CONDITIONAL CONFIDENCE INTERVALS OF PROCESS CAPABILITY INDICES FOLLOWING REJECTION OF PRELIMINARY TESTS JIANCHUN ZHANG

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1 CONDITIONA CONFIDENCE INTERVAS OF PROCESS CAPABIITY INDICES FOOWING REJECTION OF PREIMINARY TESTS by JIANCN ZANG Preeted t the Faulty f the Graduate Shl f The iverity f Texa at Arligt i Partial Fulfillmet f the Requiremet fr the Degree f DOCTOR OF PIOSOPY TE NIVERSITY OF TEXAS AT ARINGTON May 00

2 Cyright by JIANCN ZANG 00 All Right Reerved

3 T my family My wife Qizhi Ye ad daughter Jae Zhag

4 ACKNOWEDGMENTS I wuld like t give my deeet areiati t my uerviig rfer Dr. Chie-Pai a, withut hi urt ad advie, all f thee wrk are t ible t be fiihed. i tat atiee ad mtivati euraged me thrughut the etire erid f my Ph.D. tudy. I al wuld like t ay thak t Dr. D.. awki, Dr. Day Dyer, Dr. Sha Su-Mithell, ad Dr. Adrzej Krzeiwki t erve a my mmittee. I reeived lt f aademi hel frm them fr the at everal year, eeially frm Dr. awki, withut hi hel SAS rgrammig, me f the wrk are al t ible t be de. Al may thak t all the ther teaher: Dr. Carle Kig Krueger, Dr. ua Sha, Dr. Ruth Gret, Dr. rit Kjuharv, Dr David liu, et.. I will ever frget the fiaial urt I gt frm the Deartmet f Mathemati, withut the GTA urt frm the Math Deartmet, it hardly fr me t etrate my Ph.D. tudy,, may thak t the Deartmet f Mathemati f TA, ad the deartmet hair Dr. Jiaig Zhu. I al wat t ay thak t the deartmet graduate advier Dr. Jiazhg Su fr hi valued advie. Fially, I wat t ay thak t my family, my wife Qizhi, my daughter Jae ad my aret, thak them fr trutig me alway. Seial thak t my brther Jiagu Zhag ad Jiabig Zhag, thak t them fr takig are f all the thig I left behid i Chia, eeially fr takig are f my aret. Aril 9, 00 iv

5 ABSTRACT CONDITIONA CONFIDENCE INTERVAS OF PROCESS CAPABIITY INDICES FOOWING REJECTION OF PREIMINARY TESTS JIANCN ZANG, Ph.D. The iverity f Texa at Arligt, 00 Suerviig Prfer: Chie-Pai a Fidig a rdiary fidee iterval f a ukw arameter i well kw, but fidig a ditial fidee iterval fllwig rejeti f a relimiary tet i t ted, eeially fr fidig a ditial fidee iterval f the re aability idie C r Ck fllwig rejeti f me relimiary tet. Thi diertati will rvide me bai therie ad mutatial methd fr fidig uh ditial fidee iterval f the tw re aability idie. The mt bai methd ued i thi diertati i the geeral methd fr fidig a fidee iterval f a ukw arameter. Numerial methd are al ued fr fidig the value f thee ditial fidee limit. The ditial fidee iterval f the re aability idex C ad Ck are btaied. Cmutatial rgrammig de ad ther ueful ifrmati ad methd are rvided. v

6 TABE OF CONTENTS ACKNOWEDGEMENT.. ABSTRACT... IST OF FIGRES.... IST OF TABES. ⅳ ⅴ ⅸ ⅹ Chater Page. INTRODCTION Overview The Pre Caability Idie C ad Ck iterature Review CONDITIONA CONFIDENCE INTERVAS OF C The Pre Caability Idex C Bai Iferee abut C Pit Etimatr Cfidee Iterval Samlig Ditributi f Ĉ Cditial Cfidee Iterval f C Geeral Methd fr Ctrutig CCI..... vi

7 .4. Ctrutig CCI f C thrugh firt Ctrutig CCI f σ. 3.5 Examle Examle Simulated Data Examle Real Data Cmari f the egth fr CI ad CCI Rati f the wer imit Rati f the er imit Cditial Cverage Prbability f the CI Aalyi CONDITIONA CONFIDENCE INTERVAS OF Ck The Pre Caability Idex Ck 4 3. CCI f Ck Whe µ I Kw ad σ I kw CCI f Ck Whe µ I kw ad σ I Kw CCI f Ck Whe bth µ ad σ Are kw. 6 APPENDIX 3.4. Tetig fr bth Parameter Tetig fr Oe f the Tw Parameter A. IMS PROGRAM CODE. 79 B. CONDITIONA CONFIDENCE INTERVAS OF TE MEAN µ FOOWING REJECTION OF A ONE-SIDED TEST C. CONDITIONA CONFIDENCE INTERVAS OF σ FOOWING vii

8 REJECTION OF A TWO-SIDED TEST REFERENCES BIOGRAPICA STATEMENT viii

9 IST OF FIGRES Figure Page. SAS lt fr the relatihi betwee C C ad V at α= SAS lt fr the relatihi betwee C C ad V at α= SAS lt fr the relatihi betwee C C ad V at α= SAS lt fr the relatihi betwee C C ad V at α= The jit dmai f x, fr the ditial jit CDF f X ad S fllwig rejeti f tw tet The jit dmai f x, fr the ditial jit CDF f X ad S fllwig rejeti f e tet fr the mea The jit dmai f x, fr the ditial jit CDF f X ad S fllwig rejeti f e tet fr σ ix

10 IST OF TABES Table Page. Relatihi betwee ad fr differet ad mbiati Relatihi betwee ad fr differet ad mbiati 35.3 Atual verage rbability f 90 eret mial fidee iterval.. 4 x

11 CAPTER INTRODCTION. Overview The re aability aalyi ha bee rved t be a very ueful tl i rdut quality trl. There are may re aability idie PCI whih are ued urretly i idutry, t ame a few, C, Ck, Cm ad Cmk, et., but the mt mmly ued tw idie are C ad Ck. Sme bk have bee ublihed related t the PCI refer t: Ktz & velae 998, Ktz & Jh 993, et., they diued the bai iferee abut the PCI, iludig it etimatr, fidee iterval ad tetig hythee. I me ratial ituati, the ivetigatr may have rir ifrmati abut the ukw arameter, but hehe i uertai abut the ifrmati. A relimiary tet a be ued t relve the uertaity. We ider the ditial fidee iterval f the PCI after rejetig the ull hythee f the relimiary tet. Fr the ae that me ull hythee have bee rejeted, the rigial uditial fidee iterval i lger valid. S, t fid a ditial fidee iterval i a trit tw-tage iferee redure: firt, tet the hythee f relimiary tet, if the tet reult i rejetig me f ull hythee, the g t the ed tage t btai a ditial fidee iterval fllwig rejeti f the ull hythee. I thi aer, we will maily diu thi tye f rblem, the ditial fidee iterval f the re aability idie.

12 . The re aability idie C ad Ck The re aability idie are meaure f the quality f a maufaturig re, they have bee ular i idutry fr mre tha 40 year. I 90, Bell abratrie, a leader i the ue f tatiti t trl ad imrve quality, bega the firt eriu ivetigati it the aliati f tatiti thery t lt amlig ad the ue f igifiae thery i re trl. Shewhart, alg with Dr. J.M. Jura ad Dr. W. Edward Demig, develed mt f the early therie ad et f tatitial quality trl. The et f re aability idex wa firt itrdued by the Jaaee. At the begiig f 970, there were ly five re aability idie, kw a the rigial Jaaee re aability idie, ad thee ilude the tw mt mm re aability idie C ad Ck. The re aability idex C wa the mt rigial re aability idex, whih wa itrdued by Jura et al Thi re aability idex ly ut fr the re variati, ad igre the re mea. S, the re aability idex C meaure the tetial aability, whih i defied ly by the atual re read, it de t reflet the imat f hiftig the re mea the re aability t rdue qualified rdut. T verme the weake f C, the re aability idex Ck wa reated i Jaa i late 970, ad it take bth the re variati ad the re mea it iderati..3 iterature Review Befre I tarted thi diertati, a literature review f related ti, whih ilude me bk ad lt f aer, had bee mleted. The bai ifrmati abut the re aability idie, whih ilude the hitry ad me bai aalyi f the re aability idie, a be fud i the tw

13 bk writte by Ktz & velae 998 ad Ktz & Jh 993. Sie the aability idie C ad Ck ivlve tw re arameter: the re variae σ ad the re mea µ, therefre, we will diu everal aer abut the ditial fidee iterval f thee arameter fllwig rejeti f relimiary tet. The firt ad al the mt imrtat aer i writte by Meek & D Agti 983, they diued the ditial fidee iterval f the rmal mea µ fllwig rejeti f a e-ided tet. I rder t aly the geeral methd fr fidig a fidee iterval f a ukw arameter, the bk writte by Bai & Egelhardt 99 wa arefully examied. Sie the relimiary tet i thi diertati al ivlve me tw-ided tet, the aer writte by Arabatzi, Gregire ad Reyld 989 wa al reviewed, i thi aer, they diued me aet f the ditial fidee iterval f the rmal mea fllwig rejeti f a tw-ided tet. Sme ther aer related t the ditial fidee iterval were al ublihed, thee ilude the aer writte by Chiu & a 994, i thi aer, they diued the ditial fidee Iterval f the exetial ale arameter fllwig rejeti f a relimiary tet. Chiu & a 995 diued the ditial fidee iterval f the exetial lati arameter fllwig rejeti f a re-tet. I Chiu & a 999, they give the ditial iterval etimati f the rati f variae met fllwig rejeti f a re-tet. 3

14 CAPTER CONDITIONA CONFIDENCE INTERVAS OF C. The Pre Caability Idex C Cider a meauremet X f a rdut, the maufature uually require that the rdut mut meet me eifiati fr the meauremet X. If we et a lwer eifiati limit S ad a uer eifiati limit S fr X, the the value f X utide thee limit will be termed frmig NC Ktz & Jh, 993. A idiret meaure f the tetial ability aability t meet the requiremet S X S i the re aability idex C, whih i defied a C = S S 6 Sue X fllw a rmal ditributi with mea μ ad variae σ, if the exeted value f X i equal t the mid-it f the eified iterval, i.e., µ = S + S, let d = S - S, the the exeted rrti f NC rdut i Φ-dσ, i term f C, it i Φ-3C. Frm abve we ee that the bigger the value f C, the maller rrti f NC rdut. The relatihi betwee C ad the rrti f NC rdut i traight frward, therefre, the re aability idex C i widely ued i rdut quality trl. 4

15 . Bai Iferee abut C.. Pit Etimatr Sue we have a radm amle X, X,, X take frm Nμ, σ, the the mt mmly ued etimatr f μ i the amle mea X, ad the mt mmly ued etimatr f σ i the amle tadard deviati S, i.e., ˆ = X = X j j ad ˆ = S = j X j X Sie C = here i σ, therefre S S 6 = d 3, the ly arameter eed t be etimated Ĉ = d 3ˆ d = 3S i a it etimatr f C. Sie it i well kw that E[ ], we mut have, E[ Ĉ ] C. S Therefre, thi i a biaed etimatr. The bia will be give i eti.3. 5

16 6.. Cfidee Iterval The uditial fidee iterval f C a be derived diretly frm the uditial fidee iterval f σ. Sie the radm amle X, X,, X i take frm Nμ, σ, therefre, - S i ditributed a. Thi lead t the reult that a 00-α% fidee iterval fr σ i,,, where i the berved value f S. Sie ˆ C = d ˆ 3, therefre, a 00-α% fidee iterval f C i,, 3, 3 d d ad equetly, a 00-α% fidee iterval f C i,, 3, 3 d d C C ˆ, ˆ,,

17 .3 Samlig Ditributi f Ĉ Sie ur gal i t ivetigate the ditial fidee iterval f C, ad a it etimatr f C i Ĉ, we firt eed t fid the rbability deity futi df ad the umulative ditributi futi CDF f Ĉ. ere, we ll ue the trafrmati methd t derive the df f Ĉ. Sie S the radm amle X, X,, X i take frm Nμ, σ, we have that - d d i ditributed a. Sie Ĉ = =, therefre, Ĉ i ditributed a 3S 3 S C. ig trafrmati methd, we a derive the df f Ĉ. et X ~ ad Y = fx = C C gy = - Y. Sie X > 0, the trafrmati X = X i e-t e, the Jabia f the trafrmati i J = d g Y dy = C Y C Y Sie f X x x e, x > 0, x therefre, we have f y f [ g Y] Y X J 7

18 8 ] [ y C y C e y C y C Simlify the abve equati, we get the uditial df f Ĉ : 3 ˆ y e y y f, y > 0 where i the amle ize. Oe we have the df f Ĉ, we a ue it t alulate the mea ad the variae f Ĉ by firt arryig ut it rth mmet abut the rigi. T imlify the alulati, here we take the advatage f the Chi-Square ditributi f X t alulate the rth mmet f Ĉ abut the rigi: ] 3 [ ˆ r r S d E C E = r r C ] [ r S E = r r C dx e x x x r 0 = r C r r

19 9 Therefre, we btai the rth mmet f Ĉ abut the rigi: r r r C r C E ] ˆ [ S, the mea f Ĉ : f C b C C E ] ˆ [ ad the variae: 3 ˆ f C b C Var where bf i a ubiaed fatr give by b f Thu, the bia f Ĉ i C E C C b ] ˆ [ ˆ = f C b

20 Sme f the abve frmula a al be fud i Ktz & velae 998, age Cditial Cfidee Iterval f C The mt mm tet hythei i re aability aalyi i t tet : the re i t aable v. : the re i aable. A we already kew, large C value reult i mall rrti f NC rdut. Therefre, fr the C idex, thi lead t the hythei : C v. : C. Fr eah value f, there rred ather value σ, uh that = d3σ. T tet : C i equivalet t tet : σ σ. If the ull hythei i t rejeted, whih mea the re i t aable, the we t. A t aable re mea a large rrti f frmig rdut will hae, i thi ae, the mt imrtat ad eeary thig we eed t d i t fid ut the weake f the re ad try t imrve it. Therefre, further tatitial aalye eed t be de befre we imrve the re. If the ull hythei ha bee rejeted, whih mea the urret C value i withi ur aetable regi, i thi ae, a fidee iterval i eeded t give mre ifrmati abut the ible regi f C. Thi fidee iterval i differet frm the reviu uditial fidee iterval CI, it i the ditial fidee iterval CCI fllwig rejeti f the ull hythei : C. We ll rejet at level α if the tet tatiti V = - ;, i.e. Cˆ ; 0

21 .4. Geeral Methd fr Ctrutig CCI I eti.3, we have derived the uditial df f Ĉ : 3 ˆ y C e y y f, y > 0 The ditial df f Ĉ fllwig rejetig the ull hythei : C i the truated ditributi f the abve df, where the tet tatiti fr igifiae level α mut t exeed the ritial value ;, i.e., ;, ad thi i equivalet t Ĉ ; 3 d = ; =. The ditial df f Ĉ fllwig rejetig the ull hythei : C. i give by 3 3 ˆ y C C dy e y e g y C dy e y e, = The CDF f the ditial ditributi f Ĉ fllwig rejetig the ull hythei : C.i give by

22 y y e dy Gˆ, y y e dy [Reult.4.] et G; dete the ditial CDF f the Ĉ ditributi fllwig rejetig the ull hythei : C, ad ue h ad h are futi that atify Gh; = α ad Gh; = - α fr eah i the arameter ae f, where 0 < α <, 0 < α < with α + α <. et be a berved value f Ĉ, the we have the fllwig. If h ad h are ireaig futi f, the the luti f G; = - α ad G; = α trut a 00- α - α% ditial fidee iterval f.. If h ad h are dereaig futi f, the the luti f G; = α ad G; = - α trut a 00- α - α% ditial fidee iterval f. The abve reult fllw diretly frm the geeral methd therem give i Bai & Egelhardt 99. The geeral methd tate that, if a tatiti fr a arameter exit with a ditributi that deed thi arameter but t ay ther uiae arameter, the we a ue the geeral methd t fid a

23 fidee iterval f thi arameter. We refer thi tatiti t be a uffiiet tatiti r me reaable etimatr uh a a ME, but thi i t required. I ur ae, the ditial CDF f ĉ P deed ly C but t ay ther uiae arameter, we a ue the geeral methd t fid a CCI f C. Sie we a t lve fr h ad h exliitly i thi ae, it hard fr u t rve i thery whether the tw futi h ad h are ireaig r dereaig, therefre we tate the abve reult with the tw ible ae. Reult.4. hw theretially that we a fid a ditial fidee iterval f C uig the ditial CDF f ĉ P. But i ratie, thi redure i mewhat mliated, ie the ditial CDF f ĉ P i t a mmly ued ditributi futi, there i exitig muter rgram whih a be ued diretly t alulate the rbabilitie f thi ditributi. S ext, we ll ue ather methd t fid a ditial fidee iterval f C..4. Ctrutig CCI f C thrugh firt Ctrutig CCI f σ There are tw mai advatage fr dig thi way. Firt, ie the etimatr f σ i S, ad S fllw a ditributi, we a ue the umulative ditributi futi z ; υ f the hi-quare ditributi with υ degree f freedm t give the ditial CDF f S ditributi, a well a the tw imliit futi whih tai the uer r lwer limit f σ ad the berved tatiti value t fr S. Thi might hel a lt later whe we alulate the tw ditial fidee limit fr give examle uig mathematial ftware uh a IMS umerial library. Sed, e we btai the CCI f σ, we al btai the CCI f σ r, a well a the CCI f fr ay r. Thi 3

24 reult may have me mre aliati fr ther tatitial aalyi related t r the ditial fidee iterval f. Fr the radm amle X, X,, X draw frm Nμ, σ, a level α tet fr tetig : v. : ha the ritial regi K = { S : - S ; } The ull hythei i rejeted ly if S K, ad a ditial fidee iterval f σ i muted ly if the ull hythei ha bee rejeted. The ditial df f S a be exreed i the fllwig way f f 0, D, if therwie ; where f i the uditial df whih i determied by the wer f the tet give by S ~, D i D = P ; = P ; =, ; 4

25 where ad i the CDF f hi-quare ditributi with - degree f freedm. der, max{d} =. Whe 0, the value, the wer D arahe. The uditial ditributi f S i imilar t the ditributi exet it ha a ukw arameter σ. T exre the uditial CDF f S i term f, we have Ft = P S t S = P[ t] = t, t 0 Therefre, the ditial CDF f : i give by S fllwig rejetig the ull hythei F t F t D = t, with ; t ; [Reult.4.] et 0 < α <, 0 < α < with α + α <, ad t be a berved value f S, let dete the CDF f hi-quare ditributi with - degree f freedm. If the berved value t reult i rejetig the ull hythei :, the the luti f 5

26 t ; = α.4. ad t ; = - α.4. trut a 00- α - α% ditial fidee iterval f. We ue the geeral methd therem t hw thi reult. et Ft; dete the ditial CDF f the S ditributi, ad ue h ad h are futi that atify Fh ; = α ad Fh ; = - α fr eah, where 0 < α <, 0 < α < ad α + α <, let t be a berved value f S. It eay t hek i thi ae that bth h ad h are ireaig futi f thi i imilar t the ituati fr trutig CCI f the rmal mea μ rvided by Meek ad D Agti i 983. I fat, fr mall fixed value α r α, if ireae, the bth h ad h ireae, ad the rate f hagig fr h ad h are alway maller tha that f. Thi reult i urted bth aalytially ad umerially. Fllwig the therem give by Bai & Egelhardt 99, we a btai a ditial fidee iterval f by lvig the tw equati Ft; Thi give the abve reult. = α ad Ft; = - α. Oe we btai a ditial fidee iterval f ditial fidee iterval f C i give by 6 a, ditial fidee iterval f σ i fllwed a,. Thu, a, a

27 d 3 d, 3.5 Examle.5. Examle, Simulated Data Sue a ertai rdut maufatured frm a fatry ha the fllwig eifiati fr the meauremet: The eified lwer limit i 84.5, the eified uer limit i Sue the re mea i kw a 84.75, ad the mahie ued i the fatry a maufature the rdut that i rmally ditributed with mea ad tadard deviati Ardig t the abve ituati, we a ue a imle SAS rgram t imulate the maufaturig redure ad btai a ertai umber f imulated bervati data imu; d k= t 48; y= *rmal843; utut; ed; r rit; title 'imulated data'; ru; The reult f the abve SAS rgram give u the fllwig data et with ttal 48 bervati

28 Fr the abve data et, after we did me bai aalyi, we get the amle mea X ad the amle tadard deviati S Sie i thi ae d = S - S =.0*0.5 = 0.5, the it etimate f C i Ĉ = d 3 = 0.5 3* =.8 A 95% uditial fidee iterval f σ i,,, = 48 * *0. 095, 47, ,0.05 = , 0.03 Therefre, a 95% uditial fidee iterval f C i,, Cˆ, Cˆ 8

29 = 47,0.05 Cˆ ,0.975, Cˆ =.45,.9 Fr reaable value f C, Mtgmery 985 remmeded miimum value fr C a.33 fr a exitig re, ad.50 fr a ew re. Fr the ree related t eetial afety, fr examle, i maufaturig f blt whih are ued i bridge truti, a miimum value f.50 i remmeded fr a exitig re ad.67 i remmeded fr a ew re. I ur ae, ue we require a miimum value f the C a.33. The we eed t trut a tetig hythei a : C. 33 v. : C. 33. Fr the tetig hythei : C. 33 v. : C. 33, it i equivalet t the hythei : σ σ v. : σ < σ, where σ = d 3 = 0.53*.33 = We ll rejet at level α = 0.05 if the tet tatiti V = - 47,0.05 = 3.7. Nw the tet tatiti: V = - = 47* =

30 therefre, we rejet the ull hythei : C. 33 r : σ 0.53 at the level α = Ardig t reult.4., a 97.5% ditial uer fidee limit f σ fllwig rejeti f the ull hythei : C. 33 r : σ 0.53 at level α = 0.05 a be btaied by lvig the equati t ; = α = 0.05 where t = i the berved value f S, ad σ = 0.53 i the value f σ uder the ull hythei. The abve equati a be lved by uig IMS umerial library refer t the attahed FORTRAN de i Aedix A. The luti f the abve equati i = Similarly, a 97.5% ditial lwer fidee limit f σ fllwig rejeti f the ull hythei : C. 33 r : σ 0.53 at level α = 0.05 a be btaied by lvig the equati t ; = - α = The luti f the abve equati i =

31 Thu, a 95% ditial fidee iterval f σ fllwig rejeti f the ull hythei : C. 33 i give by:, = , ad equetly a 95% ditial fidee iterval f C fllwig rejeti f the ull hythei : C. 33 a be determied a d d C, C =, 3 3 =.09,.9 Cmare the abve ditial fidee iterval f C t the uditial fidee iterval f C we fud reviuly, we a ee that the legth f the ditial fidee iterval f C i relatively lger, ad it ver the whle legth f the uditial fidee iterval f C. The mt iteretig thig i, the ditial fidee iterval f C ver bak me regi f the C value whih ha bee rejeted by the hythei tet reviuly. The rea fr thi kid f reult i t quite ure, thi a be due t the tye Ⅰ errr. Further aalyi ee eti.6 ad eti.7 hw that thi i a mm ituati fr the ditial fidee iterval f C fllwig rejeti f a e-ided tet. That i, the ditial fidee iterval f C ver the uditial ditial fidee iterval f C, ad i mt f the ae, the ditial fidee iterval f C ver bak me value f C whih have bee rejeted by the ull hythei f the tet. Thi ituati i quite imilar t the e diued by Meek & D Agti 983. I their ae, they diued

32 the ditial fidee iterval f the rmal mea fllwig rejeti f a e-ided tet..5. Examle, Real Data The fllwig data et it f weight meauremet i ue fr 60 majr league baeball ee i Bai & Egelhardt 99, age 69, rblem Sue the re mea f thi rdut i kw a 5.5, the eified uer limit ad lwer limit are 5.45 ad 4.85 reetively It reaable that, fr thi rdut, we allw mre deviati frm the lwer ide f the re mea. Therefre, d = S - S = 0.60*0.5 = After dig me bai aalyi t the abve data et, we btai the mea ad the tadard deviati a X = 5.0 ad S = , the it etimate f C i Ĉ = d 3 = * =.54 A 95% uditial fidee iterval f σ a be determied by,,,

33 = 60 * * , 59, ,0.05 = , A 95% uditial fidee iterval f C i,, Cˆ, Cˆ = 59,0.05 Cˆ ,0.975, Cˆ =.6,.8 Sue the miimum required value f the re aability idex C fr thi re i.00, the a tet hythei fr tetig the value f C a be truted a : C. 00 v. : C. 00. Fr the tet hythei : C. 00 v. : C. 00, It i equivalet t the hythei : σ σ v. : σ < σ, where σ = d3c = 0.303*.00 = We ll rejet at level α = 0.05 if the tet tatiti V = - 59,0.05= Nw the tet tatiti: V = - = 59* =

34 Therefre, we rejet the ull hythei : C. 00 r : σ at level α = Next, we will trut a 95% ditial fidee iterval f C fllwig the rejeti f the ull hythei : C. 00. Ardig t reult.4., a 97.5% ditial uer fidee limit f σ fllwig rejeti f the ull hythei : C. 00 r : σ at level α = 0.05 a be btaied by lvig the equati t ; = α = 0.05 where t = i the berved value f S, σ = 0.000, = 60, ad, The abve equati a al be lved by uig IMS umerial library. The luti f the equati i = Fr the luti f a 97.5% ditial lwer fidee limit f σ fllwig rejeti f the ull hythei : C. 00 r : σ at level α = 0.05, ardig t reult.4., we a fid the luti by lvig the equati t ; = - α =

35 But thi time the IMS umerial library i uable t reah a luti due t flatig errr. Further aalyi ee i eti.6 hw thi lwer limit d exit, ad the rati f the uditial lwer limit ver ditial lwer limit equal t.0. Therefre, ardig t the reult i eti.6, we btai the ditial lwer fidee limit f σ a = Fially, a 95% ditial fidee iterval f σ fllwig rejeti f the ull hythei : C. 00 i give by:, = , ad equetly a 95% ditial fidee iterval f C fllwig rejeti f the ull hythei : C. 00 i give by d d C, C =, 3 3 =.5,.8.6 Cmari f the egth fr CI ad CCI I rder t mare the legth f the ditial fidee iterval f C t the legth f the uditial fidee iterval f C, we will derive frmula fr the rati f the ditial fidee limit f C ver the uditial fidee limit f C. 5

36 6.6. Rati f the wer imit The 00- α% uditial uer limit f σ, deted by, i frmed by lvig the equati t.6. The 00- α% ditial uer limit f σ, deted by, fllwig rejeti f the ull hythei : C v. : C i btaied by lvig the equati t.6. Frm.6., we have: t t.6.3 lug.6.3 it.6., we get et ad ubtitutig t i the demiatr, we have

37 .6.4 where V, ad V = - = - t et C ad C be the uditial ad ditial 00- α% lwer fidee limit f C reetively, the C C Frm equati.6.4, we ee that the rati f the ditial 00- α% lwer fidee limit f C ver the uditial 00- α% lwer fidee limit f C ly deed the value f the eified igifiae level α, the amle ize ad the rati f the ritial value ver the berved hi-quare tet tatiti f the relimiary tet fr tetig : C C0 r : σ. Fr the give examle.5. i eti.5, we have α= 0.05, =48;, 9. 96, V = =.877; therefre, ue IMS 47,0.05 umerial library t lve the equati.6.4, we btai the rati f the ditial 97.5% lwer fidee limit f C ver the uditial 97.5% lwer fidee limit f C i thi ae a C C Thi reult hw that i the give examle, the ditial 97.5% lwer fidee limit f C i relatively maller tha the uditial 97.5% lwer fidee limit f C. The abve reult mathe the reult we btaied i eti.5. 7

38 Fr the examle.5., we have α= 0.05, =60;, , 59,0.05 V = =.6379, therefre, ue IMS umerial library t lve the equati.6.4, we get the rati f the ditial 97.5% lwer fidee limit f C ver the uditial 97.5% lwer fidee limit f C i thi ae a C C 0.98 Thi reult al mathe the reult we btaied i examle.5.. I the examle, we gt the uditial 97.5% lwer fidee limit f C a.5 ad the ditial 97.5% lwer fidee limit f C a.6, whih lead t the rati f the ditial uer fidee limit ver the uditial uer fidee limit t a tiy differee ur betwee the tw reult jut beaue f the rudig errr. Table. give the relatihi betwee ad fr differet amle ize ad igifiae level mbiati. Figure. ad Figure. are the SAS lt hwig the relatihi. It quite lear frm the table ad the tw grah that the rati f C C i alway le tha r equal t, whih mea the ditial lwer fidee limit f C i alway maller tha r equal t the uditial lwer fidee limit f C. Whe the amle ize ad the rati V get bigger ad bigger, igifiae level α get bigger. arahe. beme bigger t whe the 8

39 9 Table. Relatihi betwee ad fr differet ad mbiati = = = = 0.05 = = = = = = = = = =

40 30 _. 0 at i gi f i ae l evel _ Figure. SAS lt fr the relatihi betwee C C ad V at α=

41 3 _. 0 at i gi f i ae l evel _ Figure. SAS lt fr the relatihi betwee C C ad V at α= 0.05.

42 .6. Rati f the er imit The 00- α% uditial lwer fidee limit f σ, deted by, i frmed by lvig t.6.5 The 00- α% ditial lwer fidee limit f σ, deted by, fllwig rejeti f the ull hythei : eqati C i give by lvig the t.6.6 Similarly, after me algebrai maiulati, we have.6.7 where V ad C C The abve reult hw that the rati f uditial 00-α% uer fidee limit f C ver the ditial 00-α% uer fidee limit f C deed ly the value f the igifiae level α, the amle ize ad 3

43 the rati f the ritial value ver the berved hi-quare tet tatiti f the relimiary tet. Fr the examle.5. i eti.5, α = 0.05, = 48; therefre, 67. 8, V =.877. ig IMS, we btai the rati f 47,0.975 the ditial 97.5% uer fidee limit f C ver the uditial 97.5% uer fidee limit f C fr thi ae a C C Thi reult hw that i the give examle, the ditial 97.5% uer fidee limit f C i almt uhaged mare t the uditial 97.5% uer fidee limit f C. Thi reult al it with the reult we btaied i examle.5.. Fr the examle.5., α = 0.05, = 60; therefre, 59,0.975 = 8., V = ee, uig IMS, we btai the rati f the ditial 97.5% uer fidee limit f C ver the uditial 97.5% uer fidee limit f C a C C.0000 Table. hw the relatihi betwee ad fr differet amle ize ad igifiae level mbiati. Figure.3 ad Figure.4 are the SAS lt hwig the relatihi. Frm the table ad lt, we ee that the rati f C C i till alway le tha r equal t, ad i mt f the ae, thi rati i equal r almt equal t. Thi reult tell u that the ditial uer fidee limit 33

44 f C i almt uhaged mare t the uditial uer fidee limit f C i mt f the ae. Thi ituati i quite differet frm the rati f the lwer fidee limit. Fr the lwer fidee limit ae, the rati varie a lt frm ae t ae. T ummarize the reviu reult, we may lude that, fllwig rejeti f a e-ided tet, the rati f are all equal t r early equal t, therefre, the ditial uer fidee limit are almt uhaged mare t the uditial uer fidee limit. But the rati f are alway le tha r equal t, the ditial lwer fidee limit are alway le tha r equal t the uditial lwer fidee limit. Thi lead t a mm lui that, fllwig thi tye f rejeti, the legth f the ditial fidee iterval f C i alway t le tha that f uditial fidee iterval, ad i mt ae, the ditial fidee iterval f C ver the whle legth f the uditial fidee iterval f C. Thi reult al verified the reult we btaied i examle.5. ad.5.. Whe the amle ize, the rati ad the value f ad beme bigger ad bigger, the ditial fidee iterval f C will beme ler ad ler t the uditial fidee iterval f C. It huld be ted that the ditial verage rbability f the uditial fidee iterval i maller tha that f the ditial fidee iterval, thi i diued i the ext eti. 34

45 35 Table. Relatihi betwee ad fr differet ad mbiati = = = = 0.05 = = = = = = = = = =

46 36 _ at i gi f i ae l evel _ Figure.3 SAS lt fr the relatihi betwee C C ad V at α=

47 37 _. 00 at i gi f i ae l evel _ Figure.4 SAS lt fr the relatihi betwee C C ad V at α= 0.05.

48 .7 Cditial Cverage Prbability f the CI Aalyi I ae the uditial fidee iterval f C are ued t arximate the ditial fidee iterval f C thi hae whe we igre the reult f a relimiary tet, it i eeary t give the atually verage rbability that i rvided at the mial uditial - *00 eret level. Frm eti.6, we kw that the ditial lwer fidee limit f C i relatively maller tha the uditial lwer fidee limit f C, ad the ditial uer fidee limit f C almt kee uhaged mare t the uditial uer fidee limit f C. Therefre, the ditial fidee iterval f C i alway lger tha the uditial fidee iterval f C, ad the ditial fidee iterval f C almt ver the whle legth f the uditial fidee iterval f C. Thu, if we ue the uditial fidee iterval f C t arximate the ditial fidee iterval f C, the atually verage rbability will alway be le tha r equal t the mial level. d Sie C =, ad atually we derived the ditial fidee iterval 3 f C thrugh the ditial fidee iterval f σ, therefre, whe we ue a uditial fidee iterval f C t arximate the ditial fidee iterval f C, the atual verage rbability f the fidee iterval f C i equal t the atual verage rbability f the rredig fidee iterval f σ rvided at the ame mial uditial level. Thi give the fllwig reult. [Reult.7.] Fr a equal tail - *00 eret level uditial fidee iterval f C, the ditial verage rbability i give by 38

49 39 Prf: We ly eed t rve the abve frmula fr the atual verage rbability f the equal tail uditial fidee iterval f σ. Sie the equal tail - *00 eret level uditial fidee iterval f σ i, Plug the abve uer ad lwer limit it equati.4. ad.4., we have α = ; t = ; t = where V i defied a ame a i eti.6, ad - α = ; t

50 40 = ; t = Therefre, the atual ditial verage rbability i - α - α =.7. Table.3 give the atual ditial verage rbability f a mial 90 eret fidee iterval fr differet amle ize ad rati mbiati. The reult hw that whe bth the amle ize ad rati are mall, the atual ditial verage rbability f a mial 90 eret fidee iterval i very mall. Whe either the amle ize r the rati r bth f ad get bigger ad bigger, the atual ditial verage rbability f a mial 90 eret fidee iterval get bigger ad bigger, ad evetually thi atual verage rbability arahe 90 eret the mial eretage level.

51 4 Table.3 Atual verage rbability f 90 eret mial fidee iterval

52 CAPTER 3 CONDITIONA CONFIDENCE INTERVAS OF Ck 3.. The Pre Caability Idex Ck At the begiig f Chater, we diued that if µ, the exeted value f X meauremet f the rdut, r imly the re mea, i equal t the midit f the eified iterval, the the exeted rrti f NC rdut i equal t Φ-3C. But if the exeted value f X i t equal t the midit f the eified iterval, i.e., µ S + S, the the exeted rrti f NC rdut will be bigger tha Φ-3C. The abve ituati fte hae i ratie. Fr examle, if we wat t rdue a air f axle ad bearig buhig, the the atual meauremet f diameter fr the axle huld be alway le tha the meauremet f diameter fr the bearig buhig, therwie, the axle a t be fitted i the bearig buhig. Therefre, whe we maufature the axle, after we eified the target value f the diameter, rmally we allw mre deviati fr the lwer limit f the diameter. But fr the bearig buhig, it i jut the ite, that i, we allw mre deviati fr the uer limit f the diameter. I thi ae, it i t uitable fr u t till ue the re aability idex C fr either the re f maufaturig the axle r the re f maufaturig the bearig buhig. Thu, we itrdue ather re aability idex Ck t verme thi drawbak, If we ider the effet f the value f the re mea μ, the the re aability idex Ck i defied a 4

53 Ck = mi S, S 3 Sie mia,b = fr ay a 0 ad b 0, therefre b a - b - a + Ck = S S S S 3 d = S S 3 = S S d C r imly Ck = d m 3 = m d C where m S S i the midit f the eified iterval. Nte, we aume that S μ S i ur diui. If μ i utide f the eified iterval, the by the iitial defiiti f Ck, the value f Ck wuld be egative, ad the re wuld learly be iadequate fr trllig the quality f the rdut. 43

54 m Sie we alway have, therefre, Ck C, with equality if d ad ly if µ = S S. Similar t the idex C, the maller value f the Ck rred t wre quality f the rdut. The re aability idex Ck ivlve bth the re mea ad the re variae. Whe the tw arameter are all ukw, ad a radm amle X, X,, X i take frm Nμ, σ ditributi, the a etimatr f μ i the amle mea X, ad a etimatr f σ i the amle tadard deviati S. Therefre, a it etimatr f Ck i give by ˆ C k d ˆ S S 3 ˆ = d X S S 3S = d X m 3S Sie X ad S are mutually ideedetly ditributed, it i till ible fr u t alulate the mea ad the variae f Ĉ k by firt arryig ut it rth mmet abut the rigi, like we did i Chater fr the idex C reultig frmula fr Ck refer t: Ktz & velae 998, age 55. But thi redure i muh mre mliated ad it ivlve ather tye f ditributi whih i alled flded ditributi. If we ider bth the mea µ ad the variae σ a ukw arameter, the the truti f uditial fidee iterval f Ck 44

55 i diffiult due t the fat that the ditributi f Ĉ k ivlve the jit ditributi f tw -etral t-ditributed radm variable. N igle tehique i idered bet i ratie at thi time Ktz & velae 998, age 57. Althugh the exliit exrei f uh a fidee iterval i almt imible, but theretially, thi fidee iterval i till ible t be determied fr artiular rblem. The idea i t exted the geeral methd t the tw arameter ae. That i, firt try t fid the jit fidee regi f the tw arameter µ ad σ, ad the ue thi jit fidee regi t btai a fidee iterval f Ck, thi methd i diued i eti 3.4. I the ae that we have me uertai rir ifrmati abut the value f µ ad σ, we will ue relimiary tet fr tetig thee tw arameter. We will adt tw equetial tet fr tetig µ ad σ earately, itead f tetig µ ad σ jitly. The ditial fidee iterval f Ck will be idered fllwig rejeti f ay f the equetial tet. I thi hater, we will diu the ditial fidee iterval f Ck fr the fllwig three differet ae: The mea µ i kw, the variae σ i ukw The mea µ i ukw, the variae σ i kw 3 Bth the mea µ ad the variae σ are ukw 3. CCI f Ck Whe µ I Kw ad σ I kw If the re mea µ i kw, the fr the re aability idex Ck, there i ly e ukw arameter σ. Thi ituati i imilar t the e fr fidig the ditial fidee iterval f the re aability idex C. I thi ae, the relimiary tet huld be truted a : C C v. : k 45

56 C, r, imly ue the arameter σ: : v. :, where k C the value f σ a be determied by the frmula C = d m 3 [Reult 3..] If a re ha a kw mea µ ad a ukw variae σ, the a 00- α - α% ditial fidee iterval f Ck fllwig rejeti f the ull hythei : fllwig iterval C C r : a be determied uig the k d m 3, d m 3 where, i a 00- α - α% ditial fidee iterval f σ fllwig rejeti f the relimiary tet fr tetig : v. :. The value i a 00- α% ditial uer fidee limit f σ whih i determied by equati.4., ad the value f i a 00- α% ditial lwer fidee limit f σ whih i determied by equati.4.. alulati. The rf f the abve reult fllw the reult.4. ad me imle [Examle 3..] Cditial fidee iterval f Ck aalyi fr the data et i examle.5. whe µ i kw ad σ i ukw I examle.5., ue the re mea µ i kw a 5.5, ad the eified uer limit ad lwer limit are 5.45 ad 4.85 reetively,, d = S - S = 0.30, m = S + S = 5.5. Frm the amle, we btai the 46

57 amle mea ad the amle tadard deviati a X = 5.0 ad S = , thu, the it etimate f Ck i Ĉ k = d m 3S = * =.03 Sie i thi examle, the re mea i t equal t the midit f the eified iterval, it bviu that the value f the it etimate f Ck i maller tha the value f the it etimate f C Cˆ d Sue the miimum required value f the re aability idex Ck fr thi re i 0.80, the we eed t trut a tet hythei a : C 0.80 v. : C k k Fr the tet hythei : C v. : C 0. 80, It i equivalet k t the hythei : σ σ v. : σ < σ, where σ = d - µ - m 3C = *0.80 = We ll rejet at level α = 0.05 if the tet tatiti V = - 59,0.05 = Nw the tet tatiti: k V = - = 59* =

58 Therefre, we rejet the ull hythei : C r : σ at level α = k Ardig t reult.4., a 97.5% ditial uer fidee limit f σ fllwig rejeti f the ull hythei : at level α = 0.05 a be btaied by lvig the equati t ; = α = 0.05 where t = i the berved value f S, σ = , = 60, ad, e IMS umerial library t lve the abve equati, we btai the luti = Fr the luti f a 97.5% ditial lwer fidee limit f σ fllwig rejeti f the ull hythei : C r : σ at level α = 0.05, ardig t reult.4., we a btai the value f by lvig the equati k t ; = - α = Fr the abve equati, the IMS umerial library i uable t reah a luti due t flatig errr. S we ue the reult i Chater, that i, eti.6 equati.6.7, t fid the rati f the uditial lwer fidee limit 48

59 f σ ver the ditial lwer fidee limit f σ firt i thi ae, α = 0.05, = 60, 8., ad V =.84. The reult hw that thi rati 59,0.975 = = Sie the uditial lwer fidee limit f σ i eay t btai, ad it i equal t frm reviu examle, therefre, the ditial lwer fidee limit f σ i equal t = Fially, a 95% ditial fidee iterval f σ fllwig rejeti f the ull hythei : C i give by: k, = , ad equetly, a 95% ditial fidee iterval f Ck rejeti f the ull hythei : C i give by k fllwig d m d m C, C =, 3 3 = 0.5, CCI f Ck Whe µ I kw ad σ I Kw I me ituati, if we have eugh ifrmati abut the variae f a re, i.e. the variae σ f the re a be regarded a kw. The fr the re aability idex Ck, there i ly e ukw arameter, the re mea µ. If the meauremet f a re fllw a rmal ditributi, the a it etimatr f µ i the amle mea X. Therefre, a it etimatr f the re aability idex Ck beme 49

60 Ĉ k = d X m 3 = d X m, 3 d X m, 3 if if X X m m I thi ae, fidig a ditial fidee iterval f the re aability idex Ck i really a matter f fidig a ditial fidee iterval f the re mea µ. Fr the re mea µ, if a radm amle X, X, X draw frm the re fllw a rmal ditributi Nμ, σ, the a 00-α% uditial fidee iterval f µ i give by the iterval X z, X z After we btai a uditial fidee iterval f µ, the a uditial fidee iterval f Ck a be eaily determied by uig the d m frmula Ck =, ie the ly ukw arameter i thi frmula i µ. 3 The tet hythei fr the arameter Ck fr thi ae µ i ukw, σ i kw a be truted a : C C v. : C k C, r equivalet t k the hythei : µ = µ v. : µ µ, where m d 3 C if m, ad m d 3 if m. Fr the ame value f C, whether we he the C value µ by uig the diti m r m deed rir ifrmati. Fr examle, if we allw mre deviati frm the lwer ide f the mea, the 50

61 we eed t ue the diti m. That i, we he md3c. Otherwie, we ue the diti m ad he m d 3 C. A mm rule f hw t ue the abve relimiary tet i that, if the ull hythei i t rejeted, the we ue µ0 a a etimate f µ t give a etimate f Ck, there i eed t trut a ditial fidee iterval f Ck i thi ae. But if the ull hythei i rejeted, we huld ue x a a etimate f µ t give the etimate f Ck, ad the we eed t fid a ditial fidee iterval f Ck fllwig rejeti f the ull hythei : r equivaletly : µ = µ. C C, k A we already kw, whe the re variae σ i kw, the re aability idex Ck tai ly e ukw arameter, the re mea µ. Therefre, i rder t fid a ditial fidee iterval f Ck, we ly eed t fid a ditial fidee iterval f the mea µ. Arabatzi, Gregire ad Reyld 989 ivetigated the ditial fidee iterval f the rmal mea fllwig rejeti f a tw-ided tet whe σ i kw, althugh I d t quite agree with the mai reult they have fr the ditial fidee iterval f µ, but me artial reult are till ueful. Next, we ll fllw the geeral methd t fid a ditial fidee iterval f µ fllwig rejeti f the ull hythei : µ = µ. If a radm amle X, X,, X i take frm a rmal ditributi Nμ, σ, where µ i ukw ad σ i kw. The a level α tet fr tetig : µ = µ v. : µ µ ha the ritial regi K X : X z 5

62 where z i the quatile f the tadard rmal ditributi. The ull hythei i rejeted if X K muted ly after we rejeted the ull hythei., ad a ditial fidee iterval f µ i The ditial df f X a be exreed a x f f x D, if x z 0, therwie where f x i the uditial df f X, ad D i the wer f the tet whih i give by D = x z P = z z = z z where, ad i the CDF f the tadard rmal ditributi. der, D =. Whe, D arahe. The ditial CDF f X a be exreed a F x z z x z z x z z, if x z, if x z 5

63 The abve frmula imlie that if the ull hythei i rejeted by a mall bervati f X, i.e., if a be exreed a F x x z x z z, the the ditial CDF f X x = z z 3.3. If the ull hythei i rejeted by a large bervati f X, i.e., if x z, the the ditial CDF f X a be exreed a F x x z z z z x z z = z z 3.3. It quite bviu frm equati 3.3. ad 3.3. that the ditial CDF f X deed ly the arameter µ, but t ay ther uiae arameter, we a ue the geeral methd metied befre i Chater t fid a ditial fidee iterval f µ. Ad i thi ae, it till a be verified umerially that the tw futi h ad h truted by the fllwig equati F h ; 53

64 ad F h ; are ireaig futi. Therefre, aly the geeral methd fr fidig a fidee iterval f a ukw arameter, we get the fllwig reult. [Reult 3.3.] Sue the radm amle X, X, X i take frm a rmal ditributi Nμ, σ, where µ i ukw ad σ i kw. et 0 < α <, 0 < α < with α + α <, ad x be a berved value f X. et dete the CDF f the tadard rmal ditributi. If the berved value x reult i rejetig the ull hythei : µ = µ at level by the diti x z, the the luti f ad x u z z u u = α x l z z l l = - α trut a 00- α - α% ditial fidee iterval 54 l, u f µ. Otherwie, if the berved value x reult i rejetig the ull hythei : µ = µ at level by the diti x z, the the luti f x u z u z u z z u u 3.3.5

65 ad x l z l z l z z l l trut a 00- α - α% ditial fidee iterval l, u f µ. The abve equati lk like mliated, but if we ue IMS umerial library, we till a lve thee equati fr the ditial lwer ad uer fidee limit f µ. Oe we btai the ditial fidee iterval f µ a, l u, t btai a ditial fidee iterval f Ck jut fllw me imle alulati. We ll ue a examle t illutrate the abve redure fr fidig a ditial fidee iterval f Ck fllwig rejeti f the ull hythei : µ = µ. [Examle 3.3.] Cditial fidee iterval f Ck aalyi whe µ i ukw ad σ i kw Cider the ame data et a i examle.5.. Sue that the eified target value f the re i 5.5, the eified uer limit ad lwer limit are 5.45 ad 4.85 reetively. Therefre, d = S - S = 0.60*0.5 = 0.30, ad m = S + S = *0.5 = 5.5. Sue the tadard deviati f thi re i kw a σ = 0.06 the re mea µ i ukw. After dig me bai aalyi t the data et, we btai the amle mea X = 5.0 ad the amle tadard deviati S =

66 Sie i thi ae X = 5.0 > m = 5.5,, a it etimate f Ck fr thi re i Ĉ k = = = d x m 3 d x m *0.06 =.33 by A 95% uditial fidee iterval, f µ a be determied l u x z, x z = 5..96* , 5..96* = 5.958, 5.6 d m Sie Ck =, ue the abve ifrmati f the uditial 3 fidee iterval f µ, we a determie a 95% uditial fidee iterval C, C f Ck. Sie i thi ae the value f m i lated the left k k ide f the abve iterval,, therefre, a 97.5% uer fidee limit f l u Ck hae at , ad a 97.5% lwer fidee limit f Ck hae at 5. 6, therefre u l 56

67 C, C = k k d u m d l m, 3 3 = * , * 0.06 =.4,.4 Nte, if m haeed t be iide f the iterval,, the the uer l u fidee limit f Ck equal t the value f C = d3σ, ad the lwer fidee limit f Ck a be determied by e f the tw value ad u whih ha the lger ditae frm the midit it m f the eified iterval. l S far, all the abve reult baed the ituati that there i relimiary tet fr the re mea ha bee erfrmed. If fr ay rea a relimiary tet fr tetig : µ = µ v. : µ µ ha bee de the value f µ0 deed the rir ifrmati f the re mea µ, r f the re aability idex Ck, whih a be btaied frm reviu exeriee, the the abve reult f the uditial fidee iterval f Ck i lger valid. S we eed the fllwig redure t fid a ditial fidee iterval f Ck. I thi examle, the target value f re i eified a µ = 5.5, if the tadard deviati f the re i kw a σ = 0.06, the we exet the d m value f Ck fr thi re a Ck = = *0.06 = 3.. S rmally, if we have ay rir ifrmati whih hw that the re mea will be arud the value f 5.5, the we will trut a tet hythei : C. v. : C., r equivaletly t : 5. 5 v. : k k If the ull hythei i t rejeted, the we aet the tet value µ = 5.5 a 57

68 kw. That i, we aet fr thi re, the re mea µ i equal t 5.5, ad equetly the re aability idex Ck i equal t., ditial fidee iterval i eeded. If the ull hythei i rejeted, the we eed t ue Ĉ k = d X m 3 t give a it etimate f Ck, ad the trut a ditial fidee iterval f Ck fllwig rejeti f the tet. Fr the tetig hythei : C. v. : C. r equivaletly : 5. 5 v. : 5. 5 i thi examle, we ll rejet at level α = 0.05 if the value f the tet tatiti X fall it the fllwig rejet regi K x : x z k k Nw the tet tatiti x = 5.0, ad the value z = * = Therefre, we have x Z : 5. 5 at level α = 0.05., ad we rejet the ull hythei Next, we will trut a 95% ditial fidee iterval f the re mea µ fllwig rejeti f the ull hythei : Sie i thi examle the rejeti f the ull hythei i aued by a mall value f X, i.e., the rejeti i due t x Z. Ardig t reult 3.3., a 97.5% ditial uer fidee limit f µ fllwig rejeti f 58

69 the ull hythei : 5. 5 at level α = 0.05 a be determied by lvig the equati x u z z u u = α =0.05 where x = 5.0 i the berved value f X, µ = 5.5, σ = 0.06, = 60, ad z.96. e IMS umerial library t lve the abve equati fr u, we get the luti u = 5.7 T btai a 97.5% ditial lwer fidee limit f µ fllwig rejeti f the ull hythei : 5. 5at level α = 0.05, ardig t reult 3.3., we a fid l by lvig the equati x l z z l l = - α = equati a Agai, ue the IMS umerial library, we btai the luti f the abve l = Therefre, a 95% ditial fidee iterval f µ fllwig rejeti f the ull hythei : 5. 5 i give by:, = 4.954, 5.7 l u 59

70 After we btai a ditial fidee iterval f the re mea µ, w we a determie a ditial fidee iterval f the re aability idex Ck fllwig rejeti f the ull hythei : C. r equivaletly : Sie i thi ae, the value f m = 5.5 i iide f the ditial fidee iterval,, therefre, a 97.5% ditial uer l u fidee limit f Ck hae at the value f m 5. 5, ad a 97.5% ditial lwer fidee limit f Ck hae at the value f Thu, a 95% ditial fidee iterval f Ck fllwig rejeti f the ull hythei : C. a be determied a k k l C, C = k k d l m d m m, 3 3 = * , * 0.06 = 0.58,.67 The relatihi betwee the ditial fidee iterval f Ck ad the uditial fidee iterval f Ck fr the ae that µ i ukw ad σ i kw a al be aalyzed by uig a imilar methd diued i eti.6. Exet i thi ae, we eed firt t fid the relatihi betwee the ditial fidee limit f µ ad the uditial fidee limit f µ. Sie thi redure i mliated, we will t diu i detail at thi time. I me eial ae, we till eed t tet a e-ided hythei fr the re aability Ck, thi ilude the fllwig tw differet ituati, : Ck C v. : Ck > C r : Ck C v. : Ck < C. T fid a ditial fidee iterval f Ck fllwig rejeti f the abve ull hythee 60

71 fllw a imilar redure diued i thi eti. Firt, we eed t fid a ditial fidee iterval f the re mea µ fllwig rejeti f the d m tet, ad the we ue the relatihi Ck = t btai a ditial 3 fidee iterval f Ck. Aedix B give me brief reult fr fidig ditial fidee iterval f the re mea µ fllwig rejeti f e-ided tet. 3.4 CCl f Ck Whe bth µ ad σ Are kw Previuly, we diued the ditial fidee iterval f the re aability idex Ck fr the tw differee ae: either µ i kw ad σ i ukw r µ i ukw ad σ i kw. But i mt ituati, bth the true value f the tw arameter µ ad σ are ukw. S ext, we ll diu the ditial fidee iterval f Ck whe bth µ ad σ are ukw. The tetig hythee we eed t ider fr thi ituati deed hw muh rir ifrmati we have. If we have rir ifrmati fr bth arameter µ ad σ, the we eed t trut tetig hythee fr the tw arameter µ ad σ. But i me ae, we ly have ifrmati fr e f the tw arameter. If thi i the ae, the we ly eed t trut e tetig hythei. Next, we will diu thee tw differet ae Tetig fr bth Parameter A we metied at the begiig f Chater 3, if bth the mea µ ad the variae σ f a re are ukw arameter, ad we have uertai rir ifrmati fr bth µ ad σ, the we will tet the arameter µ ad σ earately uig tw equetial tet. The ditial fidee iterval f Ck will be idered fllwig rejeti f ay f the tw tet. The redure i 6

72 give a the fllwig. Firt, tet the hythei : σ = σ v. : σ σ, if the ull hythei i t rejeted, we regard σ a give σ = σ, ad the tet : µ = µ v. : µ µ fr the arameter µ, thi tet i a rmal tet ie σ i give. If the ull hythei : µ = µ i al t rejeted, the we ue µ ad σ a tw etimate f µ ad σ t give the etimate f Ck, ditial fidee iterval f Ck i eeded. But if the ull hythei : µ = µ i rejeted, we ue x ad σ a tw etimate t give the etimate f Ck. Ad the we will fid a ditial fidee iterval f µ fllwig rejeti f the ull hythei : µ = µ f the tw equetial tet. Fially, we ue the abve ditial fidee iterval f µ tgether with the value f σ ie σ = σ i regarded a kw i thi ae t btai a ditial fidee iterval f Ck. The redure fr fidig thi ditial fidee iterval f Ck i almt the ame a the e we diued i the lat eti eti 3.3. If the ull hythei f the firt tet fr tetig : σ = σ v. : σ σ ha bee rejeted, i thi ae, we eed t ue the amle tadard deviati a a etimate f σ, ad the regard σ a ukw t trut the ed hythei : µ = µ v. : µ µ fr tetig the re mea µ. Thi time the tet i a t-tet ie σ i ukw. If the ull hythei f the ed tet i t rejeted, we eed t ue µ ad a tw etimate f µ ad σ t give the it etimate f Ck, ad the try t fid a ditial fidee iterval f σ fllwig rejeti f the ull hythei : σ = σ methd refer t Aedix C, ditial fidee iterval f σ fllwig rejeti f a tw-ided tet. The ditial fidee iterval f Ck fllwig rejeti f the ull hythei : σ = σ f the tw equetial tet a be btaied a fllwig. We regard µ a kw µ = µ ad σ a ukw ad ue the ditial fidee iterval f σ tgether with the kw value f µ µ = µ t trut a ditial fidee iterval f Ck thi redure i imilar t the e we diued i 6

73 eti 3., exet the ditial fidee iterval f σ i thi ae fllw rejeti f a tw-ided tet. [Examle 3.4.] I examle.5., ue the re mea µ ad variae σ are all ukw. The eified uer limit ad lwer limit are 5.45 ad 4.85 reetively,, d = S - S = 0.30, m = S + S = 5.5. Frm the amle, we get the amle mea ad amle tadard deviati a X = 5.0 ad S = Sue frm rir ifrmati, we kw that the re tadard deviati might be arud 0.05, ad the re mea µ might be arud 5.0. S we ue tw equetial tet t tet : σ = 0.05 v. : σ 0.05 ad : µ = 5.0 v. : µ 5.0 fr the re tadard deviati ad mea earately. Fr the hythei : v. : 0. 05, we will rejet the ull hythei at level α = 0.05 if the tet tatiti V = - ;. Nw i thi examle, the berved tet tatiti ;, r V = - V = - = 60 * = ad = ; 59;0.975 level α = = 8., we rejet the ull hythei : at 63

74 After we teted fr the re variae, w we eed t tet fr the re mea µ. Sie the ull hythei i the firt tet ha bee rejeted, the re variae i w ukw. Fr tetig hythei : 5. 0 v. : 5. 0, ie the true value f σ i ukw, we ue a tw-ided t-tet. We ll rejet the ull hythei : 5.0 at level α = 0.05 if the tet tatiti X, S fall i the fllwig rejeti regi K x, : x t Nw frm the amle: x = 5.0, = , x = = 0.0 ad t =.00* =.0 Thu, we have t, ad we d t rejet the ull x hythei : 5. 0 at level α = Therefre, we regard 5. 0 a kw fr thi re. Fr the ditial fidee iterval f Ck fllwig rejeti f the ull hythei : σ = σ f the tw equetial tet, we ue the ditial fidee iterval f σ tgether with the kw value f µ µ = 5.0 t trut it. S ext, we eed t fid a ditial fidee iterval f σ fllwig rejeti f the ull hythei : σ =

75 65 Sie the rejeti i due t a mall berved value f, i.e., ; thi huld be alway the ae i aalyi f thi tye f ditial fidee iterval f Ck, beaue if the rejeti i due t a large berved value f, the the re i bviuly t aable, there i eed t trut a ditial fidee iterval befre we imrved the urret re. Ardig t reult C. i aedix C, a 97.5% ditial uer fidee limit f σ fllwig rejeti f the ull hythei : σ = 0.05 at level α = 0.05 a be btaied by lvig the equati ; ; = α = 0.05 where t = i the berved value f S, σ = 0.05, = 60,, ad, 8.. e IMS umerial library t lve the abve equati, we get the luti a = T btai a 97.5% ditial lwer fidee limit f σ fllwig rejeti f the ull hythei : σ = 0.05 at level α = 0.05, ardig t reult C., we a fid the ditial lwer limit by lvig the equati ; ; = - α = The luti f the ditial lwer fidee limit f σ i

76 = Therefre, a 95% ditial fidee iterval f σ fllwig rejeti f the ull hythei : σ = 0.05 at level α = 0.05 i give by:, = 0.064, l u Cequetly, a 95% ditial fidee iterval f Ck fllwig rejeti f the ull hythei : σ = σ f the tw equetial tet a be determied a C, C = k k d m d m, 3 u 3 l = , 3* * =.05,.34 If the ull hythei f the ed tet fr tetig : µ = µ v. : µ µ i al rejeted, I thi ae, bth µ ad σ eed t be idered a ukw w, ad the ditial fidee iterval f Ck huld be idered fllwig rejeti f the tw relimiary tet. Thi ituati i mre mliated tha all the ae we diued befre. I rder t fid a ditial fidee iterval f Ck, we huld firt ider t fid a jit fidee regi f µ ad σ. Next, we ll give me bai aalye fr hw t fid a ditial fidee iterval f Ck i thi ituati. 66

77 Fllwig the geeral methd, i rder t fid a ditial jit fidee regi f µ ad σ, firt we eed t fid the ditial jit CDF f X ad S, we tart with fidig the uditial jit df f X ad S. If a radm amle X, X,, X i take frm a rmal ditributi Nμ, σ, the X ~ Nμ, σ, S ~, ad al X ad S are ideedet. Fllw Arabatzi, Gregire ad Reyld 989, the uditial jit df f X ad S a be exreed a f x, g x q [ x ] e fr x, 0, where i the Gamma futi. The ditial jit df f X ad S fllwig rejeti f the tw tet fr tetig : σ = σ v. : σ σ ad : µ = µ v. : µ µ a be exreed a f x, f x, D, 0, if x, K therwie where K i the ritial regi f the tw tet determied by the itereti f x t ad ; r ;, whih i al the ttal haded e regi f I, II, III ad IV hw i figure 3.; D i the 67

78 ttal uditial rbability f x, fallig it the abve ritial regi, whih i determied by the fllwig duble itegral. D f x, dxd K The ditial jit CDF f X ad S fllwig rejeti f the tw tet fr tetig : σ = σ v. : σ σ ad : µ = µ v. : µ µ a be exreed a F f x, dxd x,, fr x, K D 3.4. Figure 3. The jit dmai f x, fr the ditial jit CDF f X ad S fllwig rejeti f tw tet. The lie AB ad AC are determied by t. The lie DE ad FG are determied x by ; ad ; It huld be tied that the alulati f the duble itegral f x, dxd i equati 3.4. are quite differet whe the air f 68

79 bervati x, fall it differet regi f I, II, III r IV hw i figure Frm equati 3.4., it quite bviu that the ditial jit CDF f X ad S ly deed the tw ukw arameter µ ad σ. Althugh the exrei f the ditial jit CDF f X ad S i mewhat mliated, but with werful muter rgram, it till ible t alulate the umulated rbability fr ay berved value f x,, if the tw arameter µ ad σ are give. Nw we have all the ifrmati we eed fr fidig a ditial jit fidee regi f µ ad σ, amely, the ditial jit CDF f X ad S whih ly deed the tw ukw arameter but t ay ther ukw uiae arameter. Fr ay berved value f x,, if a ditial jit fidee regi f µ ad σ exit, it uld be fud by uig the abve ifrmati. Next, we ll try t exted the geeral methd f fidig a fidee iterval fr a ukw arameter t the tw arameter ae. Sue K i e relatively mall regi f X ad S uh that P [ X, S ], if we regard x, a radm tatiti ad let µ, σ hage jitly, the the tatemet x, i equivalet t the tatemet F x, fr me α ad α uh that. Therefre, if the iequality F x, ha a luti fr the regi f µ, σ, the thi luti huld trut a 00-α% jit fidee regi f µ ad σ. I ther wrd, if we lug ay air f µ, σ value it the abve iequality ad make the iequality a true tatemet fr a air f berved tatiti x ad. the thi air f µ, σ value huld be i a 00- α - α% ditial jit fidee regi f µ ad σ whih i related t thi berved air f tatiti x ad. I thi way, we a exted the geeral methd t the tw arameter ae, ad btai the fllwig reult. 69

80 [Reult 3.4.] Sue a radm amle X, X, X i take frm a rmal ditributi Nμ, σ, where µ ad σ are bth ukw. et 0 < α <, 0 < α < uh that 0 < - α - α <. et x ad be the berved value f X ad S, ad let F x, dete the ditial jit CDF f X ad S ee equati If the berved value f x ad reult i rejetig the tw ull hythee : µ = µ ad : σ = σ at level, the the luti f F x, 3.4. fr all air f µ, σ trut a 00- α - α% ditial jit fidee regi f µ ad σ. The reultig jit fidee regi f the luti f equati 3.4. i t eay t figure ut, but we may thik i the fllwig way t get a rugh iture. I equati 3.4., if we fix e f the tw ukw arameter, ay σ, at e value σ, the the rblem beme t fidig a ditial fidee iterval f e igle ukw arameter. By the geeral methd we diued i Chater, the luti huld be a fiite iterval if the value f σ i withi the jit fidee regi. If we hage σ t ather fixed value σ, the the luti f µ i ather fiite iterval if σ i till i the jit fidee regi. Same ituati hae whe we fix µ at e value ad try t fid the luti f σ. S, we may lude that the luti f equati 3.4. i jut e eted regi f µ ad σ, ad thi regi huld tai the air f berved value f x,. I rder t verify thi, we may take a examle uig the ame data et a i examle.5.. Sue the re mea ad variae are all ukw, ad we are itereted i the value f σ = σ = 0. ad µ = µ = 5.5, we take tw 70

81 equetial tet fr tetig : σ = 0. v. : σ 0. ad : µ = 5.5 v. : µ 5.5. Sie the berved tatiti are = ad x = 5., the tw ull hythei are all rejeted at level α = 0.05 by the abve tw berved tatiti. Ad i thi ae, the air f berved tatiti x, fall it the rejet regi I a hw i figure The ttal ritial regi i determied by the lie 0. 08, 0.8 ad x t , x t Sie i thi ae, the berved air f tatiti x, fall i regi I, therefre, the duble itegral i the umeratr f equati 3.4. a be writte a f x, dxd = x 0 f x, y dxdy ad the wer D a be exreed a D = , f x, dxd + f x, dxd + f x, dxd f x, dxd Thu, the ditial jit CDF f X ad S fllwig rejeti f the tw equetial tet a be exreed a F x, f x, dxd D 7

82 = , f x, dxd f x, dxd x f x, y dxdy f x, dxd f x, dxd If we lug the uditial jit df f x, f X ad S it the abve CDF ad imlify, we btai the ditial CDF f X ad S a F x, , e 60 x 59 x x 59 y y e 60 0 dxd dxd dxdy dxd dxd where e 60 x 59 i ued ly fr imlifiati f the exrei. Fr the berved air f tatiti x, = 5., , the iequality fr a 95% ditial jit fidee regi f µ ad σ equati 3.4. a be writte a

83 0.08 5, e 60 x dxd e 60 x dxd dxd dxd dxd Nw if we fix the value f σ at 0.065, the the abve iequality ly tai e ukw arameter µ, thi ituati i imilar t the e fr fidig the ditial fidee iterval f a igle arameter. ig werful muter rgram, we a btai the luti f µ a a iterval. If we hage the value f σ t ather umber 0.06, we a btai ather luti f iterval if σ = 0.06 i till withi the jit fidee regi f µ ad σ. After we btaied the ditial jit fidee regi f µ ad σ, the ditial fidee iterval f Ck fllwig rejeti f the tw tet fr tetig : σ = σ v. : σ σ ad : µ = µ v. : µ µ a al be determied, but the mutati i till very mliated, we eed t ue werful muter rgram t alulate it Tetig fr Oe f the Tw Parameter I me ituati, we may have uertai rir ifrmati e f the tw ukw arameter. If thi i the ae, the we a ly trut e relimiary tet. Firt, ider the ae that we have me rir ifrmati abut the re mea µ, ad we tet the hythei : µ = µ v. : µ µ. If the ull hythei i t rejeted, the we regard µ = µ a kw, ditial fidee iterval f Ck i eeded the ditial fidee iterval f Ck i thi ae fllw t rejetig a relimiary tet, whih i t i 73

84 ur ti. If the ull hythei : µ = µ i rejeted, the we eed t ue x ad a tw etimate f µ ad σ t give a it etimate f Ck. The ditial fidee iterval f Ck fllwig rejeti f the ull hythei : µ = µ a be btaied by uig a imilar redure diued i eti The ditial jit df f X ad S fllwig rejeti f the ull hythei : µ = µ a be exreed a f x, f x, D, 0, if x, K therwie where f x, i the uditial jit df f X ad S; K i the ritial regi f the tet whih i determied by t, i.e., the regi I ad II x hw i figure 3.; D i the ttal uditial rbability f x, fallig it the abve ritial regi, whih i determied by the fllwig duble itegral. D f x, dxd K I thi ituati, D i al the wer f the tet fr tetig : µ = µ v. : µ µ, whih a be alulated by uig the -etral t-ditributi, that i D = X t S P P X t S = t t 74

85 where i the CDF f the -etral t-ditributi with - degree f 0 freedm ad with -etrality arameter. It quite bviu that D ivlve the tw ukw arameter µ ad σ. Figure 3. The jit dmai f x, fr the ditial jit CDF f X ad S fllwig rejeti f e tet fr the mea : µ = µ v. : µ µ. The lie AB ad AC are determied by x t. The ditial jit CDF f X ad S fllwig rejeti f the ull hythei : µ = µ a be exreed a F f x, dxd x,, fr x, K D 3.4. The alulati f the duble itegral f x, dxd i equati 3.4. i till quite differet whe the air f bervati x, fall it differet regi f I ad II hw i figure Thi ditial jit CDF f X ad S deed ly the tw ukw arameter µ ad σ but t ay ther uiae arameter, we a fllw the ame redure diued i eti 3.4. t fid a ditial jit fidee regi f µ ad σ fllwig rejeti f the ull 75

86 hythei : µ = µ. After we btaied the jit fidee regi f µ ad σ, we a ue it t btai a ditial fidee iterval f Ck. I ae we ly have uertai rir ifrmati abut the re variae σ, the we eed t tet the hythei : σ = σ v. : σ σ. If the ull hythei i t rejeted, the we regard σ = σ a kw. The fidee iterval f Ck i thi ae will t be diued at thi time, ie there i tet hythei ha bee rejeted. Thu, ditial fidee iterval f Ck i eeded. If the ull hythei : σ = σ i rejeted, the we eed t ue x ad a tw etimate f µ ad σ t give a it etimate f Ck. The ditial fidee iterval f Ck fllwig rejeti f the ull hythei : σ = σ a be truted imilarly t the reviu ae, exet the rejeti regi i differet. The ditial jit df f X ad S fllwig rejeti f the ull hythei : σ = σ a be exreed a f x, f x, D, 0, if x, K therwie where f x, i the uditial jit df f X ad S. K i the ritial regi f the tet whih i determied by ; ad ;, that i, the regi I ad II hw i figure 3.3. D i the ttal uditial rbability f x, fallig it the abve ritial regi, whih i al determied by the duble itegral. D f x, dxd K 76

87 The ditial jit CDF f X ad S fllwig rejeti f the ull hythei : σ = σ a be exreed a F f x, dxd x,, fr x, K D Figure 3.3 The jit dmai f x, fr the ditial jit CDF f X ad S fllwig rejeti f e tet fr σ : σ = σ v. : σ σ. The lie AB ad CD are determied by ad ; ; The ame ituati a i the reviu tw ae, the alulati f the duble itegral f x, dxd i equati are differet whe the air f bervati x, fall it differet regi f I ad II hw i figure A we a hek, thi ditial jit CDF f X ad S ly deed the tw ukw arameter µ ad σ. Therefre, we a ue reult 3.4. t fid a ditial jit fidee regi f µ ad σ fllwig rejeti f the ull hythei : σ = σ. Oe the ditial jit fidee regi f µ ad σ i 77