A New Approach to Moments Inequalities for NRBU and RNBU Classes With Hypothesis Testing Applications

Transcription

1 Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 7 A New Appoach to Momets Iequaltes fo NRBU ad RNBU Classes Wth Hypothess Testg Applcatos L S Dab Depatmet of Mathematcs aculty of Scece Al-Azha Uvesty [Gls Bach] Nas Cty Cao Egypt E-mal: lamaa_deyab@yahoocom Abstact-- I ths atcle ew momet equaltes ae deved fo ew eewal bette tha used NRBU) ad eewal ew bette tha used RNBU) classes of lfe dstbutos demostateg that f the mea lfe s fte fo ay of them the all hghe ode momets exst Next based o these equaltes ew testg pocedues fo testg expoetalty agast ay oe of the above classes ae toduced ad studed showg that they ae smple tha most eale oes ad hold hgh effcecy fo some commoly used alteatves ad compaed the two testes elatve to the tests of Mahmoud et al ) ad ) The powe ad ctcal values of the poposed statstc ae calculated Real examples ae peseted to elucdate the use of the poposed tests fo pactcal elablty aalyss allythe poblem case of ght cesoed data s also hadled wth some applcatos Abstact-- ew eewal bette tha used NRBU) eewal ew bette tha used RNBU) momets U -Statstc lfe testg effcecy Mote Calo method powe ad cesoed data INTRODUCTION AND MOTIVATIONS I ths wok we focus o the NRBU ad RNBU popetes These popetes play a mpotat ole fomulatg epa o eplacemet polces Mahmoud et al ) cosdeed the poblem of testg expoetalty agast the NRBU class of lfe dstbuto based o U-test statstc by usg expected value method Mahmoud et al ) studed two test statstcs fo NRBU ad RNBU classes of lfe dstbutos as alteatves based o momets equaltes Recetly Mahmoud et al 5) studed testg expoetalty agast RNBU based o a U-statstc fo cesoed ad ucesoed data Dab et al 6) vestgated the test statstc fo testg expoetalty agast NRBU based o a goodess of ft appoach Ahmad et al 7) studed a ew appoach to momets equaltes fo testg expoetalty vesus IR NBU ad NBUC classes of lfe dstbutos wth hypotheses testg applcatos Now let be a o egatve adom vaable wth dstbuto ad a suvval fuctos by o pactcaltes s ofte assumed but eed ot be) absolutely cotuous wth pobablty desty fucto f ad has mea ad vaace Cosde a devce wth lfe legth ad lfe dstbuto The devce s eplaced statly upo falue by a sequece of mutually depedet devces These devces ae depedet of the fst ut ad detcally dstbuted wth the same dstbuto Whe the eewal of the system s cotued deftely the statoa lfe dstbuto of a devce opeato at tme x s W x The coespodg eewal suvval fucto s W x Whee u) du < s the mea lfe of the adom vaable I fact stochastc compaso betwee the adom vaable T small wth dstbuto ad ts eewal adom vaable T wth lfe dstbuto W fo whch W W ) desty fucto fucto w ad eewal suvval W leads to the followg deftos Defto A adom vaable T o ts dstbuto s sad to have ew eewal bette tha used st deoted by NRBU) popety f Tt TW whee T t s the codtoal vaable of T gve t wth dstbuto t) p T t T Ths defto meas that T s y NRBU f x gves t) W t) y t y ) Relato ) ca have the fom y t) W t) y t ) Itegatg both sdes of elato ) wt y ove W x t) W W t) ) e the eewal dstbuto s NBU ad s deoted by RNBU The coespodg dual classes of lfe dstbuto ae ew eewal wose tha used ad eewal ew wose tha used paet popety deoted by NRWU ad RNWU They ae defed by evesg the equalty sg of ) ad )The Decembe IJENS I J E N S

2 Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 8 elato betwee the above deftos as easly see as follows NRBU RNBU The pupose of the cuet vestgato s to povde ew momets equaltes of the above two classes that wll geeally asset that f < the the momets would exst fo ay oe of these classes I ths spt the momet equaltes developed secto ae used to costuct test statstc fo the two poblems secto These test statstcs ae based o sample momets of the agg dstbutos They ae smple to deve calculate study ad have hgh effceces fo some of the well kow alteatves elatve to the othe moe complcated tests so compaed the ptma asymptotc effcecy PAE) of the two tests elatve to the tests of Mahmoud et al ) ad ) Usg Mote Calo methods ctcal values ad the powes of the poposed test statstc peseted hee ae easy to obta fo dffeet choces of the odes of momets Some eal examples gve as applcatos ally secto 4 we cosde the poblem the case of ght-cesoed data ad selected ctcal values ae tabulated wth few applcatos NEW MOMENT INEQUALITIES Ths secto s dvded to two ma subsectos The fst oe s coceed wth the costucto of the momet equalty fo the NRBU dstbuto I the secod subsecto smla pocedue s developed fo the RNBU dstbuto Momet Iequalty fo the NRBU Dstbuto The fst esult povdes momets equalty fo the NRBU dstbuto I ths as well as subsequet esults all momets ae assumed to exst ad ae fte Lemma o all o-egatve tege ad s NRBU we get ) ) ) ) ) ) ) whee E ) Poof Sce s NRBU the x x dxdy Y x W dxdy Thus the left had sde of ) s equal to u) dudy Ey I Y > dy ) E Y ) ) ) Whle the ght had sde of ) s equal to ) ) ) ) ) Y u dudy fom ) ) ad 4) the esult follows ) ) ) ) 4) Coollay Let the ) educes to the same fom of the test of Mahmoud et al ) Momet Iequalty fo the RNBU Dstbuto Next let us look at the RNBU class Lemma o all o-egatve tege ad s RNBU we get ) ) ) Poof Sce s RNBU the x W x dxdy ) ) ) ) ) ) 5) x W W dxdy Aga the left had sde of 6) s equal to 6) Y u dudy E > ) ) y I Y y dy ) E Y ) ) ) ) ) 7) Whle the ght had sde of 6) s equal to > ) > ) ) ) x I x dx E y I Y y dy ) ) ) ) ) ) 8) fom 7) ad 8) the esult follows Coollay Let the 5) educes to the same fom of the test statstc of Mahmoud et al ) APPLICATION TO HYPOTHESES TESTING Ths secto s dvded to thee ma subsecto Decembe IJENS I J E N S

3 Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 9 the fst subsecto we cosde two hypothess testes the fst oe coceed wth the costucto of the poposed NRBU test statstc as a U-Statstc ad the othe coceed wth RNBU test statstc as a U-Statstc I the secod subsectos we dscussed asymptotc effcecy fo NRBU ad RNBU test I the thd subsectos we peseted the powe ad uppe pecetle values fo ad 99 of the poposed statstc as well Testg agast NRBU Alteatves Oe of the oldest poblems lfe testg s to test ) H : s expoetal agast H : s NRBU ad ot expoetal Usg Lemma we may use as a measue of depatue fom H favo of H Sce ) ) ) ) ) ) ) ) set ) The we estmate by ˆ ˆ Takg we get ) ˆ 4 6 ) Note that 4 ) 4) 6 the ˆ s a classcal U-Statstc Thus ) E ) Ude H 4 ) 4 6 Theefoe 4 E ) E 4) 6 Hece E ) 4 E 6 4 ) 4 Testg agast RNBU Alteatves Aothe poblem s to test ) H : H : s expoetal agast s RNBU ad ot expoetal Usg Lemma ) wth we ca popose the followg measue of depatue Sce ) ) ) ) ) ) ) ) ) 5) whe s estmated by ˆ 4 6 Next set 4) whch ca be estmated by ˆ ˆ 4 set 4 ) 6 Ude H 4 ) Thus 4 E )) E ) Theefoe E ) 56 6) 7) It s clea fom ) ad 7) that the testes based o the momet equaltes cotas two vaables whch ae smple tha those gve by Mahmoud et al ) Asymptotc Relatve Effcecy I ths secto we calculate the Ptma asymptotc effceces PAE) of the above test statstc ˆ ad ˆ ) ad 7) So to assess the qualty of these pocedues we compae these tests to some othe classes Hee we choose the tests V ad K peseted by Mahmoud et al ) ad ) fo NRBU) popety by usg expected value method ad RNBU) by usg momet equalty method espectvely The compasos ae acheved by usg Ptma asymptotc elatve effcecy PARE)whch s defed as follows: Decembe IJENS I J E N S

4 H Let Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 T ad T be two test statstcs fo testg c x costat the the PARE of wth c a abtay T elatve to T s defed by ) / )/ ) )/ e T T ) whee ad ) / E T ) lm ) va T ) lm s the ull vaace These calculatos ae doe usg the followg thee alteatves Lea falue ate famly Makeham famly x exp x x e )) x Webull famly exp x ) x Dect calculatos of the asymptotc effceces of the two test ˆ ad ˆ fo NRBU ad RNBU classes ae summazed Table The effceces Table show clealy that ou U-Statstcs ˆ ad ˆ pefom well fo ad tha V ad exp x x /) x Table Effcecy LR Makeham webull ˆ ˆ V K K I Table we gve the ptma asymptotc elatve effceces of ˆ ad ˆ wt V ad K Relatve effcecy E ˆ V ) E ˆ K ) Table LR Makeham webull om Table oe ca see that the ou two test statstcs ˆ ad ˆ have hgh elatve effceces fo all gve alteatves 4 Mote Calo Ctcal Values fo ˆ Statstc We have smulated the uppe pecetle values fo 9%95%98 % ad 99% fo small samples of szes 55)5 ad 6 4 sample szes The calculatos ae based o 5 smulated samples Table gves these pecetle values of the statstc ˆ that gve ) Decembe IJENS I J E N S

5 Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 Table Ctcal values fo the uppe pecetles of ˆ 9% 95% 98% 99% It s otced fom Table that the ctcal values ae ceasg as the cofdece level ceasg ad deceasg as the sample sze ceasg 5 The Powe Estmates I ths secto we peset a estmato of the powe of the poposed test statstc ˆ s calculated at sgfcace level 5 ad fo commoly used dstbutos elablty These dstbutos ae Lea falue ate famly exp x x /) x Paeto famly Gamma famly u e u du / ) x x Wth paametes values values 5 ad 5 wth sample szes ad The estmates of the powe ae peseted Table 4 Note that H s attaed at ) s attaed at ) ad s attaed whe ) Table 4 Alteatve Dstbutos: LR paeto ad Gamma L R paeto Gamma Decembe IJENS I J E N S

6 Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 om Table 4 oe ca see that ou test has a good powes fo all alteatves ad the powes ceases as the sample sze ceases The powe s gettg as smalle as the NRBU appoaches the expoetal dstbuto 6 Some Applcatos We peset ths secto some of a good eal examples to elucdate the applcatos of ou test the case of o cesoed data at 95% cofdece level Example Cosde the data Abouammoh et al 994) these data epeset set of 4 patets suffeg fom blood cace Leukema) fom oe of msty of health hosptal Saud Aaba ad the odeed values yeas ae: It was foud that ˆ 57 ad ths value s less tha the ctcal value of the Table The we accept H whch states that the data set have expoetal popety Example Cosde the followg data set s fom Kotz ad Johso 98)ad epesets the suvval tmes yeas) afte dagoss of 4 patets wth a ceta kd of leukema It was foud that ˆ 57 ad ths value s less tha the ctcal value of the Table The we coclude that ths data set have expoetal popety 4 Expected Depatue Test fo Cesoed Data I ths secto a test statstc s poposed to test H vesus H wth adomly ght-cesoed data Such a cesoed data s usually the oly fomato avalable a lfe-testg model o a clcal study whee patets may be lost cesoed) befoe the completo of a study Ths expemetal stuato ca fomally be modeled as follows Suppose obects ae put o test ad deoted the tue lfe tme We assume that be depedet detcally dstbuted d) accodg to a cotuous lfe dstbuto Let Y Y Y be d) accodg to a cotuous lfe dstbuto G Also assume that s ad Y s ae depedet I the adomly ght-cesoed model we obseve the pas ) whee m Y ) ad f f thobsevato sucesoed ) Y thobsevato scesoe d) Let ) < ) < ) < < ) deote the odeed s ad ) s the the coespodg to ) espectvely Usg the cesoed data ) Kapla ad Mee 959) poposed the poduct lmt estmato )/ ) ) ) ) : ) 4) Now fo testg H ˆ : agast H ˆ : > usg the adomly ght-cesoed data We poposed the followg test statstc ) dx dy y ˆC x y whee ad s the poduct lmt estmato gve 4) C o computatoal puposes ˆ the last above equato may be ewtte as: ˆC ) 5 4) whee k M ) C M k ) k ) ) k M ad dx ) ) v) ) Cv ) ) ) v Q) ) CQ ) ) ) Q ) C k k k Table 4 below gves the ctcal values pecetles of test fo sample szes 55)) 78 5 eplcatos C ˆ ad 86 o ) y dy Decembe IJENS I J E N S

7 Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 Table 4 ctcal values fo pecetles of C ˆ test 9% 95% 98% 99% Table 4 show that the ctcal values ae ceasg as the cofdece level ceasg ad almost deceasg as the sample sze ceasg ally wet peset two eal examples medce to llustate the applcatos of ou test cesoed data at 95% cofdece level Example 4 Cosde the data fom Susala ad Vayz 978) whch epeset 8suvval tmes moths) of patets of melaoma Out of these 46 epesets o-cesoed data ad 5ode cesoed data ae values: Now takg to accout the whole set of suvval data both cesoed ad ucesoed) ad computg the statstc fom 4) cesoed data we get ˆC 64 whch s geate tha the ctcal value of the Table 4 at 95% uppe pecetle The we accept H whch states that the data set have NRBU popety ad ot expoetal Example 4 O the bass of ght cesoed data fo lug cace patets fom Pea ) These data cossts of 86 suvval tmes moth) wth ght cesoed The whole lfe tmes o-cesoed data) ad cesoed obsevatos ae: Now takg to accout the whole set of suvval data both cesoed ad ucesoed) ad computg the statstc fom 4) cesoed data we get ˆC 8667 whch s geate tha the ctcal value of the Table 4 at 95% uppe pecetle The we accept H whch states that the data set have NRBU popety ad ot expoetal ukow paametes Comm Statst Smula Compu 7 - [] Gubbs E 97) ducal bouds o elablty fo the two paamete egatve expoetal dstbuto Techomet [4] Kapla EL ad Mee P958) Nopaametc estmato fom complete obsevato J Ame Statst Assoc [5] Kotz S ad Johso NL 98) Ecyclopeda of Statstcal Sceces Wley New Yok [6] Mahmoud M A W EL-ashy S M ad Dab L S ) A o-paametc test of ew eewal bette tha used class of lfe dstbutos Poceedg of Iteatoal Cofeece o Mathematcs teds ad developmets Cao EGYPT) 4 9- [7] Mahmoud M A W EL-ashy S M ad Dab L S ) Momet equaltes fo testg ew eewal bette tha used ad eewal ew bette tha used classes IJ Rel Appl [8] Mahmoud M A W EL-ashy S M ad Dab L S 5) Testg eewal ew bette tha used lfe dstbutos based o U-test Appl Math Model [9] Pavu R J Edgema R L ad Scott R C 99) Quadatc statstcs fo the goodess of ft test of vese Gaussa dstbuto IEEE Tas Rel 4 8- [] Pea A E )Goodess of ft tests wth cesoed data Statscedu peatakspesetedtalk actoel [] Susala V ad Vayz J 978) Empcal bays estmatos of a suvval fucto ght cesoed obsevato A Statst [] Shapo S S Goodess of t Tests 995) I the expoetal dstbuto theoy methods ad applcatos Balaksha N ad Basu A P Edtos Godo ad Beach Amstedam REERENCES [] Dab LSEl-ashySM ad Abdul AlmNA 6) Testg NRBU class of lfe dstbutos usg a goodess of ft appoach ItJ Rel Appl 7No 9-77 [] Edgma R L Scott RC ad Pavu RJ 988) A modfed Kolmogoov-Smov test fo the vese Gaussa desty wth Decembe IJENS I J E N S