ON TESTING EXPONENTIALITY AGAINST NBARFR LIFE DISTRIBUTIONS
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1 STATISTICA, ao LII,. 4, ON TESTING EPONENTIALITY AGAINST NBARR LIE DISTRIBUTIONS M. A. W. Mahmoud, N. A. Abdul Alm. INTRODUCTION AND DEINITIONS Tesg expoealy agas varous classes of lfe dsrbuos has go a good deal of aeo. Wh respec o esg agas IR, see Proscha ad Pyke (967), Barlow (968), ad Ahmed (975) amog ohers. or esg agas IRA, see Deshpade (983), Lmk (989), Aly (989), ad Ahmed (994). or esg agas NBU, see Hollader ad Proscha (97), Koul (977), Kumazawa (983) ad Ahmed (994). or esg agas NBUE, NBUR ad NBAR classes, we refer o Klefsjo (98, 98), Deshpade e al. (986), Aboammoh ad Ahmed (988), Loh (984) ad Hed e al. (). Recely Mahmoud ad Abdul Alm () suded esg expoealy agas NBURR based o a U-sasc for cesored ad ocesored daa. Now le T be a o egave radom varable wh lfe dsrbuo (), where ( ) for < ad () may o be zero. The correspodg survval fuco of ew sysem (), for > ad he desy fuco s gve by f (). The falure rae a me s defed by r ( ) f ( )/ ( ),. I he log ru, f a devce s replaced by sequece of muually ad decally dsrbued, he remag lfe dsrbuo of he sysem uder operao a me s gve by saoary reewal dsrbuo as follows: W where ( ) ( du,, s he mea lfe of he radom varable T, ( du. The correspodg reewal survval fuco s gve by W ( ) ( du. The desy fuco of he reewal dsrbuo W () s gve by
2 6 M. A. W. Mahmoud, N. A. Abdul Alm w ( ) d d d ( du ( ) W ( ),. d The falure rae of he reewal dsrbuo W () s gve by w ( ) ( ) r ( ) ( ) for, W ( ) ( du where () s he mea remag lfe dsrbuo of a used u a me. Defo.. s ew beer ha average reewal falure rae (NBARR) f r () r W ( du,. Equvalely r () lw ( ), where W () ( du,.e he falure rae of a ew sysem s less ha he average reewal falure rae of a used sysem. Defo.. s ew worh ha used average reewal falure rae (NWARR) f r () r W ( du,. Equvalely r () lw ( ),.e he falure rae of a ew sysem s greaer ha he average reewal falure rae of a used sysem (see Abouammoh ad Ahmed, 99). Theorem.. The lfe dsrbuo or s survval havg NBARR ff ( du e r (),. Proof. Le be a lfe dsrbuo wh falure rae r(.), s NBARR meas r () lw ( ) he r ( ) lw ( ), W r () e. ( )
3 O esg expoealy agas NBARR lfe dsrbuos 6 Ths s equvale o he form r () ( du e,. () If () s sasfed, he s easy o proof he NBARR propery. By usg sead of, he proof of he followg heorem ca be coduced. Theorem.. The lfe dsrbuo or s survval havg NWARR ff ( du e r (),. The ma purpose of hs arcle s esg H : s expoeal agas H : NBARR ad o expoeal, based o a radom sample,,... from a coous lfe dsrbuo (ocesored daa), ad also (cesored daa) based o Z, ), =,,3,, where Z m(, Y ) ad ( f f Z Z Y, where, Y, Y,... Y be..d accordg o a dsrbuo G.. TESTING AGAINST NBARR CLASS OR NONCENSORED DATA Noparamerc esg for classes of lfe dsrbuos have bee cosdered by may auhors (see Ahmed, 975, 994, 995; Ebrahm e al., 99; Hed, 999; Hed e al., ; Mahmoud ad Abdul Alm, ). I hs seco we derve a oparamerc U-sasc es for esg : H : s expoeal agas H : NBARR ad o expoeal. or more deals abou U-sascs see Lee (989). Here, he problem s based o sample,,... from. Sce s NBARR, hs meas ( du r() f () e e for all, we use he followg measure of deparure from H E( e f () ( d f () () e d ( ) ( ) d ( ).
4 6 M. A. W. Mahmoud, N. A. Abdul Alm Noe ha uder H ad uder H,. To esmae le,,... be a radom sample from ; he (), () ad f() wll be emprcally esmaed. So he emprcal form of s as follows: fˆ () j ˆ ( ) ( ). e j I j () Sce (Hardle, 99) p j f ˆ () f (), as, herefore we ca wre f( ) (, ) e ( ) I( ) ad defe he symmerc kerel (, ) (, ),! R where, he summao over all arragemes of,, he o U-sasc ˆ s equvale U ( R, j ). (3) Sce he order of he kerel (3) s wo, hs procedure s smple o calculae. I also has asympoc properes. The followg heorem summarzes he asypoc ormaly of ˆ. Theorem.. ) As,, s asympocally ormal wh mea ad varace ha s as (4). Uder H, /. ) If s couous NBARR, he he es s cosse. Proof. ) Usg sadard heory of U-sasc (Lee, 989), we eed oly evaluae he asympoc varace, whch s equal o VarE (, ) E (, )
5 O esg expoealy agas NBARR lfe dsrbuos 63 Recall he defo of, ), hus s o dffcul o show ha ( xf () E (, ) e d( x) d( x) xd( x). Smlarly, we have f () E (, ) e xd( x) xd( x) d( x). Hece, uf () f () Var e d( ud( e ( ) ud( (4) Uder H, Var e /. ) ca be wre he form = ( () e ( )) d ( ). f ) f ) Le D( ) () e ( ). Sce s NBARR ad couous, he D()> ad sce s o expoeal he D()> for a leas oe, call. Se f ad ( ) ( ). Thus f ) ( f () ( D ) () e ( ) ) e ( ) D( ) ad ( ) ( ), ad sce s he po of crease of, hus >. To coduc he es, calculae ˆ ad rejec H Z, he sadard ormal varae a level. f hs value exceeds Lower ad upper percele pos of he sasc ˆ s based o 5 smulaed samples from he sadard expoeal dsrbuos of order 5()5 are compued as able 3.
6 64 M. A. W. Mahmoud, N. A. Abdul Alm 3. TESTING AGAINST NBARR CLASS OR CENSORED DATA I hs seco, a es sasc proposed o es H versus H wh radomly rgh cesored samples. I he cesorg model, sead of dealg wh,..., we observe he par ( Z, ), =,,3,, where Z m(, Y ) ad f Z, f Z Y, where,... deoe her rue lfe me from a dsrbuo ad Y, Y,... Y be..d accordg o dsrbuo G. Also s ad Y s are depede. Le Z( ) Z() Z()... Z( ) deoe he ordered Z s ad () s he correspodg o Z (), respecvely. Usg he Kapla ad Meer (958) esmaor he case of cesored daa ( Z, ), =,, : ˆ ( ) ˆ ( ) ( Z ( ) ) ( ),, Z ad Taer (983), hazard rae esmae wh cesored daa (), rˆ( ) R k Z K Rk, where: R k s he dsace bewee po ad s k-h eares falure po K(.) s a fuco of bouded varao wh compac suppor o he erval [-,]. The he proposed es sasc s gve by c ˆ e f () ( du d ( ) ( du d ( ) (5) or compuao use, ˆc where j j k. m c ˆ (5) ca be wre as j ( k ) ( k ) C Z Z C Z Z ( k ) () ˆ ( m ) ( mm ) Z ( ) f C C e, ( m) k/ k ( j ) mm ( j ) ( mm ) j k C k ad d Z ) ( Z ) ( Z ). Table 4 gves he perceles of ( k ) ( j j j ( j ) ( j ) c ˆ for sample sze 5()5, 6, 7, 8, 8.
7 O esg expoealy agas NBARR lfe dsrbuos ASYMPTOTIC RELATIVE EICIENCY AND POWERS Sce he above es s ew ad o oher ess kow for NBARR we compare our es o he smaller classes ad choose NBU, NBUR ad NBURR classes proposed Ahmed (994), Hed e al. () ad Mahmoud ad Abdul Alm (). We choose he followg aleraves: ) Lear falure rae famly : ( ) exp /,, ) Makeham famly ) Pareo famly ( ) e ( ) ( ) ( e / v) Webull famly ( ) e,, ) v) Gamma famly ( ) u e u du / ( ),, Noe ha H s aaed a, v) ad v), s aaed a ) ad ), ad s aaed whe ). Drec calculaos of he asympoc effceces of he NBARR class compared wh NBU (Ahmed, 994), NBUR (Hed e al., ) ad NBURR (Mahmoud ad Abdul Alm, ) able. or he prevous aleraves, he powers for he proposed es are abulaed as able usg smulaed umber of sample 5 for sample szes, ad 3 ad values,3 ad 4. TABLE Powers for NBARR class es LR N Powers Powers Powers 3 = = = Pareo 3 = = = Webull Gamma 3 3 = = =3 =3 =4 =4 I s clear from able ha our es has a good powers specally he case of Webull ad Gamma famles. Table shows ha our es has much hgher asympoc effcacy for he lear falure rae ad Makeham famles compared wh oher wo ess (Ahmed, 994 ad Hed e al., ). Also shows accepable AE for Webull ad Gamma famles.
8 66 M. A. W. Mahmoud, N. A. Abdul Alm I seco ad he es sascs () ad (5) for ucesored ad rgh cesored daa respecvely are derved. Usg (), (5) ad ables 3 ad 4 applcaos medcal scece are preseed o llusrae he heorecal resuls seco 5. TABLE Asympoc relave effcecy of o ( ), ( ) ad ( 3) of Ahmed (994), Hed e al. () ad Mahmoud ad Abdul Alm () Effcecy Lear falure rae Makeham 3 Pareo 4 Webull 5 Gamma ˆ (Ahmed, 994) () ˆ (Hed e al., ) () ˆ (3) () e(, ) () e(, ) (3) e(, ) APPLICATIONS Cosder he daa Abouammoh e al. (994). These daa represe a se of 4 paes sufferg from blood cacer (Leukema) from oe of Msry of Healh Hospals Saud Araba ad he ordered values are: I was foud ha he es sasc for he daa se, by formula () s ˆ ad exceeds he crcal value of able 3. The we rejec he ull hypohess of expoealy. Cosder he daa Susarla ad Varyz (978). These daa represe 8 survval mes of paes of melaoma. Of hem 46 represe whole lfe mes (o-cesored daa) ad he ordered values are: The ordered cesored observaos are:
9 O esg expoealy agas NBARR lfe dsrbuos Now gorg cesored daa, oe ca apply he mehodology of seco o es he hypohess Ho: he survval mes are expoeal agas H : he survval mes follow barfr ad o expoeal. Compug ˆ from (), we ge ˆ = exceeds he crcal po able 3 a 95% upper perceles. The we accep H whch saes ha he se daa have barfr propery. A smple compuer program s wre o calculae c ˆ for hese daa ad he c value we ge leads o a ˆ =.6768x- less ha he crcal value able 4 a 95% upper percele. The we rejec H : whch saes ha he se of daa have barfr propery. 6. CONCLUSIONS Tesg expoealy agas NBARR dsrbuos s cosdered. The perceles ad powers of our es are abulaed. Comparsos bewee our es ad ess of Ahmed (994), Hed e al. () ad Mahmoud ad Abdul Alm () are gve. Tes for hs problem whe rgh cesored daa s avalable s hadled. Our sudy explaed ha our es performs hgher AE wh respec o Ahmed (994) ad Hed e al. () ess. I gves a very good powers for he mos commo aleraves. Deparme of Mahemacs Al-Azhar Uversy, Egyp MOHAMED ABDUL WAHAB MAHMOUD NASSER ANWER ABDUL ALIM
10 68 M. A. W. Mahmoud, N. A. Abdul Alm APPENDI TABLE 3 Crcal value for ˆ N
11 O esg expoealy agas NBARR lfe dsrbuos 69 Crcal values of TABLE 4 c ˆ -(6,7,8,8)
12 63 M. A. W. Mahmoud, N. A. Abdul Alm ACKNOWLEDGMENTS The auhors would lke o hak he edor ad he referee for useful remarks ad commes. REERENCES A. M. ABOUAMMOH, A. N. AHMED (99), O reewal falure rae classes of lfe dsrbuos. Sascs & Probably Leer 4, pp. -7. A. M. ABOUAMMOH, A. N. AHMED (988), The ew beer ha used falure rae class of lfe dsrbuo. Adv. Prob.,, pp A. M. ABOUAMMOH, S. A. ABDULGHAN, I. S. QAMBER (994), O paral ordergs ad esg of ew beer ha reewal used classes. Relably Eg. Sys. Safey, 43, pp I. A. AHMED (975), A oparamerc es for he moocy of a falure rae fuco, Commucao Sascs, 4, pp I. A. AHMED (994), A class of sascs useful esg creasg falure rae average ad ew beer ha used lfe dsrbuo, J. Sas.. Pla. If., 4, pp I. A. AHMED (995), Noparamerc esg of class of lfe dsrbuos derved from a varably orderg, Pasakhya Samkkha,, pp E.E. ALY (989), O esg expoealy agas IRA alerave, Merka, 36, pp R. E. BARLOW (968), Lkelhood rao ess for resrced famles of probably dsrbuos, Aals of Mahemacal Sascs 39, pp J. V. DESHPANDE (983), A class of ess for expoealy agas falure rae average. Bomerka, 7, pp J. V. DESHPANDE, S. C KOCHAR, H. SINGH (986), Aspecs of posve agg, Joural of Appled Probably, 8, pp N. EBRAHIMI, M. HABIBULLAH, E. SOI (99), Tesg expoealy based o Kullback-Lebler formao, Joural of he Royal Sascal Socey, 54 B, pp W. HARDLE (99), Smohg Techques Wh Implemeao I S, Sprg-Verlag, New York. M. I. HENDI (999), Tesg geeral harmoc ew beer ha used expecao usg keral mehod, Egyp. compuer. Sc. J.,, pp. -6. M. I. HENDI, H. ALNACHAWATI, M. N. AL-GRAIAN (), Tesg NBUR ad NBAR classes of lfe dsrbuos usg keral mehods, Arab J. Mah. Sc., 6, pp M. HOLLANDER,. PROSCHANE (97), Tesg wheher ew beer ha used, Aals of Mahemacal Sascs, 43, pp E. L. KAPLAN, P. MEIER (958), Noparamerc esmao from complee observaos, J. Amer. Assoc, 53, pp B. KLESJO (98), HNBUE survval uder some shoch models, Scadava Joural of Sascs, 8, pp B. KLESJO (98), The HNBUE ad HNWUE classes of lfe dsrbuos, Naval. Res. Logscs Quarely. 4, pp H. L. KOUL (977), A ew es for ew beer ha used, Comm. Sas. Theor. Mah, 6, pp Y. KUMAZAWA (983), Tesg for ew s beerha used, Comm. Sas. Theor. Mah.,, pp. 3-3.
13 O esg expoealy agas NBARR lfe dsrbuos 63 A. J. LEE (989), U-Sascs, Marcel Dekker, New York, NY. W. A.LINMK (989), Tesg for expoealy agas mooo falure raeaverage alerave, Comm. Sas. Theor. Mah., 8, pp W. Y. LOH (984), A ew geeralzao of he class of NBU dsrbuo, IEEE Tras. Rel., R3, pp M. A. W. MAHMOUD AND N. A. ABDUL ALIM, O esg expoealy agas NBURR class of lfe dsrbuos, Submed for publcao.. PROSCHAN ad R. PYKE (967), Tess for mooo falure rae, Proc. 5h BereklySymp., pp V. SUSARLA, J. VANRYZIN (978), Emprcal bayes esmao of survval fuco rgh cesored oservaos, Aals of Sascs, 6, pp M. A. TANNER (983), A oe o he varable kerel esmaor of he hazard fuco from radomly cesored daa, Aals of Sascs,, pp SUMMARY O esg expoealy agas NBARR lfe dsrbuos Ths paper cosders esg expoealy agas ew beer ha average reewal falure rae (NBARR) aleraves. The perceles of hs es sasc are abulaed for sample szes 5()5. Pma s asympoc effceces relave o he ess of he ew beer ha used (NBU), ew beer ha used falure rae (NBUR) ad ew beer ha used reewal falure falure rae (NBURR) (Ahmed, 994; Hed e al., ad Mahmoud ad Abdul Alm, ). The powers of hs es are also calculaed for some used lfe dsrbuos. The problem whe rgh-cesored daa s avalable s hadled. Praccal applcaos of our ess he medcal sceces are prese.
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