Modified Inverse Weibull Distribution

Transcription

1 J. Sa. Appl. Po. No. 5-5 NSP Moded Ivee Webull Dbuo Muhaad Shuab Kha ad Robe K School o Maheacal ad Phcal Scece he Uve o Newcale Callaha NSW 8 Auala Eal Adde huab.a@al.co obe.k@ewcale.edu.au Receved Apl 4 Reved Ma AccepedMa 4 Abac A eealzed veo o ou paaee oded vee webull dbuo D oduced h pape. h dbuo eealze he ollow dbuo Moded Ivee oeal dbuo Moded Ivee Raleh dbuo Ivee webull dbuo. We povde a copeheve decpo o he aheacal popee o he oded vee webull dbuo alo wh elabl behavou. We deve he oe oe eea uco ad eae he ode ac. We popoe he ehod o au lkelhood o ea he odel paaee ad oba he obeved oao a. KewodRelabl uco; oe eao; oe eea uco; lea quae eao; ode ac; au lkelhood eao. Ioduco he vee webull dbuo he le e pobabl dbuo whch ued he elabl eee dcple. he vee webull dbuo ca be ued o odel a vae o alue chaacec uch a a oal ueul le ad wea-ou peod. Relabl ad alue daa boh o le e ad evce ecod whch oe odeled b he le e dbuo uch a he vee oeal vee Raleh vee Webull dbuo. I h eeach we have developed a ew elabl odel called oded vee webull dbuo. h pape ocue o all he popee o h odel ad pee he aphcal aal o oded vee webull elabl odel. h pape pee he elaohp bewee hape paaee ad ohe popee uch a o- elabl uco elabl uco aaeou alue ae cuulave aaeou alue ae odel. h le e dbuo capable o odel o vaou hape o a ad alue cea.he popoed odel ca be ued a a aleave o vee eealzed oeal vee eealzed Raleh vee eealzed webull dbuo. he cuulave dbuo uco CD o he Ivee webull dbuo deoed b ad deed a Iw Joual o Sac Applcao & Pobabl --- A Ieaoal NSP Naual Scece Publh Co. Iw. he CD ve equao. becoe decal wh he CD o Ivee Raleh dbuo o ad o cocde wh he Ivee oeal dbuo. I he pobabl heo o ac he webull ad vee webull dbuo ae he al o couou pobabl dbuo whch have he capabl o develop a ohe le e dbuo uch a oeal eave oeal Raleh vee Raleh dbuo ad webull ale alo kow a pe I II ad III eee value dbuo. Recel Aa e al. [] popoed a Moded Webull dbuo. Soe wok ha alead bee doe o o Ivee Webull dbuo b M. Shuab Kha e al. [9-]. hee dbuo have eveal aacve popee o oe deal we ee o []-[8] [-7].I h pape we oduce ew ou paaee dbuo called Moded Ivee Webull dbuo

2 6 M. Shuab Kha ad Robe K Moded Ivee... D wh ou paaee ad. Hee we povde he acal popee o h elabl odel. he oe eao oe eea uco ad au lkelhood eae MLE S o he ukow paaee ae deved. he apoc codece eval o he paaee ae dcued. he u ad au ode ac odel ae deved. he jo de uco o Moded Ivee Webulldbuo D ae deved. he he Ioao a alo dcued. Moded Ivee Webull Dbuo he pobabl dbuo o D ha ou paaee ad. I ca be ued o epee he alue pobabl de uco PD ve b. MODIIED INVERSE WEIBULL PD MODIIED INVERSE WEIBULL CD ==.5= === ===.8 ==.5= === === === ==.5=.6.4 ===.5.. ==.5= Moded Ivee Webull PD Moded Ivee WebullCD Whee he hape paaee epee he dee pae o he Moded Ivee Webull pobabldbuo. Hee a cale paaee epee he chaacec le ad alo pove a locao paaee alo called a uaaee e alue-ee e o u le. he Moded Ivee Webulldbuo ad o be wo-paaee whe. he pd o he Moded Ivee Webulldbuo ve.. Sce he eco. o he value o ad ae alwa he ae o he D... how he dvee hape o he Moded Ivee WebullPD wh =.5.5 o ad he value o. I poa o oe ha all he ue baed o he aupo ha.he cuulave dbuo uco CD o he Moded Ivee Webull dbuo deoed b ad deed a. Whe he CD o he Moded Ivee Webulldbuo ha zeo value he epee o alue copoe b. I he Moded Ivee WebullCD called u le. Whe he e o ad epee he chaacec le... how he pecal cae o Moded Ivee WebullCD wh ad o he value o ad =.5

3 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo I clea o he.. ha all cuve eec a he po o.4979 he chaacec po o he Moded Ivee WebullCD. Relabl Aal he Moded Ivee Webulldbuo ca be a ueul chaacezao o le e daa aal. he elabl uco R o he Moded Ivee Webull dbuo deoed b R alo kow a he uvvo uco ad deed a R. Oe o he chaacec elabl aal he hazad ae uco deed b h he hazad uco H o he Moded Ivee Webull dbuo alo kow a aaeou alue ae deoed b h ad deed a / R h. I poa o oe ha he u o h he pobabl o alue pe u o e dace o ccle. heoe.he hazad ae uco o a Moded Ivee Webull dbuo ha he ollow popee I he alue ae ae a he MIRD I he alue ae ae a he MIED I he alue ae ae a he IWD. Poo. I he alue ae ae a he MIRD h MIR. I he alue ae ae a he MIED h MIE.4 I he alue ae ae a he IWD

4 H R h 8 M. Shuab Kha ad Robe K Moded Ivee... h IW.5 ue. lluae he elabl pae o a Moded Ivee Webull dbuo a he value o he hape paaee. MODIIED INVERSE WEIBULL R MODIIED INVERSE WEIBULL H.7.8 === === ==.5=.6 === === ==.5= ===.4. ==.5=. === ==.5= Moded Ivee Webull PD Moded Ivee WebullCD 5 MODIIED INVERSE WEIBULL CH 5 5 === === === ==.5= ==.5= Moded Ivee WebullCD I poa o oe ha R... how he Moded Ivee WebullR wh ad =.5.5. I clea ha all cuve eec a he po o.95 he chaacec po o he Moded Ivee WebullR. Whe =.5 he dbuo ha he decea HR. Whe = he HR eadl decea whch epee eal alue. Whe he H couall cea bewee.. 5 ad he decea aaeou alue ae bewee.6 whch epee wea-ou alue. he HR o he D a ve equao. becoe decal wh he HR o Moded Ivee Raleh dbuo o ad o cocde wh he Moded Ivee oeal dbuo. So he Moded Ivee Webulldbuo a ve leble dbuo... how he Moded Ivee Webull H wh ad =.5.5.

5 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo... 9 he Cuulave hazad uco CH o he Moded Ivee Webulldbuo deoed b ad deed a H H l.6 I poa o oe ha he u o H ae he cuulave pobabl o alue pe u o e dace o ccle... how he Moded Ivee WebullCH wh ad =.5.5. I poa o oe ha a ceae he pae o CH cl decea. 4Sacal popee h eco la he acal popee o he D. 4.Quale ad eda he quale q o he D he eal oluo o he ollow equao l q 4. q q he above equao ha o cloed o oluo q o we have dee cae b ubu he paaec value he above quale equao 4.. So he deved pecal cae ae. he q-hquale o he MIRD b ubu q 4 l q. he q-hquale o he IWD b ubu l q q. he q-hquale o he IRD b ubu q l q 4. he q-hquale o he IED b ubu q l q o b ubu q l q B pu q. 5 equao 4. we ca e he eda o D 4.Mode

6 M. Shuab Kha ad Robe K Moded Ivee... he ode o he D ca be obaed a a oluo o he ollow o-lea equao wh epec o 4. he above equao 4. ha o a uabuou oluo he eeal o. he eeal o ha he ollow pecal cae I we pu ad he we have IRD cae h cae equao 4. ake he ollow o 6 4 Solv h equao we e he ode a Mod I we pu he we have MIED cae h cae equao 4. ake he ollow o Solv h equao we e he ode a Mod I we pu he we have IWD cae h cae equao 4. ake he ollow o Solv h equao we e he ode a Mod Such ha kow ha RD ca be deved o IWD whe heeoe he RD becoe Mod 4. Moe he ollow heoe ve he h oe o D heoe 4.I ha he D he h oe o a ve a ollow o o o 4.

7 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo... he poo o h heoe povded Apped. Baed o he above eul ve heoe 4. he coece o vaao coece o kewe ad coece o kuo o D ca be obaed accod o he ollow elao CV 4.4 CS CK 4.6 he coece o vaao he qua ued o eaue he coec o le e daa. he coece o kewe he qua ued o eaue he kewe o le e daa aal. he coece o kuo he qua ued o eaue he kuo o peaked e o he o he le e dbuo. So he above odel ae helpul o acce hee chaacec. 4.4Moe Geea uco he ollow heoe ve he oe eea uco o D. heoe 4. I ha he D he oe eea uco o a M ve a ollow o M o 4.7 o he poo o heoe 4. povded Apped. Baed o he above eul ve heoe 4. he eaue o ceal edec eaue o dpeo coece o vaao coece o kewe ad coece o kuo o D ca be obaed accod o he above elao. 5 Lea quae eao Cae A Le... be a ado aple o Moded Ivee Webull dbuo wh cd ad uppoe ha... deoe he odeed aple. o aple o ze we have E = he lea quae eao LSE S ae obaed b z Q = - 5. I cae o D Equao 5. becoe 4

8 M. Shuab Kha ad Robe K Moded Ivee... - = Q 5. o ze Equao 5. wh epec o ad we deeae wh epec o hee paaee whch lead o he ollow equao l 5.5 Cae B Le... be a ado aple o D Moded Ivee Webull dbuo wh cd o aple o ze we have he lea quae eao LSE S ae obaed b z Q o D equao 5.6 becoe = Q 5.6 o ze equao 5.6 wh epec o ad we deeae wh epec o hee paaee whch lead o he ollow equao l l l 5.9 o he wo equao 5.7 ad 5.8 we e ˆ R 5. R ˆ 5.

9 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo... Subu 5.7 ad 5.8 o 5.9 we e a o-lea equao. B olv he obaed o-lea equao wh epec o we e R ˆ. A ee uch o-lea equao ha o cloed o oluo. So we have o ue a uecal echque uch a Newo Rapho ehod o olve. Cae C Le... be a ado aple o D wh cd ad uppoe ha... deoe he odeed aple. o aple o ze we have.4 -. = E... he ak eeo ad coelao ehod o D ae obaed b u he cd hee u le zeo ad l l l l 5. Le l l l a b l a l l l l l l l l l ˆ 5. b l l l l l l l l ˆ 5.4 he coelao coece o D b ak above aupo l l l l l l l l l l l l cc 5.5 he adad eo o eae o D b ak above aupo k S. l l l l l l l l 5.6 he coece o deeao o D b ak above aupo

10 4 M. Shuab Kha ad Robe K Moded Ivee... R. l l l l l l l l l l l l Ode Sac Le... be he ode ac he he pd o ve b C 6. he jo pd o ad ve b u u C u u 6. Whee C ad C 6.Dbuo o Mu ad Mau Le... be ve ado vaable. Hee we dee M... ad Ma.... We d he dbuo o he oded Ivee Webull dbuo o he u ad au obevao Y ad Y heoe 6.Le... ae depedel decall dbued ado vaable o oded Ivee Webull dbuo wh oupaaee hav pobabl de uco pd ad cuulave dbuo uco Poo o he u ad au ode ac o he ou paaee oded Ivee Webulldbuo D pd ve b Cae A Mu Ode Sac 6.. he u ode ac o he MIRD b ubu 6.4. he u ode ac o he MIED b ubu 6.5. he u ode ac o he IWD b ubu he u ode ac o he IRD b ubu

11 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo he u ode ac o he IED b ubu 6.8 Cae B Mau Ode Sac 6.9. he au ode ac o he MIRD b ubu 6.. he au ode ac o he MIED b ubu 6.. he au ode ac o he IWD b ubu he au ode ac o he IRD b ubu he au ode ac o he IED b ubu 6.4 heoe 6. he ou paaee D oded Ivee Webulldbuo o he eda o ve b Poo o he eda ode ac o he ou paaee oded Ivee Webulldbuo D pd ve b 6.5 We have dee cae b ubu he paaec value he above eda ode ac o equao 6.5. So he deved pecal cae ae

12 6 M. Shuab Kha ad Robe K Moded Ivee.... he eda ode ac o he MIRD b ubu 6.6. he eda ode ac o he MIED b ubu 6.7. he eda ode ac o he IWD b ubu he eda ode ac o he IRD b ubu he eda ode ac o he IED b ubu Jo Dbuo o he h ode Sac ad he h ode ac S he jo pd o ad S wh ad u ve b u u 6.B ak ad 6. he ad a jo de ca be we a 6. heoe 6. B u 6. he jo de uco o oded Ivee Webulldbuo D pd ve b Poo he jo pd o ad S wh ad u ve b

13 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo he u ad au ode ac o he jo de uco o he MIRD b ubu 6.4. he u ad au ode ac o he jo de uco o he MIED b ubu 6.5. he u ad au ode ac o he jo de uco o he IWD b ubu he u ad au ode ac o he jo de uco o he IRD b ubu 4 6.7

14 8 M. Shuab Kha ad Robe K Moded Ivee he u ad au ode ac o he jo de uco o he IED b ubu Mau Lkelhood Eao o he D Code he ado aple... co o obevao whe equao. o hee paaee o oded Ivee Webulldbuo D pd ake a pobabl de uco. he lkelhood uco o equao. ak deed a L ; B ak loah o equao 7. deea wh epec o ad equa o zeo we oba he ea equao ae L l ;... l 7. L l 7. L l l l 7.4 L l 7.5 L l L l l l l L l L l l

15 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo... 9 l L l L l l B olv equao ad 7.5 hee oluo wll eld he ML eao ˆ ˆ adˆ. 7. he Ioao a o hed Suppoe a ado vaable wh pobabl de uco. whee... k he oao a I he k k ec a wh elee. he lo lo I IJ E 7.6 j I he de. ha ecod devave lo / j o all ad j he he eeal eo lo I IJ E j o he hee paaee Moded Ivee Webulldbuo D pd all he ecod ode devave ae e. hu we have L... ; he vee dpeo a l L l L l L l L l L l L V V E l L l L l L B olv h vee dpeo a hee oluo wll eld he apoc vaace ad covaace o hee ML eao o ˆ ˆ adˆ. o he wo paaee Moded Ivee Raleh dbuo he pecal pe o Moded Ivee Webull dbuo whe ad o he wo paaee Moded Ivee oeal dbuo he pecal pe o Moded Ivee Webull dbuo whe all he ecod ode devave ae e. B u 7.7 appoael % codece eval o ca be deeed a 7.7 ˆ Vˆ Z / Z / ˆ Vˆ ad ˆ Z / Vˆ 7.8 Whee Z / he uppe h pecele o he adad oal dbuo.

16 M. Shuab Kha ad Robe K Moded Ivee... 8 Cocluo I h pape we oduce he ou paaee Moded Ivee Webulldbuo ad peeed heoecal popee. h dbuo ve leble elabl odel ha appoache o dee le e dbuo whe paaee ae chae. o he aaeou alue ae aal obeved ha ha cea ad decea alue ae pae o le e daa. Apped A he Poo o heoe 4. d B ubu o equao. o he above elao we have d A Cae A I h cae ad. he oe qua A Hee equao A ake he ollow o d d d A Cae B I he ecod cae we aue ha ad d B ubu w he we e 4 A4 Cae C I he hd cae we aue ha ad d

17 M. Shuab Kha ad Robe K Moded Ivee Webull Dbuo... B ubu w he we e 4 A5 he Poo o heoe 4. d e M B ubu o equao. o he above elao we have d M X A6 B ak aupo ha he u le zeo d M X Cae A I h cae ad. B u equao A equao A6 ake he ollow o d M X X M A7 Cae B I h cae ad. B u equao A equao A6 ake he ollow o d M X M X A8 Reeece [] Aa M. Saha ad MazeZad. Moded Webull dbuo Appled Scece 9-6. [] A. M. Abouaoh&Awa M. Alh. Relabl eao o eealzed veed oeal dbuo Joual o Sacal Copuao ad Sulao [] A. lah H. Elalloukh E. Med ad M. Mlaova heoeaed Iveed Webull Dbuo Appl. Mah. I. Sc. 6 No [4] Deveda Kua ad Abhhek Sh Recuece Relao o Sle ad Poduc Moe o Lowe Recod Value o Moded-Ivee Webull Dbuo Ge. Mah. Noe No. Mach 6-. [5] Euea Paaecu Paelo Geoe Popecu PopleaCoza Maaa Popa Baea ad o-baea Eao u ecod ac o he oded-vee Webull dbuo poceed o he Roaa acade ee A No. 4. [6] Gokaa R. Aal Ch P. oko. aued Webull Dbuo A Geealzao o he Webull Pobabl Dbuo. Euopea Joual o Pue ad Appled Maheac Vol. 4 No. 89-.

18 M. Shuab Kha ad Robe K Moded Ivee... [7] GovdaMudholka DeoSvaava ad Geoe Kolla. A eealzao o he webull dbuo wh applcao o he aal o uvval daa. Joual o he Aeca Sacal Aocao [8] Hoa Pha ad Ch-Dew La. O Rece Geealzao o he Webull Dbuo. IEEE aaco o Relabl [9] Kha M.S Paha G.R ad Paha A.H. heoecal aal o Ivee Webull dbuo. WSEAS aaco o Maheac [] Kha M.S Paha G.R. he plo o obevao o he Ivee Webull Dbuo o pobabl pape. Joual o Advace Reeach Pobabl ad Sac. Vol [] Kha S.K. he Bea Ivee Webull Dbuo Ieaoal aaco Maheacal Scece ad Copue Vol. -9. [] Kha M. Shuab Paha G.R ad Paha A.H. he Ioao Ma o he Ivee Webull Dbuo Ieaoal J. o Mah. Sc. &E. Appl. IJMSEA Vol. No. III [] Lu Ch-chao A Copao bewee he Webull ad Looal Model ued o Aalze Relabl Daa. PhD he Uve o Noha 997. [4] MazeZad Aa M. Saha Paaee Eao o he ModedWebull Dbuo Appled Maheacal Scece Vol [5] Maal M. Naa ad ah H. Ea. O he oeaedwebull Dbuo. Coucao Sac - heo ad Mehod [6] M. Shakl M. Ahaullah Revew o Ode Sac ad Recod Value o Vol. VIII No. -. Dbuo. Pak.j.a.ope.e. [7] M. Z. Raqab Ieece o eealzed oeal dbuo baed o ecod ac. J. Sa. Pla & Ieece. Vol