On the Class of New Better than Used. of Life Distributions

Transcription

1 Appld Mahacal Sccs, Vol. 6, 22, o. 37, O h Class of Nw Br ha Usd of Lf Dsrbos Zohd M. Nofal Dpar of Sascs Mahacs ad Israc Facl of Corc Bha Uvrs, Egp dr_zofal@hoal.co Absrac So w rsls abo NBU3 class of lf dsrbos ar obad. Closr of h NBU3 class dr covolo ar gv. Shoc odls whr shocs ar arrvg accordg o hoogos ar sablshd. Kwords: NBU3, covolo, shoc odls, lf dsrbos. Irodco Cra classs of lf dsrbos ad hr varaos hav b rodcd rlabl, h applcaos of hs classs of lf dsrbo ca b s grg, socal, bologcal scc, aac ad borcs. Thrfor, sascas ad rlabl aalss hav show a growg rs odlg srvval daa sg classfcaos of lf dsrbos basd o so aspcs of agg s for apl Barlow ad roscha 98. Aog h os wll ow fals of lf dsrbos ar h classs of crasg falr ra IFR, crasg falr ra avrag IFRA, w br ha sd NBU, w br

2 68 Z. M. Nofal ha sd pcao NBUE, ad haroc w br ha sd pcao NBUE. For or dals, radrs ar rfrrd o Barlow ad roscha 98, d ad Mashhor993, Ch994, Blzc al.2, L al.2, Blzc al.2, L ad Kochar2, Fraco al.2, Nada al.23, L ad Zo24, ad Ahad al.26 aog ohrs. I hs sco, w rvw so of h agg crra ad hr rlaoshps. W also dscrb how h agg proprs of h orgal dsrbo ar rasford o h agg proprs of h rsdal lf. L X b a o-gav rado varabl rprsg lf of a copo. L F b h clav dsrbo fco of X ad b h rlablfco or h srvval fco of X. Th s h srvval fco of a of ag. Evdl, a sd of h phoo of agg shold b o ad fcos rlad o hs. Ths. F s sad o b F 2 f l s cocav, whr s h ds corrspodg o. 2. F s sad o hav crasg dcrasg falr ra, f s dcrasg crasg. If F s absoll coos wh ds f, h F s h class f / s crasg dcrasg. 3. F s sad o hav crasg dcrasg falr ra avrag, f / s crasg dcrasg. 4. F s sad o hav w br wors ha sd, NBU NWU f for,. 5. F s sad o hav crasg dcrasg a rsdal lf, DMRLIMRL f h a rsdal lf / s dcrasg crasg assg ha h a ss. 6. F s sad o hav w br wors ha sd pcao, NBUENWUE f for all. Th cha of plcaos bw hs classs of dsrbos s IFRA NBU NBUC NBUCA F 2 IFR DMRL NBUE NBUE GNBUE NBAFR NBUFR NBU NBUA

3 Class lf dsrbos 68 Th rvrs plcaos ar o r, for corapl; s Brso ad Sddq 969. So sos of hs classs of dsrbos ar coad Klfso 98, 983, Shadad Shahar 994, Dshpad al. 986, Aboaoh ad Ahd 988 ad Loh 984. L ad Kochar2 sdd so proprs of class NBU2. Th a h of hs s o frhr vsga h class. I sco 2 dfos ad so basc rsls ar rodcd. Sco 3 clds closr proprs dr covolo ad hoogos osso shoc odl. 2. Dfos ad Basc Rsls Dfo2..A lf dsrbos F.. F- s calld w br ha sd a hrd ordr f F dd F F dd 2. Dfo2.2.A dscr dsrbo,, s sad o b dscr NBU3 f, Covolo 3. Th Closr roprs As a pora rlabl oprao, covolo of lf dsrbo of a cra class s of pad ch ao. Th closr proprs of IFR, NBU, NBUE, ad IFRA ca b fod Barlow ad roscha 98. Th closr proprs of class NBU2 wr pod o ad X 22. I h hor w sablsh h closr propr of h NBU3 class dr h covolo oprao. Thor 3. Sppos ha ad ar wo dpd NBU3 lf dsrbos. Th hr covolo s also NBU3 roof F F z df z 3. 2

4 682 Z. M. Nofal B grag boh sds wc frs wh rspc o, scod wh rspc o h, F dd z dddf z, b Th, F F F F z df z dd F z dd F F z dd. F dd F Thor 3.2 F z dd F dd F z dddf z Sppos ha ad ar wo dpd dscr NBU3 lf dsrbos. Th hr covolo s also NBU3. roof,, 3.2 B sao rspc o, h,,

5 Class lf dsrbos 683 Whr 3.2.Shoc odls Sppos ha a dvc s sbcd o shocs occrrg radol as vs a osso procss wh cosa s. Sppos frhr ha h dvc has probabl of srvvg h frs K shocs. Th h srvval fco of h dvc s gv 3.3 For h dscr dsrbo,,, s wll ow ha proprs of ar rflcd h corrspodg proprs of h coos lf dsrbo.ths s show b Esar al. 973 for IFR, IFRA, DMRL, NBU ad NBUE classs, Klfso 98 for NBUE ad Aboaoh ad Ahd 988 for NBUFR Dfo 3.3 A dscr dsrbo or s srvval probabl fco wh f a s calld dscr NBU3 f for all,, 3.4

6 684 Z. M. Nofal Thor 3.4 Th srvval fco 3.3 s NBU3 f ad ol f has h dscr NBU3 propr roof L b h probabl ha h dvc srvvs h frs K shocs, whr Th srvval fco s dd dd dd dd dd dd dd dd d d d d d 2 W ow ha, h

7 Class lf dsrbos F dd 2 d d d d d d d dd Rfrcs. A. Aboaoh ad A. Ahd, Th w br ha sd class of lf dsrbo, Adv. Appl. robab., 2988, A. Nada,. Sgh, N. Mraad. al,rlabl proprs of rvrsd rsdal lf, Cocaos Sascs, ar A - Thor ad Mhods, 3223, B. Klfso,NBUE srvval dr so shoc odls, Scad. J. Sas., 898, B. Klfso, A sfl agg propr basd o h Laplac rasfors, J. Appl. robab., 2983,

8 686 Z. M. Nofal 5. F. Blzc, M. Fracoad J.M. Rz, O agg proprs basd o h rsdal lf of -o-of- sss, robabl h Egrg ad Iforaoal Sccs, 3999, F. Blzc, E. Oraga ad J.M. Rz, A o o sochasc coparsos of css lfs of rwal procsss, J. Appl. robab, 382, I. Ahad, A. Ahd, M.Kad ad I. El Baal,A agg oo drvd fro h crasg cov ordrg h NBUCA class, Joral of Sascal lag ad Ifrc, 3626, J. Cao ad Y. Wag,Th NBUC ad NWUC classs of lf dsrbos, J. Appl. robab., 2899, J.D. Esar, A.W. Marshall adf. roscha, Shoc odls ad war procsss, A. robab., 973, J.V. Dshpad, S.C.Kochar ad. Sgh, Aspcs of posv agg, J. Appl. robab., 23986, M.C. Brso ad M.M.Sddq,So crra for agg, J. Ar. Sas Assoc., 6499, M. Fraco, J.M. Rz ad M.C. Rz, O closr of h IFR2 ad NBU2 classs, Joral of Appld robabl, 382, M. d ad A.Mashhor,Closr of h NBUC class dr forao of paralll sss, J. Appl. robab.,3993, M. Shadad J.G. Shahar,Sochasc Ordrs ad Thr Applcaos.Acadc rss, Nw Yor994.

9 Class of lf dsrbos R.E. Barlow adf. roscha,sascal Thor of Rlabl ad Lf Tsg. To Bg wh, Slvr Sprg,MD, T. ad. X, roofs of h closr propr of NBUC ad NBU2 dr covolo, J. Appl. robab.,3922, W.Y. Loh, A w gralzao of h class of NBU dsrbos, IEEE Tras. Rlabl, Dc, vol , pp X.. L ad S.C.Kochar, So w rsls volvg h NBU2 lass of lf dsrbos, J. Appl. robab.,382, X. L ad M. Zo, rsrvao of sochasc ordrs for rado a ad aa, wh applcaos, NavalRs. Logs., o appar, X.L, Z. L ad B. Jg,So rsls abo h NBUC class of lf dsrbos, Sas. robab. L., 462, Y. Ch,Classs of lf dsrbos ad rwal cog procss, J. Appl. robab., 3994, 5. Rcvd: Spbr, 22