GOODNESS OF FIT TEST FOR THE PROPORTION OF SUCCESSES IN BINOMIAL TRIALS AND CONFIDENCE INTERVAL VIA COINCIDENCE: A CASE OF RARE EVENTS

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Cleopatra Bryant
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1 IJRRAS 30 Frury df GOODNESS OF FIT TEST FOR THE ROORTION OF SUESSES IN BINOMIAL TRIALS AND ONFIDENE INTERVAL VIA OINIDENE: A ASE OF RARE EVENTS Vctor Njmr School of Mthmtc d Stttc rlto Uvrty Ottw ON KS 5B6 d vctorjmr@gml.com ABSTRAT I th r w df th cocdc th cotxt of oml trl. W codr th ull hyoth H 0 : = = = = θ gt th ltrtv hyoth H : θ for om = whr th rolty of ucc or roorto ch rformd xrmt d θ rl cott. W lo codr tht θ mll θ 0 d th umr of xrmt lrg d m u of oo lmt Thorm to tlh tttcl tt to xm th hyoth. W how tht f th ull hyoth ot rjctd th mot lly th cocdc xctd to occur d thrfor w comut cofdc trvl for θ trm of th grld hyrgomtrc fucto cl fucto ug th vrc of th cocdc or v cocdc. Th rult r lo wrtt trm of lmtry fucto ug th ymtotc xo of th hyrgomtrc fucto. Th otd rult c for xml ud formto rtrvl hlth cr turl lgug rocg qulty cotrol dutr tc. Kyword: ocdc oo lmt thorm Hyoth tt ofdc trvl Hyrgomtrc fucto ymtotc vluto. INTRODUTION A dcrt rdom vrl hvg oml dtruto wth rmtr d dotd ~ d t rolty m fucto.m.f gv y 0 whr th umr of xrmt rformd th umr of ucc or roorto th rolty of ucc y gv rformc d!!! rrt ll ol comto orvd th outcom. It c rdly how tht th m of whl th vrc. Th c foud y oo o th c of rolty d tttc [9]. Th oml dtruto h my lcto cc ocl cc hlth cr d grg [45690]. Iformto rtrvl turl lgug rocg Dug [4] whch r tchqu frqutly ud to m trcto tw hum d comutr ffct d qulty cotrol r tycl xml mog othr lcto of th ltr dtruto [5]. To ot mgful rult whvr th oml dtruto ld utl frtl d tttcl roch mut crfully ud. Th m quto : I th tmt tmtd vlu for th rmtr cctl to jutfy th u of th oml dtruto modl d t wht xtt?. To dqutly wr th quto o hould frt rform good of ft tt d th cotruct cofdc trvl. Th mthmtcl d tttcl ly dvlod th r c ld cc ocl cc hlthcr d grg. I th rt tudy w wll focu o txt ly whch morttly ud formto rtrvl d turl lgug rocg [40] ordr to mlfy th dcrto of our mthod. 33

2 IJRRAS 30 Frury 07 Njmr Good of Ft Tt Amog dffrt tttcl tt tht my coductd txt ly th Llhood Rto Tt whch d o mxmzg th llhood fucto []. A good dcrto of th tt tht cot of comrg two oulto roorto d txt ly c foud Dug [4]. O th othr hd Wll [0] cotructd Wld cofdc trvl Wlo cor cofdc trvl d lor-ro trvl for th rmtr rlvt to txt ly d rformd good of ft d cotgcy tt to vlut th trvl. Dug [4] uggtd tht th u of oo dtruto my rovd om ft o o hd wh mll d Wll [0] th cocluo of h vtgto o othr hd uggtd tht for wd rmtr d for lrg 0 oo dtruto would chv ttr rult. Bfor w rocd to th m of th wor w hould frt gv hort dcrto of oo dtruto t mortt tool tht w r gog to u. A dcrt rdom vrl oo dtrutd wth rmtr λ f t rolty m fucto.m.f [9] 0!. 3 It dotd ~oλ d t h vry trtg rorty tht oth th m d vrc of th rdom vrl r qul d r gv y λ. I th r w codr mor grl tuto wth ddt tud whch xrmt r rformd wth rolt of ucc 3. Ad w r trtd tlhg tt tttc for vlutg th ull hyoth gt th ltrtv hyoth H 0 : = = = = θ 4 H : θ for om = 5 whr θ om rl umr. Thu th frt gol of th wor to tlh good of ft tt to xm th hyoth d th cod gol to comut cofdc trvl I for th rmtr θ f thr o vdc to rjct th ull hyoth H 0. W wll codr tht th rmtr θ mll θ 0 d cott d uggtd Wll [0]. I tht c oo dtruto vry good roxmto for th oml dtruto [9]. Th rorty ow oo lmt thorm or th lw of rr vt [9]. Ug th rorty w wll how hortly cto tht th hyoth 4 d 5 r quvlt to th hyoth d H 0: λ = λ = = λ = θ 6 H : λ θ for om = 7 whr λ = r rmtr of om oo dtruto d θ om rl cott. Oc th do w wll th ly ow rult for oo dtruto Njmr [7] to ot w rult for th oml dtruto. For tc th c of oo dtruto Njmr [7] tlhd χ good of ft tt to xm th hyoth 6 d 7 d cotructd 00 α% cofdc trvl I for θ ug th vrc of th cocdc tht w wll df ltr cto 3 th cotxt of oml trl. I th r w good of ft tt to xm th hyoth 4 d 5 crrd out d hc w 00 α% I for θ otd ug th vrc of th cocdc Njmr [7].. AROIMATION OF THE BINOMIAL DISTRIBUTION BY OISSON DISTRIBUTION I th cto w dcr th roxmto of th oml dtruto y oo dtruto. Ad w how tht th hyoth 4 d 5 d tho 6 d 7 r quvlt. I fct th ol wh th rolty of ucc mll d th umr of trl lrg. 34

3 IJRRAS 30 Frury 07 Njmr Good of Ft Tt Thorm. oo lmt thorm A 0 d uch tht th m vlu = λ cott th roxmto 0! 8 hold. Th roof c foud [9] w rt t hr. roof.. 9! Ad f lrg!!!! 0 whch oo dtruto rolty m fucto c λ d. Morovr w orv tht f 0 d th th vrc of roxmtd y whch th m of d tht dtruto who m d vrc r qul oo dtruto cto or []. Hc udr th umto th rdom vrl oo dtrutd wth m λ = ~oλ =. Lt u ow codr tht ~ =. If = for ll = th w hv from 6 d 7 tht d H 0: = = = = θ H : θ for om =. 3 Th gv d H 0: = = = = θ = θ 4 H : θ = θ for om = 5 whch r xctly 4 d OINIDENE ROBABILITY AND MOMENTS I th cto th rolty of th cocdc d th momt octd wth th cocdc r drvd trm of th hyrgomtrc fucto followg Njmr [7]. But for th drvto w hould frt df th cocdc th cotxt of oml trl d th grlzd hyrgomtrc fucto. 35

4 IJRRAS 30 Frury 07 Njmr Good of Ft Tt 36 Dfto. Lt ~ = ddt d dtclly dtrutd d. Th cocdc wll occur f ftr coutg th umr of ucc th m ll c. Thu th cocdc gv y 0 } {. 6 Dfto. Th grlzd hyrgomtrc fucto cl fucto gv y th owr r [8]! ; ; 0 x x F q q q 7 whr d q r rtrry cott γ = Γγ+ Γγ for y comlx γ wth γ 0 = d Γ th gmm fucto. Thorm. Lt = for ll = Dfto. Th udr H 0 d θ 0 d th rolty of th cocdc roxmtd y ; ; F. 8 roof. Th jot.m.f of = = = th multoml.m.f. 9 If = for ll. 0 Th 0. Ug 4 d 4 d ly Thorm yld 0 0! ] [ lm whr θ = θ. Now ug Thorm Njmr [7] gv ; ;! ] [ 0 0 lm F. 3 Hc uttutg c θ = θ gv. ; ; F 4 Th d th roof.

5 IJRRAS 30 Frury 07 Njmr Good of Ft Tt Thorm 3. Udr H 0 d θ 0 d th γ th momt μ γ octd wth th cocdc roxmtly gv y F ; ; 5 whr γ = 3. Ad th vrc roxmtd y F ; ; F ; ;. 6 roof. Udr H 0 d θ 0 d w hv lm [ ] 0 0! 7 whr for θ = θ. Followg Lmm [7] yld lm [ ] F ; ;! Hc uttutg c θ = θ gv whch xctly 5. Now w c ly Thorm 3 [7] to ot F ; ; 9 d F ; ; 30 F ; ;. 3 Hc F ; ; F ; ;. 3 Th comlt th roof. 4. GOODNESS OF FIT TEST AND ONFIDENE INTERVAL I FOR θ VIA OINIDENE I th cto good of ft tt to xm th hyoth 4 d 5 otd. Ad f thr o vdc to rjct th ull hyoth H 0 4 th 00 α% I for θ cotructd ug th trl Lmt Thorm LT. Thorm 4. If = for ll = Dfto θ 0 d d thr o vdc to rjct th ull hyoth H 0 th 37

6 IJRRAS 30 Frury 07 Njmr Good of Ft Tt H 0 : tru 33 whr gv y 4. roof. If = for ll = w hv H 0 : tru whr gv y.4 f θ 0 d Thorm. Th d th roof. 34 O my udrtd Thorm 4 th wy. If th ull hyoth ot rjctd w hll xct mor d mor cocdc to t lc w rform mor d mor xrmt. Morovr f th ull hyoth H 0 4 ot rjctd th mot lly th cocdc xctd to h Thorm 4 d hc th vrc of tht of th cocdc σ gv Thorm 3. Ad f θ 0 th y th trl Lmt Thorm [3] w hv whr σ gv Thorm 3. ~ N 35 Lt ow th ml tmt for. Ad lt Thrfor or F ; ; ˆ ˆ ˆ ˆ. 36 ˆ ˆ ~ N0 37 F ; ; ~ N0. Hvg md tht Z~N0 w c ow coduct tt followg. Th ull hyoth H 0 wll rjctd f d or ˆ F ; ; F ; ; Z

7 IJRRAS 30 Frury 07 Njmr Good of Ft Tt ˆ F ; ; F ; ; Z. 40 I th c th ull hyoth ot rjctd 00 α% I for θ c comutd. It fct gv y whr ˆ Z ˆ ˆ Z ˆ 4 ˆ ˆ ˆ ˆ ; ; ˆ F ˆ F ; ; ˆ 4 d θ gv y 3. I th c = σ c xrd trm of modfd Bl fucto of th frt d of ordr 0 d [7]. Sttg θ = θ quto A.67 orollry [7] w hv I 43 4 I0 whr I 0 d I r th modfd Bl fucto of th ordr 0 d rctvly []. Th gv whr θ = + /. ˆ ˆ ˆ 4 I ˆ I ˆ ˆ 0 ˆ FURTHER ASYMTOTI EENSIONS OF THE I FOR θ A mtod Dfto hyrgomtrc d Bl fucto r cl fucto. Thy hv vry trtg mthmtcl rort whch c ud to mlfy th rult cto 3. For tc f θ θ 0 o c vlut σ trm of lmtry fucto rthr th ctl fucto [8]. Th clld ymtotc vluto. Th ymtotc xro for σ wr drvd Njmr [7]. I th c > th ymtotc xro for σ gv y quto 5.39 Thorm 6 [7]. Suttut θ = θ 5.39 [7] d th uttut th rultg xro for σ 35 w ot mlr xro for 39 d 40. Hc w hould rjct th ull hyoth H 0 f 45 / / / ˆ Z 5/ / 3 4 d or / / / ˆ 5/ / 4 3 Z 46 whr for θ gv y 36. If y th tt w c ot rjct th ull hyoth w c th cotruct 00 α% I for θ ug 4 d whr 39

8 IJRRAS 30 Frury 07 Njmr Good of Ft Tt / 3 ˆ 5/ 4 / / / ˆ ~ ˆ. 47 I th c = th ymtotc xro for σ gv y quto A.73 Thorm 8 [7]. Suttut θ = θ A.73 [7] d th uttut th rultg xro for σ 35 w ot mlr xro for 39 d 40. Hc th ull hyoth H 0 houd rjctd f ˆ Z 3/ 48 / 4 d or ˆ Z 3 / / 4 whr for whr θ = + /. If thr o vdc to rjct t o my cotruct 00 α% I for θ ug 4 d whr ˆ 3/ ˆ. ˆ ~ / DISUSSION AND ONLUSION W dfd th cocdc Dfto d comutd t rolty of occurrc trm of th hyrgomtrc fucto ug oo lmt thorm Thorm. W hv ud oo lmt thorm to xr th vrc d momt of th cocdc trm of hyrgomtrc fucto Thorm 3. W furthr howd tht th rolty tht H 0 tru qul tht of th cocdc Thorm 4. Th c udrtood th wy. If th ull hyoth tru cot rjctd th mor cocdc wll occur w rformg th xrmt my tm. Thrfor th vrc of = gv y th vrc of th cocdc. I th c o my u th LT to tlh tttcl tt dcrd cto 4. Hyrgomtrc d Bl fucto r cl fucto hvg trtg mthmtcl rort tht d om udrttg of rgorou mthmtc. For mlfcto uro ymtotc xo of th o-lmtry fucto wr ud to xr th vrc of th cocdc trm of lmtry fucto cto 5. Th outcom of th wor c for tc ld to chv ttr rult hlth cr comuttol lgutc qulty cotrol comutr cc d o o. 7. REFERENES [] M. Armowtz I.A. Stgu Hdoo of Mthmtcl Fucto wth Formul Grh d Mthmtcl Tl. Nt. Bur. Std 964. [] M.H. DGroot M.J. Schrvh rolty d Stttc 3 rd Ed. Addo-Wly 00. [3] G. ll R.L. Brgr Stttcl Ifrc d Ed. Duxury 00. [4] T. Dug Accurt Mthod for th Stttc of Surr d ocdc omuttol Lgutc 9 Iu [5] Egrg Stttc Hdoo. [6] M.O. Flt B. Lv Stttc for Lwyr 3 rd Ed. Srgr 05 [7] V. Njmr ocdc Good of Ft Tt d ofdc Itrvl for oo Dtruto rmtr v ocdc Amrc Jourl of Ald Mthmtc d Stttc 4 Iu [8] NIST Dgtl Lrry of Mthmtcl Fucto. htt://dlmf.t.gov/5 [9] A. oul S.U. ll rolty Rdom Vrl d Stochtc roc. 4 th Ed. McGrw Hll 00. [0] S. Wll Boml ofdc Itrvl d otgcy Tt: Mthmtcl Fudmtl d th Evluto of Altrtv Mthod Jourl of Qutttv Lgutc 0 Iu

ERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca**

ERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca** ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults