Optimal Predictive Inferences for Future Order Statistics via a Specific Loss Function

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Randell Sims
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1 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 Optal Pt I utu O Statt a a Sp L ut Nla Nal Mb IAENG Ma Pugal Ktat Nal a Vla Stl Abtat T pup t pap t lp tqu apppat tut qutt pt tal pt bu tatt utu apl l t utal t ulg tbut p but all t paat a up Pt tal tatt a l u lablt pbl a t lat pbl T tat t tal a b tl tgat But t ptal ppt t tal t al aa a g pbablt tt aal pbablt tt a g aa a t b ull pl I t pap t u t pbl a p l ut tu t pa pt tal I patula t-apl pt ba t al b ata a ut pt ptg t utu bat a apl t a aalabl t al b ata l tat apl apl pt ba a pu apl ptg t utu bat a apl t a aalabl t ata l a pu apl a t-apl pt ba bt t al-alu ata tat apl a t ata a pu apl ptg t utu alu t a ut a apl t a aalabl bt t al-alu ata tat apl a t ata a pu apl W tt attt t al tbut aat u lat a/ al ag T tqu u ptzat pt tal ba ata paz ptal quatt lat btag alla tatt It all t l t ptzat pbl a pl a Illutat apl a g I T O tatt pt tal l ut ptzat P I INTRODUCTION REDICTION pap t t l uta att t pal t gg a t blgal I t t a t al pt gall g u t a Maupt Otb 7 T a uppt pat b Gat N 6936 Gat N 736 Gat N 94 a Gat N 9544 t Lata Cul S a t Natal Ittut Matat a Iat Lata NA Nal t t Statt Dpatt EV Ra Ittut Ut Lata Rga LV-5 Lata -al: al@ul M Pugal t t Ut Lata Rga LV-5 Lata al: apu@latl KN Nal t t Appl Matat Dpatt Tapt a Tluat Itutut Rga LV-9 Lata -al: ta@tl V Stl t t Balt Itatal Aa Rga LV- 9 Lata -al: t@aplll atg a t atuaal a t aua u t labl l-lgt at Autat p tl ltg a qualt tl a t gg tqu tat u pt a a ba t u pau Stattal tqu pla a l pt t g t aal a a la l al a tat pa l bg t pat tl pug at T pta tattal t atg a u b Pa la tat pt t uatal pbl patal tatt Patl p a t u ltatu t tp I t aa lablt a l-ttg lt ata a t l a t Eptal a t Wbull t a pt abut utu bat Pt tal a tut t a a aabl g pbablt tag a p ub u utu bat Ital pt a ptat pat t atg p a at ag t lt aua pt tat A tal at uuall t a upp a a l lt bt t utu alu pt t l t a pb pbablt T lt a t all pt lt pt bu T lt a b lpul tablg aat pl tg ata ul t a aabl u pt lt a lat tal Ha a M 3 Ma aut a pt t t tutg pt lt t Wbull a t lat t alu tbut Patl Ma a Sau 4 pp pt lt t Wbull a u l t t tatt al Ma 5 Atl a Raa 6 u ulat t pu a tabl at t u t ML tat t bta pt lt Lal 7 pp pt lt ba a tal appa; lt qu bt tat t ML tat a ual tgat Eglat a Ba 8-9 a tg M a Ma a pp au appat pt lt t Wbull M a Kua p a ulat ba pu tutg pt tal Wbull ppulat Tp II a T pu ba au ll tat a qu a tat p t t t ptl pt Bau a Gbb a Kat t al 3 pp appat t tutg upp pt lt a gaa tbut C t llg apl patal pbl Aa l publat: 7 bua

2 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 t qu t putat pt bu a pt tal utu alu a quatt: a u puag a gat ul l t a a l bu t alu t t ut t b pua t l tt tbut t ppulat ut pua b t u; aal aag auatug pa upp pt bu utu aat t; plag l tt g a t pt t ub alu tat ll u b t t tt t pt t aut t tat t ll b ta a p ub ut t al S applat qu a t- pt tal tat ll t a p g g ta t utu a aabl tt It ptat t t tat t tt t pap a pt tal t t b a a tal T a tat a utu babl alu; t latt a tat but u a t ubabl paat Pt tal a pu a qutt Baa t a tal a l b tut a a qutt agut T u t pap l au pt tal pu b a qutt appa; tu a t qutt pt tal I a applat tt u t a upp pt bu a l pt bu g t au aat t ptat ta t u a t t t al alu a put ppulat ptat ta t lat Cptuall t uul t tgu bt tapl pt -apl pt a t-apl pt t-apl pt t pbl t pt utu t a apl p ba al ata tat apl p I apl ut a ll utl t a t a b alu < < < ul b tt ptg t t t t alu ; t utl l atal alu l ; ub atal alu a utu tal t t -apl pt ata a pu apl a u t a pt a utu ut apl ut t a p ppulat apl ba pu pbl l tt ata ul b tt ptg t t t alu a ut t utl alu a utu apl ut ub alu b t t a utu apl ut -t-apl pt t pbl t pt utu t a apl p ba t al ata tat apl p a ll a a pu ata apl t a p ppulat apl ut a ll utl t a t a t b alu a ut pt a ll a t b alu X X a pu ata apl pbl ul b tt ptg t t t t alu ; t utl l atal alu l ; ub atal alu a utu tal t t I gal t pt a utu alzat a a quatt t llg: A tattal l t b t ppulat p tt T l uuall t a tbut pg a t paat θ I t pap attt tt t al tbut a aat u lat a/ al ag I patula t a a b a pul aalabl plt tp II apl a tuu tbut t -µ/ but bt t lat µ a al paat a u u al tbut t pbl a aat u a gup taat a ubgup t ull a gup ta µ t lat paat a t al t µ b a ptl b l t ag µ > T gup at tatl t paat pa Iat t alu pt t paat t θ It au tat l t utal t tbut p but all t paat a up I u a alla tatt a ptal quatt tbut t p t u paat a u T tqu u tutg pt tal bu paz ptal quatt lat btag alla tatt It pt a pl pu tat a b utlz b -tatta a p al putabl plt p bt pt bu a pt tal T tqu a pal a t t aat bg apl tatt t a pa g Nal t al 4-3 applabl t tattal pbl aat u a gup taat at tatl t paat pa II WITHIN-SAMPLE PREDICTION PROBLEM t-apl pt t pbl t pt utu t a apl p ba al ata tat apl p apl ut a ll utl a t a b alu ul b tt ptg t t t t alu ; t utl l atal alu l ; ub atal alu a utu tal A Lat-Sal al Dt ut C a tuat b b a lat-al al t ut b t t paat θµ µ a > a ptl paat lat a al t al aat u t gup G pt la taat: ab t a> all au tat t btaabl at pt t t tatt < < < a a apl z t uulat tbut ut µ µ µ < < < µ < > Aa l publat: 7 bua

3 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 I a utu bat t tatt t a apl z t W / S W / a alla tatt t tbut t p µ; S a ut tatt a au ll tat ba B P-La L ut W all t tal pt pbl t t tatt < t a apl z t tuat t t bat < < < < a b b Supp tat at tat a tal ta I a uuall t a t pup t tal tatt t uul at u palt l ab all bl Supp tat t palt a a l pptal t t aut b ap t tal S a a b t t pblt tal l aat t t tal tg a utg t tu µ all I at t t l t ll b a t attag t t lgt tal u apl t ll b ult a p t g pla t tal Supp tat t t aat t t tal pptal t t lgt a I t pat t l ut lal a ua paat a altat t t ba pbl au b ultplg all l at b / Tu a l t tgat t p-la l ut < θ > T pbl p b t at pt tbut ut a t l ut aat u t gup G taat Tu t pbl t t bt aat tal pt ag R θ 3 D D a t aat tal pt RθE θ {θ} a ut C Taat t L ut It ll tat t aat l ut θ a b ta a ll: V V - V && V V V V V θ & V 4 V V < V V V V > V 5 VV V V / V S / ; / S / S D R ut It ll 5 tat t aat t a θ a b p a { θ } E{ & } R θ Eθ V 6 7 tat bt a aat pt ul u b t t pbablt t t t bat < < < a µ µ µ µ µ µ 8 E R Mzat a Iaat Pt Rul T llg t g t tal ult t t T Optal pt ba Supp tat u u a a t ag t ut u u u u u u uu u u > 9 b a lt Q b t pbablt tbut ut u /u I / / < t t ptal aat la-l tal pt ba * S S Q / Q / I / / t t ptal aat la-l tal pt ba gat t a pt pt S Aa l publat: 7 bua

4 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 a P 7 Q / {& V } E & Q {& V } E & Q 3 Q q 4 q 5 W V / V 6 N E{ & & V}/ E{ & & V}/ a l l Tu E{ & & V} p a a lut t < a t / >/ It al tat t g t u alu E{ & & V} Tu tabl I / / t t u E{ & & } t g u bg t b ttg a t u t E{ & & V }/ 7 Q Q 8 tabl Clla Mu t ptal aat pt ba T u g b { θ } E{ & } R θ Eθ V 9 a t a g b a a t a g b P T ult a at 7 u a E{ & & V}/ E{ & & V}/ a a E{ & & V }/ a T ulg a / / at a a paat tal a pt pt tat / / tal pt abl t tg pt pt t ul all T Optal aat pt ba Supp tat µ a VV V V / V / ; / / Lt u au tat u u a a t ag t ut u u u u u u uu u u > b a lt Q b t pbablt tbut ut u /u I / / < t t ptal aat la-l tal pt ba * Q / Q / I / / t t ptal aat la-l tal pt ba gat t a pt pt Q / 3 P t p t T Clla Mu t ptal aat pt ba T u g b { θ } E{ & } R θ Eθ V 4 a t a g b a a t a g b 3 P t p t Clla Aa l publat: 7 bua

5 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 III EQUIVALENT CONIDENCE COEICIENT a bta a tal pt a ga t tal a a tal t tal a aluat t t T gal ult ta t llg t T 3 Equalt t ba Supp tat VV V a a t ag t ut > b 8 a lt H b t tbut ut WV /V t pbablt t ut W g b 5 T t t aat t t ptu pt tal * S S P { : < < µ } H Q / H Q / 6 P T t pg t µ g b P{ S : S < < S µ P{ : < / < } H H H Q / H Q / } 7 T pt µ T 4 Equalt t ba Supp tat VV V a a t ag t ut al > b µ µ µ µ µ µ µ 8 a lt H b t tbut ut WV /V t pbablt t ut W g b 9 T t t aat t t ptu pt tal * P { : < X < µ } H Q / H Q / 3 P t p t T 3 T a 6 3 a t a a t at tat a a t at pptalt aat t l tg a utg lat t l l ag t lgt tal p a ttg tptat tal pt IV NEW-SAMPLE PREDICTION PROBLEM -apl pt ata a pat apl a u t a pt a utu ut apl ut t a p ppulat apl ba pu pbl l tt ata ul b tt ptg t t t alu a t t utl l alu a utu apl ut ub alu b t t a utu apl ut A Lat-Sal al Dt ut C a tuat b b a lat-al al t ut b t t paat θµ µ a > a ptl paat lat a al t al aat u t gup pt la taat: ab t a> all au tat t btaabl at pt t t tatt X <X < <X a pu a apl z a ut tatt M S a au ll tat µ µ ba XX X t t ut p µ p µ / / < < < < < µ > > 3 W a tu aug tat t al t ut a u aa l u t gup G taat: a b a µ aµ b a a> T al t ut atg t ab t u t lt al gat ptal Wbull a gaa t t ut T tutu t pbl lal t t gal a Lt b a pt utu bat t tatt a apl I aatl ptabl t W M /S W µ / a aal aat ptal tal X B P-La L ut W all t tal pt pbl t t tatt a utu apl z t tuat t t bat X < X < < X < a pat apl z a b b Aa l publat: 7 bua

6 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 Supp tat al t t p-la l ut T pbl p b t at pt t ut 3 a t l ut aat u t gup G taat Tu t pbl t t ptal tal pt ag R θ 3 D D a t aat tal pt R θ E θ {θ} a ut C Taat t L ut It ll tat t aat l ut θ a b ta a ll: V V - V && V V V V V θ & V 33 V V V V M / V V V V V S < V V > V / ; 34 M / S M / S D R ut 35 It ll 34 tat t aat t a θ a b p a { θ } E{ & } R θ Eθ V 36 tat bt a aat pt ul u b t t pbablt t t t bat X < X < < X t pat a apl z a t t tatt t utu apl z µ µ µ µ µ µ 37 E R Mzat a Iaat Pt Rul T llg t g t tal ult t t T 5 Optal aat pt ba X Supp tat u u a a t ag t ut u u u u u u uu u al u > b a lt 38 Q b t pbablt tbut ut u /u I / / < t t ptal aat la-l tal pt ba X * M S M S Q / Q / 39 I / / t t ptal aat la-l tal pt ba X gat t a pt pt M S Q / 4 P t p t T Clla 5 Mu t ptal aat pt ba X T u g b { θ } E{ & } R θ Eθ V 4 a t a g b 39 a a t a g b 4 P t p t Clla T 6 Equalt t ba X Supp tat V V V a a t ag t ut al b 38 a lt / > H b t tbut ut W V V t pbablt t ut W g b Aa l publat: 7 bua

7 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 4 T t t aat t t ptu pt tal * M S M S P { : < < µ } H Q / H Q / P t p t T 3 V NEW-WITHIN-SAMPLE PREDICTION PROBLEM 43 -t-apl pt t pbl t pt utu t a apl p ba al ata tat apl p a ll a a pu ata apl t a p ppulat apl ut a ll utl t a t a t b alu a ut pt a ll a t b alu X X a pu ata apl pbl ul b tt ptg t t t t alu ; t utl l atal alu l ; ub atal alu a utu tal t t A Lat-Sal al Dt ut C a tuat b b a lat-al al t ut b t t paat θµ µ a > a ptl paat lat a al t al aat u t gup pt la taat: ab t a> all au tat t btaabl bt ut at pt t t tatt < < < a a apl z a t b alu X X a pu ata apl z a ut tatt M S a au ll tat µ µ ba a XX X t t ut p µ p µ / / < < < < < µ > > 44 W a tu aug tat t al t ut a u aa l u t gup G taat: a b a µ a µ b a a> T al t ut atg t ab t u t lt al gat ptal Wbull a gaa t t ut T tutu t pbl lal t t gal a Lt b a utu bat t tatt t a apl z I aatl ptabl t W M / S W µ / a aal aat ptal tal X B P-La L ut W all t tal pt pbl t t tatt t a apl z t tuat bt t t bat X < X < < X < t pu ata apl z a t al b tatt < < < a a apl z a aalabl Supp tat al t t p-la l ut T pbl p b t at pt t ut 3 a t l ut aat u t gup G taat Tu t pbl t t ptal tal pt ag R D θ 45 D a t aat tal pt R θ E θ {θ} a ut C Taat t L ut It ll tat t aat l ut θ a b ta a ll: θ & V 46 & & V V V - V V V V V V V V V < V V > V V V V V M / V S / ; M / S M / S D R ut It ll 47 tat t aat t a θ a b p a { θ } E{ & } R θ Eθ V Aa l publat: 7 bua

8 IAENG Itatal Jual Appl Matat 4: IJAM_ tat bt a aat pt ul u b t t pbablt t t t bat X < X < < X t pu a apl z t al b tatt < < < a a apl z a t utu t tatt > t a apl z µ µ µ µ µ µ µ µ 5 E R Mzat a Iaat Pt Rul T llg t g t tal ult t t T 7 Optal aat pt ba X Supp tat u u a a t ag t ut u u u u u u uu u al u > b a lt 5 Q b t pbablt tbut ut u /u I / / < t t ptal aat la-l tal pt ba X * M S M S Q / Q / 5 { θ } E{ & } R θ Eθ V 54 a t a g b 5 a a t a g b 53 P t p t Clla T 8 Equalt t ba X Supp tat V V V a a t ag t pbablt t ut al > b 5 a lt b t pbablt tbut ut t pbablt t ut H V /V W W g b 55 T t t aat t t ptu pt tal * M S M S P { : < < µ } H Q / H Q / 56 P t p t T 3 VI EXAMPLES A Wt-Sapl Pt O Statt Eptal tbut Lt < < < b tatt z t ptal tbut t t t p > > 57 I / / t t ptal aat la-l tal pt ba X gat t a pt pt M S Q / 53 P t p t T Clla 5 Mu t ptal aat pt ba X T u g b C t pt pbl t tuat t t bat < < < < a b b W a t ptzat t pt tal u t l ut Lt a > T t t pbablt t ut a g b Aa l publat: 7 bua

9 p p p 58 Lt V 59 a S V 6 Ug t aat bg tqu 4-3 t a tagta a tat t t t V V 6 > 6 a Γ > 63 It ll 5 a 6 tat q 64 It ll 5 a 6 tat 65 I / / < t t ptal aat la-l tal pt ba g b * S S 66 ag q 67 a ag q 68 T t aat t t ptu pt tal * S S g b } : { P µ < < H H 69 B N-Sapl Pt O Statt Eptal tbut C t pbl pt t t tatt a utu apl z t ptal tbut t t pbablt t ut 57 t tuat t t bat X < X < < X t pu ata apl z a b b W a t ptzat t pt tal u t l ut Lt XX X X T t t pbablt t ut X X X g b p 7 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 Aa l publat: 7 bua

10 T pbablt t ut t t tatt t utu apl z g b p 7 Lt V 7 a X X S V 73 Ug t aat bg tqu 4-3 t a tagta a tat t t t V V 74 p > 75 a Γ > 76 It ll 5 a 74 tat q 77 It ll 5 a 74 tat 78 I / / < t t ptal aat la-l tal pt ba X g b * X S X S 79 ag q 8 a ag q 8 T t aat t t ptu pt tal * X S X S g b } : { P < < H H 8 C N-Wt-Sapl Pt O Statt Eptal tbut C t pt pbl t t tatt a apl z t ptal tbut t t pbablt t ut 57 t tuat bt t t bat < < < < t ata apl a t t bat X < X < < X t pu ata apl z a b b W a t ptz t pt tal u t l ut Lt XX X X T t t pbablt t ut X X X g b p 83 Lt a > T t t pbablt t ut a g b IAENG Itatal Jual Appl Matat 4: IJAM_4 6 Aa l publat: 7 bua

11 p p p 84 Lt V 85 a X X S V 86 Ug t aat bg tqu 4-3 t a tagta a tat t t t V V 87 > 88 a Γ > 89 It ll 5 a 87 tat q 9 It ll 5 a 87 tat 9 I / / < t t ptal aat la-l tal pt ba X g b * S S 9 ag q 93 a ag q 94 T t aat t t ptu pt tal * S S g b } : { P < < H H 95 VII CONCLUSION AND DIRECTIONS OR UTURE RESEARCH Tatall t tat a lp a plt a pt atg t t p a gl tat alu a g z Ital pt a ptat pat t atg p a at ag t lt aua pt tat Pt tal a t quat pt utat apl aat pt I t pap a lp pt tqu apppat tutg qutt pt tal pt IAENG Itatal Jual Appl Matat 4: IJAM_4 6 Aa l publat: 7 bua

12 IAENG Itatal Jual Appl Matat 4: IJAM_4 6 bu tatt utu apl a a p l ut l t utal t ulg tbut p but all t paat a up T t-apl pt tqu ta t aut l ut al-alu ata t -apl pt tqu ta t aut l pu pt bat a t -tapl pt tqu ta t aut bt pu a ut al-alu ata T tqu a b u t tut pt tal abta t alu t a apl utu bat T ult a b u t pt t ttal uat t a Tp II g l ttg pt a t pt t lt a -ut--: t T putat pu a b al pga a plt patal u It ll b t tat t pap a b bta tug ulat but ulat ult a utabl; t a t at ttal a ll a patal pt aaltal lut ul b u t a aalabl T ult t pap p u aaltal lut I a tattal pbl t aabl attt t ul tat a aat t pt t a ta gup taat I a g pbl at a ut tatt t ll tat t la aat ul ba t ut tatt tall plt t la all aat ul u aupt T ult a b u t tat t t a a aat ul ag aat ul ba ut tatt t a ag all aat ul I t pap tattal pt pbl a aat t pt t a ta gup taat a tut t ptal aat tal pt T t u tat t aat bg apl tatt a l ut t ptal quatt all t lat u paat t pbl T t a pal a gal at applabl t tattal pbl aat u a gup taat at tatl t paat pa T tqu pp t pap a b appl t tutg pt tal a t latal tbut ACKNOWLEDGMENT T a a uppt pat b Gat N 6936 Gat N 736 Gat N 94 a Gat N 9544 t Lata Cul S a t Natal Ittut Matat a Iat Lata REERENCES K Pa T uatal pbl patal tatt Bta l 3 pp 6 9 J K Patl Pt tal a Cuat Statt T a Mt l 3 pp G J Ha a W Q M Stattal Ital: A Gu Patt N : Wl 99 4 N R Ma a S C Sau O aluat aat aua l a a Wbull tbut Bta l 56 pp N R Ma Waat p ba t apl bat a Wbull ppulat IEEE Taat Rlablt l R9 pp C E Atl a Raa A upp lt t au utu bat a tp I t-alu tbut Bta l 59 pp J Lal O t tat a l t ulg l tbut Wbull Tt l 5 pp M Eglat a L J Ba Pt lt a t-apl pbl t plt Wbull ata Tt l pp M Eglat a L J Ba O pt lt apl a Wbull t-alu tbut Tt l 4 pp K W tg M Ma a N R Ma O tutg pt tal apl a Wbull t alu tbut Tt l pp R W M a D Kua Pt lt t Wbull tbut utlzg ulat Cputatal Statt & Data Aal l 7 pp D K Bau a R D Gbb O- pt tal at lat p bat a gaa ppulat at a lat" Tt l 48 pp K Kat T Mat a S Mu Nal ba t a gaa tbut: pt a tla tal a t-tgt lablt Tt l 5 pp N A Nal a K N Nal Iaat bg tqu a t tattal applat C Vlu Ctbut Pap t 5 S t Itatal Stattal Ittut la Hl: ISIItatal Stattal Ittut 999 pp Aalabl: ttp://tat/99/p-g/at /aat/ 9p 5 N A Nal a E K Vaa Ip D Statt Rga: SIA Izgltba l 4 6 N A Nal G Bz M Pugal a K N Nal Ip tat tat tat t a aat bg tqu WSEAS Taat Matat l 7 pp N A Nal M Pugal G Bz K Ct J Kat a K N Nal Iaat bg tqu a t applat pt ptzat tattal Aaltal a Stat Mlg Tqu a Applat K Al- Bga D a W Kttblt E LNCS l 648 Bl Hlbg: Spg-Vlag pp N A Nal M Pugal K Ct G Bz U Rz a K N Nal Pt l lt a pa pat g pl t uppt ata a pa qupt Aaltal a Stat Mlg Tqu a Applat K Al-Bga D a W Kttblt E LNCS l 648 Bl Hlbg: Spg-Vlag pp N A Nal M Pugal K Ct G Bz a K N Nal Optzat tattal a a aat bg tqu Ltu Nt Egg a Cput S: Wl Cg Egg WCE 3 Ju - Jul L UK pp N A Nal M Pugal K Ct a K N Nal Plag pt atgu aat tutu a aag tla appa Ltu Nt Egg a Cput S: Wl Cg Egg WCE 3 Ju - Jul L UK pp N A Nal a M Pugal Ip tat tat tat t a a tqu aat bg Stat Ctl C M E Cata Ia Publ: S pp 6793 N A Nal M Pugal K N Nal a U Rz Optzat pt tal tatt ba ata Ltu Nt Egg a Cput S: Pg T Wl Cg Egg WCE 6-8 Jul L UK pp N A Nal K N Nal a M Pugal Stattal utu ut t applat t ata a lablt Ltu Nt Egg a Cput S: Pg T Wl Cg Egg WCE 6-8 Jul L UK pp Aa l publat: 7 bua

Handout 30. Optical Processes in Solids and the Dielectric Constant

Handout 30. Optical Processes in Solids and the Dielectric Constant Haut Otal Sl a th Dlt Ctat I th ltu yu wll la: La ut Ka-Kg lat Dlt tat l Itba a Itaba tbut t th lt tat l C 47 Sg 9 Faha Raa Cll Uty Chag Dl, Dl Mt, a lazat Dty A hag l t a gat a a t hag aat by ta: Q Q