ON TOTAL TIME ON TEST TRANSFORM ORDER ABSTRACT

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Oswald Robbins
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1 V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe ON OAL IME ON ES RANSORM ORDER V. M. Chacko Depame of Sascs S. homas Collee hss eala-68 Emal: chackovm@mal.com ABSRAC I hs pape ode of wo lfeme adom vaable based o cove oal me o es C asfom ad ceas cove oal me o es IC asfom of he dsbos ae odced ad he mplcao wh sochasc ode ad hazad ae ode ae poved. INRODUCION Sochasc odes ad eqales ae be sed a a acceleaed ae mch dvese aea of pobably ad sascs. hs pape odces he sochasc ode of wo lfe dsbos based o cove oal me o es C asfom ad ceas cove oal me o es IC asfom. he smples way of compa wo dsbo fcos s by compaso of assocaed meas. oweve sch a compaso s based o oly wo sle mbe he meas ad heefoe s ofe o vey fomave. Whe oe wshes o compae wo dsbo fcos ha have he same mea o ha ae ceeed abo he same vale oe s sally eesed he compaso of he dspeso of hese dsbos. I may saos applcaos oe has moe dealed fomao fo he compaso of wo dsbo fcos ha ake acco vaos foms of possble kowlede abo he wo dely dsbos see Shaked ad Shahkma 994. oal me o es asfom plos ae sefl fo aalyz o-eave daa. he plos help choos a mahemacal model fo he daa ad povde fomao abo fale ae. Also complee daa ca be aalyzed ad hee s a heoecal bass fo sch a aalyss see Balow ad Campo 975. As s sefl aalyz complee daa we ca ode he dsbos accod o of especve dsbos. ocha e al. defed asfom ode ad Shaked ad Shahkma 7 sded eplcly. Na e al. 8 povded applcaos of of ode elably aalyss. B f he mea vales of he wo dsbos ae same we eed o o fo vaably meases fo ode. Cove ad ceas cove ode s sally sed o ode wo dsbos accod o he vaably of he adom vaables. I hs pape we odce he ode of wo dsbos based o cove ad ceas cove whch ca be sed o ode wo dsbos accod o he of he cove ad ceas cove fcos of he especve adom vaables. Whe we cosde cesoed daa C ad IC s moe sable fo ode wo dsbos accod o he vaably. I seco he oos of sal sochasc ode ad ae befly ecalled. I seco 3 he defo of ode s ve. I seco 4 he cocep of C ode ad IC ode ae povded ad some mplcaos bewee sochasc ode ad hazad ae ode wh C ad IC ode ae poved. Coclsos ae ve a las seco. 5

2 V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe. SOCASIC AZARD RAE AND MEAN RESIDUAL LIE ORDER Le ad be wo adom vaables sch ha P P. he s sad o be smalle ha he sal sochasc ode deoed by s.. I meas ha s less lkely ha o ake lae vales whee lae meas he vale eae ha ad ha hs s he case fo all s.. s same as P P. Le ad has dsbos ad especvely depede of each ohe. Le f h f ad h be he hazad ae fcos of ad whee f ad ae he pobably desy fcos of ad especvely. Clealy hhe he hazad ae smalle he shold be sochascally. Defo. Le ad ae wo o-eave adom vaables wh absolely coos dsbos ad especvely depede of each ohe. s sad o be smalle ha hazad ae ode deoed by h. f h f h. Aohe mpoa ode s mea esdal lfe ode. he defo of mea esdal lfe s ve below. Defo. If s a o-eave adom vaable wh a svval fco ad a fe mea μ he mea esdal lfe of a s defed as m E [ ] ad ohewse. Clealy he smalle he mea esdal lfe fco s he smalle shold be some sochasc sese. Le m f ad m be he mea esdal lfe fcos of ad especvely. Defo.3 Le ad be wo o-eave adom vaables wh absolely coos dsbos ad especvely depede of each ohe. s sad o be smalle ha mea esdal lfe ode f m f m deoed by ml. Moe deals of sochasc odes ca be see Shaked ad Shahkma 994. Now we ecall he ode he follow seco. 3. OAL IME ON ES RANSORM Le ad have dsbos ad especvely depede of each ohe. ve a sample of sze fom he o-eave adom vaables ad le... k... ad... k... be he ode sascs coespod o he samples. o he h fale fom dsbos ad ae especvely

3 V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe 5 Defe ad d d whee ad } : f{ ad }. : f{ he fac ha a.s. ad a.s. mples by lveko Caell heoem lm d d ad lm d d fomly ]. [ We defe asfom of as d ]. [ ad asfom of as d ]. [ We defe he follow ode of wo adom vaables wh absole coos dsbo fcos ad especvely. Clealy lowe he empcal of s lowe he ha of oly whe he vale of k ad ae lowe ha ha of k. ha s. Now we ecall he follow. Defo 3. Le ad be wo o-eave adom vaables wh absole coos dsbos ad especvely. s sad o be smalle ha ha of he oal me o es asfom ode f ] [. We deoe he ode as. Moe deals of ode ca be see Shaked ad Shahkma 7 ad s applcao Chacko e al.. I he follow seco we odce he Cove ode ad Iceas cove ode whch ake a acco of vaably of adom vaables. 4 CONVE AND INCREASIN CONVE Le... whee s a cove fco. he defe fo d whee

4 V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe lm as lm d d say. Moe eeally defe fo evey cove fco fcos d Smlaly we ca defe lm d d : R R say. d d d say. Le ad be wo adom vaables sch ha fo all cove : R R ad all samples of sze. he s smalle ha some sochasc sese sce s aveae of oal obseved cove asfomed me of a es. he vales ae less lkely o ake lae vales ha vales. heefoe we defe he follow cove ode. Defo 4. Le ad be wo o-eave adom vaables wh absolely coos dsbo fcos ad especvely. If [] ad s cove fco he s smalle ha cove ode deoed as. Rohly speak cove fcos ae fcos ha ake o hem elavely lae vales ove eo of he fom a b fo a < b. Now we odce he ceas cove ode. Defo 4. Le ad be wo o-eave adom vaables wh absolely coos dsbo fcos ad especvely. If [] ad s ceas cove fco he s smalle ha ceas cove ode deoed. IC Rohly speak s boh smalle ad less vaable ha some sochasc sese. Eample 4. Le ~ Ep ad ~ Ep ad = a cove fco. ad []. ˆ ˆ whe. ece we ca coclde ha ad f ˆ ˆ. Aa C C 53

5 V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe d d e e d e d e * ** ad whee * f{ : } ad * * f{ : } he [] f ad * * *. Now we pove he follow heoem whch ves he mplcao of sochasc ode ad cove ode f he epecaos of adom vaables ae fe. heoem 4. Le ad be wo o-eave adom vaables hav absolely coos dsbo fcos ad especvely. Le be a cove fco : R R. If E ad E he s mples C. Poof: Clealy de he saed codos ad [] P P P d whee s a cove fco. heefoe. ece he poof. C P d Now we pove he follow heoem whch ves he mplcao of hazad ae ode ad cove ode f he epecaos of adom vaables ae fe. heoem 4. Le ad be wo o-eave adom vaables hav absolely coos dsbo fcos ad especvely. Le be ad cove fco : R R If E ad E he h mples C. Poof: Clealy de he saed codos h f d h h P e P e f he by above heoem. ece he poof. C h d I a smla way we ca pove he mplcaos of sochasc ode ad ceas cove ode ad hazad ae ode ad ceas cove ode be eplac he fco by a ceas cove fco. he esls ae saed below who poof. heoem 4.3 Le ad be wo o-eave adom vaables hav absolely coos dsbo fcos ad especvely. Le be a ceas cove fco : R R. If E ad E he mples. heoem 4.4 Le ad be wo o-eave adom vaables hav absolely coos dsbo fcos ad especvely. Le be a ceas cove fco : R R If s C E ad E he mples. h IC 54

6 V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe 5. CONCLUSIONS he ma advaae of cove ad ceas cove ode elao s o ode wo adom vaables accod o he vaably ad closeess o eve whe cesoed daa s avalable. I eeds fhe sdy o eploe he close popees as ohe ode behavos. he cocave ad ceas cocave ode ca be defed easly. Aaloos esls ae sah fowad. he esls have heoecal ad paccal applcaos elably heoy. REERENCES. Balow R. E. ad Campo R. A. 975 oal me o es pocesses ad applcao o fale daa aalyss Reseach epo No. ORC 75-8 Uvesy of Calfoa.. lefsjo B. 98 O ae popees ad oal me o es asfom Scad. J. Sascs Chacko V.M. Paveea P.C. ad M. Maohaa Ae popees of sem-makov sysem ad oal me o es asfom Poc. Ieaoal Coess of Mahemacas Absac< dsa Books ydebad A ocha S. C. L.. ad Shaked M. he oal me o es asfom ad he ecess wealh sochasc odes of dsbos Adv. Appl. Pob Shaked M. ad Shahkma J Sochasc Odes ad he applcaos Academc Pess Newok. 6. Shaked M. ad Shahkma J.. 7 Sochasc Odes Spe Newok. 7. Na. U. Sakaa P.. ad Vesh ma B. 8 asfoms of ode ad he mplcaos elably aalyss J. Appl. Pob

Suppose we have observed values t 1, t 2, t n of a random variable T.

Suppose we have observed values t 1, t 2, t n of a random variable T. Sppose we have obseved vales, 2, of a adom vaable T. The dsbo of T s ow o belog o a cea ype (e.g., expoeal, omal, ec.) b he veco θ ( θ, θ2, θp ) of ow paamees assocaed wh s ow (whee p s he mbe of ow paamees).