studs of th stators ar vstgatd o scto 6. Fall th papr s coclud.. Sst Rlablt R Lt X ad Y b two dpdt GIED rado varabls wth paratrs, ad, rspctvl. Th rlablt of th sst s dfd as follows R P Y < X dd, Mau lklhood stato Assu that two dpdt rado sapls X, X,..., X ad Y, Y,..., Y ar obsrvd fro GIED,, ad GIED, rspctvl. Th lklhood fucto of, ad for th obsrvd sapls s L data;,, [

3 O Estato of Strss Strgth Modl for Gralzd 57 [ 4 Thrfor, th -lklhood fucto of, ad wll b L. 5 Th stators, ad of th paratrs, ad rspctvl ca b obtad as th soluto of th lklhood quatos Fro Equatos 7 ad 8, th stators of ad ar gv b [, 9 ad [ whr s th soluto of th olar quato [ Oc th stators of ad, ar drvd ad usg th varac proprt of th MLEs, th MLE of R bcos [ [ R. Assug that th scal paratr s kow, w hav [, 3 [, 4

4 58 Joural of Rlablt ad Statstcal Studs, Dc. 3, Vol. 6 ad R [ [. 5 Wh th scal paratr s kow ad qual to o, t ca b asl show that th rado varabl [ U s dstrbutd as potal rado varabl wth a. Slarl, [ V s dstrbutd as potal rado varabl wth a. Thrfor, [ [ ~ χ ad ~ χ Gupta ad Kudu,. Accordgl, R ca b wrtt as R F, whr F has a Fshr dstrbuto wth ad dgrs of frdo rspctvl ad thrfor, th pdf of R s gv b u Γ u f u R, 6 Γ Γ u u u whr < u <. Basd o ths forato a τ % cofdc trval of R ca b obtad as follows,, 7 F,,τ F,, τ R R whr F,, τ ad F,,τ ar th lowr ad uppr τ th prctl of a Fshr dstrbuto wth ad dgrs of frdo rspctvl 4. Asptotc dstrbuto ad cofdc trval of R Basd o th asptotc proprts ad th gral codtos of th MLEs, ad Lha, 999, th asptotc dstrbuto of th MLEs datl follows fro th Fshr forato atr of, ad. That s, wh, ad p, < p <, t follows that D,, N3, Σ3 8 whr 3 I I I Σ3 I Ω I I I3, 9 I3 I3 I33 ad th atr I Ω s th Fshr forato atr of th paratr vctor Ω,,, ad th th lt s gv b th scod partal drvatvs

5 O Estato of Strss Strgth Modl for Gralzd 59 I ll Ω ω ω,,,, 3. Fro th asptotc proprts of th MLEs of, ad, o ca asl gt, D R R N, ψ whr R R ψ E E, A -τ% approat cofdc trval of R ca b costructd basd o th asptotc rsults obtad. Ths asptotc cofdc trval s gv b R ±Z τ ψ, whr ψ s th asptotc stadard dvato of R. 5. Basa Estato of R I ths scto, th Bas stator of R dotd as R BS s obtad udr th assupto that th shap paratrs ad ar dpdt rado varabls wth pror dstrbutos Γ a, b ad Γ a, b wth pdf's rspctvl a b a b π ; Γ a >, 3 ad a b a b π ; Γ a >. 4 Basd o th abov assuptos ad fro Equato 4, th ot dst of th data, ad ca b obtad as L data,, L data;, π π. 5 Thrfor, th ot postror dst of ths data, ad gv th data ca b obtad as follows L data,, π π L, data, 6 L data,, π π dd Th postror pdf's of ad ar π ~ Gaa a, b T, 7 ad π ~ Gaa a, b T. 8 rspctvl, whr T [ ad [ T. Sc ad ar assud to b dpdt ad usg Equato 6-8, th postror pdf of R bcos a a r r π r, K, < r <, 9 a a [ r b T r b T whr Γ a a a a K b T b T. Γ a Γ a

6 6 Joural of Rlablt ad Statstcal Studs, Dc. 3, Vol. 6 Thrfor, th Basa stator of R udr squard rror loss fucto s gv b R BS E R, rπ r, dr. 3 Th Bas stat of R udr squard rror loss caot b coputd aaltcall. Altratvl, urcal soluto basd o MATHEMATICA progra s plod to valuat R BS for dffrt valus of th paratrs. 6. Sulato stud I ths scto, Mot Carlo sulato s prford to tst th bhavor of th proposd stators for dffrt sapl szs ad for dffrt paratr valus. Th prforacs of th MLEs ad th Bas stats ar copard trs of bass ad a squars rrors MSEs. Bas stats, ar coputd basd o two tp of prors, o-foratv prors, whr a a b b., Cogdo,, Kudu ad Gupta, 5. Iforatv prors, whr t s assud that thr ar so pror forato about th paratrs ad a a, b, b for apl,, > a a 3, b b. Th sulatos ar basd o rplcatos ad th rsults ar prstd Tabls. I Tabl, w obta both CI EX ad CI AS usg Equatos 7 ad rspctvl. Both cofdc trvals ar basd o MLEs of R. th rsults ar show tabl. All sulatos ar basd o th followg sapl szs;, 5, 5, ad 5 ad w assu.5,.5,.5, 4. 5, ad. 5, rspctvl. Tabl 3 shows th rsults of Bas stato of R. It ca b otd that v for sall sapl szs, th prforac of th Bas stator s bttr tha th MLE of R trs of bass ad MSEs. It s also obsrvd that wh, crass, th MSE ad bass dcras for both MLE ad Basa stators. I addto, t s otd that for fd sapl szs ad as th paratr crass MSE ad bass dcras for both MLE ad Basa stato thods. Th cofdc trvals CI, prfors qut wll as th sapl szs AS crass, whl CI EX hav largr trval lgth coparg to CI AS. It s also obsrvd that for th MLE of R thr s ovrstato for paratr valus of < R <.5, whl for valus.5 R, thr s udrstato of th tru valu of R.

7 O Estato of Strss Strgth Modl for Gralzd 6 Tabl : MLE stato of R wh s fd ad qual to o ad scal paratr s kow., R R ML Bas MSE , , , , , , Tabl : Eact ad asptotc cofdc trvals of R basd o MLEs ad at sgfcac lvl.5 ad scal paratr s kow, R CI EX CI AS..3,.36.37, , ,.456,.5.355, , , , , , , , , ,.439, , , ,.83.63, , , ,.3.456, , ,.396 5, , , , , , ,.789

8 6 Joural of Rlablt ad Statstcal Studs, Dc. 3, Vol. 6 Tabl cot.: Eact ad asptotc cofdc trvals of R basd o MLEs ad at sgfcac lvl.5 ad scal paratr s kow, R CIEX CIAS..398, , ,.45.97, , , , , , ,.8.733, , , ,.438.3, ,5.5.44, , , , , , , , ,.4.334, , , , , , , ,.7574 Tabl 3: Basa stato of R wh th populato valu of s o ad th scal paratr s kow No-foratv prors Iforatv prors, R a a b b. a a 3, b b Bas MSE Bas MSE , , , , , ,

9 O Estato of Strss Strgth Modl for Gralzd Cocluso I ths papr, th probl of statg PrY <X for th gralzd vrtd potal dstrbuto has b addrssd. Th asptotc dstrbuto of th au lklhood stator has b usd to costruct cofdc trvals whch fucto wll v for sall sapl szs. It has b obsrvd that th Bas stators bhav qut slarl to th MLEs. Morovr, th MSE of th stats of R dcrass as th paratr crass for fd,. Furthr, wh,, crass th MSEs of all th stators dcrass rapdl. Th prforac of th Bas stators s also qut wll ad th MSEs of th Bas stators ar sallr tha th MSEs of MLEs. Fall, th avrag lgths of all trvals dcras as, crass. Rfrcs. Abouaoh, A.M. ad Alshgt A.M. 9. Rlablt stato of gralzd vrtd potal dstrbuto, Joural of Statstcal Coputato ad Sulato, 79, p Al, M., Pal, M., ad Woo, J.. Estato of PrY < X wh X ad Y b to dffrt dstrbuto fals. Joural of Probablt ad Statstcal Scc, 8, p Babr, D Th ara abov th ordal doac graph ad th ara blow th rcvr opratg graph, Joural of Mathatcal Pscho,, p Barbro A. 3. Ifrc o Rlablt of Strss-Strgth Modls for Posso Data, Joural of Qualt ad Rlablt Egrg, Artcl ID 5353, 8 pags. 5. Cogdo, P.. Basa Statstcal Modlg. Wl, Nw York. 6. Gogo J. ad Borah M.. Estato of Rlablt for Multcopot Ssts Usg Epotal, Gaa ad Ldl Strss-Strgth Dstrbutos, Joural of Rlablt ad Statstcal Studs, 5, p Gupta R. D. ad Kudu D.. Gralzd potal dstrbuto: Statstcal Ifrc, Joural of Statstcal Thor ad Applcatos,, p Kotz, S., Lulsk, Y. ad Psk, M. 3. Th Strss-Strgth Modl ad ts Gralzatos: Thor ad Applcatos. Nw York: World Sctfc. 9. Kudu, D. ad Gupta, R. D. 5. Estato of PY <X for gralzd potal dstrbuto. Mtrka, 6, p Rubo. F. J. ad Stl, M.F.J.. Basa frc for PX < Y usg astrc dpdt dstrbutos, oural of Basa Aalss, 73, p Sgh B., Rath S. ad Sgh G.. Ifrtal aalss of th r-odld Strss-strgth sst rlablt wth Applcato to th ral data, Joural of Rlablt ad Statstcal Studs, 4, p Wog A.. Itrval stato of PY<X for gralzd Parto dstrbuto, Joural of Statstcal Plag ad Ifrc, 4, p