(Estadística) - Pruebas de Bondad de Ajuste de Una Distribución de Poisson y Binomial Con Ji-Cuadrado

 

Ltrg. Kvæm Froóeccg

^rue`fs he `gmhfh he fduste Hkstrk`uakñm he ^gkssgm y Mgrlfc

Estf prue`f se utkckzf pfrf heterlkmfr heterlkmfr sk umf luestrf he vfcgres g`servfhgs, g`servfhgs, he fcoumf fcoumf lfmerf fceftgrkf es aglpftk`ce g mg agm cf bkpñtesks he que se extrfdg he umf pg`cfakñm he vfcgres que estæ hkstrk`ukhf mgrlfclemte g he jgrlf hksaretf. ^gr tfmtg cfs bkpñtesks sgm4  Bg4 cf hkstrk`uakñm he cgs hftgs es aglpftk`ce g prgvkeme he umf –R“ hkstrk`uakñm (Mgrlfc, Bglgoëmef g Umkjgrle, ^gkssgm, `kmglkfc).  Bf4 cf hkstrk`uakñm he cgs hftgs MG es aglpftk`ce g prgvkeme he umf –R“ hkstrk`uakñm (Mgrlfc, Bglgoëmef g Umkjgrle, ^gkssgm, `kmglkfc).

Zflgs f edelpckjkafr ec afsg he umf hkstrk`uakñm hksaretf (^gkssgm) em cf aufc se qukere heterlkmfr sk cgs hftgs prgvkemem g se fdustfm f hkabf hkstrk`uakñm. EDEL^CG. Cfs cceofhfs he cgs ackemtes fc  Hu`ei‛s Jggh Lfriet  em   em Xfccfse, Jcgrkhf. Hfhg que reakëm bf bf`khg fcoumgs prg`celfs he persgmfc, cgs oeremtes sgckaktfm cgs

servkakgs he umf elpresf he agmsuctgrìf pfrf que ces fyuhe em cf prgorflfakñm he cgs elpcefhgs he afdfs. Hespuës he revksfr ec fvfmae he cfs jkcfs em cfs afdfs, cf elpresf he agmsuctgrìf suoerkræ um prgaehklkemtg pfrf cf prgorflfakñm he cgs elpcefhgs he afdfs. Este prgaehklkemt prgaehklkemtg g se `fsf em um fmæcksks fmæcksks lftelætkag he cfs jkcfs y sñcg es fpckaf`ce fpckaf`ce sk ec môlerg he cceofhfs he ackemtes hurfmte um heterlkmfhg cfpsg he tkelpg skoue umf hkstrk hks trk`ua `uakñm kñm he ^gkssg ^gkssgm. m. ^gr ^gr tfmtg, tfmtg, fmtes fmtes he pgmer pgmer em lfrabf lfrabf ec prgaeh prgaehklk klkemt emtg g he  prgorflfakñm, bf`ræ que reagceatfr hftgs sg`re cfs cceofhfs he cgs ackemtes y refckzfr umf  prue`f estfhìstkaf pfrf ver sk es rfzgmf`ce supgmer que cfs cceofhfs he cgs ackemtes skouem umf hkstrk`uakñm he ^gkssgm. Cfs cceofhfs he cgs ackemtes f cf tkemhf se hejkmem em tërlkmgs he afmtkhfh he ackemtes que emtrfm em cf tkemhf hurfmte kmtervfcgs he 5 lkmutgs. ^gr tfmtg cf bkpñtesks mucf y fctermf em este estuhkg sgm cfs skoukemtes4 >  B  4 Cf afmtkhfh he ackemtes que emtrfm cf tkemhf hurfmte kmtervfcgs he 5 lkmutgs tkeme umf hkstrk`uakñm he prg`f`kckhfh heem ^gkssgm.

 B f4 Cf afmtkhfh he ackemtes que emtrfm em cf tkemhf hurfmte kmtervfcgs he 5 lkmutgs mg tkemem umf hkstrk`uakñm he prg`f`kckhfh he ^gkssgm. Aglg puehem hfrse auemtf, em este afsg fc estuhkg se kmteresf quehfrse agm cf bkpñtesks mucf, he cg agmtrfrkg ec fmæcksks lftelætkag seræ kmvfckhfhg y pgr tfmtg se temhræm que tglfr gtrfs lehkhfs pfrf ederaer sgcuakñm f cf prg`celætkaf (gtrg prgaehklkemtg he  prgorflfakñm). ^frf prg`fr cf supgskakñm supgskakñm he que cfs cceofhfs he cgs ackemtes ackemtes em cfs lföfmfs he cgs hìfs emtre selfmf skouem umf hkstrk`uakñm he ^gkssgm, um elpcefhg he cf tkemhf tglf umf luestrf fceftgrkf he 602 kmtervfcgs he 5 lkmutgs, em cfs lföfmfs he tres selfmfs agmseautkvfs. Hurfmte afhf umg he cgs kmtervfcgs he 5 lkmutgs que jgrlfm cf luestrf, ec elpcefhg reokstrf ec môlerg he cceofhfs he ackemtes.

 

Ltrg. Kvæm Froóeccg

^frf resulkr cgs hftgs, ec elpcefhg heterlkmf ec môlerg he kmtervfcgs he 5 lkmutgs em cgs que mg bu`g mkmoumf cceofhf, ec môlerg he kmtervfcgs he 5 lkmutgs em cgs que bu`g umf cceofhf, ec môlerg he kmtervfcgs he 5 lkmutgs em cgs que bu`g hgs cceofhfs, eta. Estgs hftgs se presemtfm em cf tf`cf4  Môlerg he ackemtes >

Jreauemakf g`servfhf 0

6 0 1 8 5 : ? 2 ;

2 6> 60 62 00 00 6: 60 : Ρ < 602

Cf tf`cf hf cfs jreauemakfs g`servfhfs em cfs 6> afteogrìfs. Fbgrf se usf cf prue`f he  `gmhfh he fduste pfrf heterlkmfr sk cf luestrf he cgs 602 cfpsgs he tkelpg jfvgreae cf bkpñtesks he que cfs cceofhfs tkemem umf hkstrk`uakñm he ^gkssgm. ^frf usfr cf prue`f ^frf prue`f he `gmhfh `gmhfh he fduste, fduste, se meaeskt meaesktfm fm agmskh agmskherfr erfr,, cfs jreaue jreauemak makfs fs esperfhfs pfrf afhf umf he cfs 6> afteogrìfs, `fdg cf supgskakñm he que cf hkstrk`uakñm he cfs cceofhfs skof umf hkstrk`uakñm he ^gkssgm. Es heakr, sk em refckhfh cfs cceofhfs he cgs ackemtes skouem umf hkstrk`uakñm he ^gkssgm, se meaesktf afcaucfr ec môlerg esperfhg he cfpsgs he tkelpg tkelpg em cgs que cceofræm aerg ackemtes, ackemtes, um ackemte, ackemte, hgs ackemtes, ackemtes, etaëterf. Cf jumakñm he prg`f`kckhfh he ^gkssgm, que yf es agmgakhf y tkeme cf jgrlf4  x ∐ γ

 γ e j   (( x )<  x !

Em estf jumakñm, γ represemtf cf lehkf g môlerg esperfhg he cceofhfs he ackemtes em cfps cfpsgs gs he 5 lkmu lkmutg tgs, s,  x   represe represemtf mtf cf vfrkf` vfrkf`ce ce fceftg fceftgrkf rkf hec môlerg môlerg he cceofhf cceofhfss he ackemtes em um cfpsg he 5 lkmutgs y ackemtes es cf prg`f`kckh prg`f`kckhfh fh he  x  cceofhfs   cceofhfs he ackemtes em  j(x)  um cfpsg he 5 lkmutgs. Es meaesfrkg meaesfrkg estklfr estklfr cf lehkf  µ agm fyuhf he cf tf`cf he jreauemakfs pgr ec lëtghg trfhkakgmfc he hftgs grhemfhgs.  Môlerg he ackemtes  x >

Jreauemakf g`servfhf Gk 0

6

2

2

0

6>

0>

1

60

1:

8

62

?0

5

00

66>

:

00

610

?

6:

660

2

60

;:

 x¿Gk >

 

Ltrg. Kvæm Froóeccg

;

:

58

Ρ < 602

Ρ

Jkmfclemte cf lehkf se g`tkeme4 :8>/ 602 < 5 ackemtes9 agm cg aufc se estklfm cfs  prg`f`kckhfhes he ^gkssgm agm cf jgrlucf oemerfc lemakgmfhf. F jkm he jfakcktfr cgs aæcaucgs cgs vfcgres he prg`f`kckhfh y jreauemakfs esperfhfs se resulem em cf skoukemte tf`cf4  Môlerg he ackemtes x ackemtes  x > 6 0 1 8 5 : ? 2 ;

Jreauemakf g`servfhf Gk 0 2 6> 60 62 00 00 6: 60 :

 j(x)

E k

>.>>:? >.>11? >.>280 >.68>8 >.6?55 >.6?55 >.68:0 >.6>88 >.>:51 >.>1:1

>.2:0 8.160 6>.?26 6?.;:2 00.8:> 00.8:> 62.?6? 61.1:; 2.15: 8.:80

Em cf tf`cf se fpreakfm cfs prg`f`kckhfhes he ^gkssgm j(x) ^gkssgm  j(x) pfrf  pfrf afhf vfcgr he x he  x99 pgr gtrg cfhg se estklfrgm cfs jreauemakfs esperfhfs. Ec aæcaucg he eccfs se hkg luctkpckafmhg afhf vfcgr he prg`f`kckhfh j(x) prg`f`kckhfh j(x) pgr  pgr ec tgtfc he jreauemakfs g`servfhfs (602), pgr edelpcg  pfrf cf prklerf E  prklerf E k < >.>>:?ß602 < >.2:0, pfrf cf seoumhf E  seoumhf  E k < >.>11?ß602 < 8.160 y fsì suaeskvflemte. Xfl`këm es klpgrtfmte lemakgmfr que em cf tf`cf bfy 1 jreauemakfs 1  jreauemakfs esperfhfs lemgres 0 f 59 59 estf es umf agmhkakñm he cfs prue`fs he abk , cfs jreauemakfs esperfhfs mg he`em ser  lemgres f 59 pfrf estf sktufakñm se he`em forupfr cfs jreauemakfs pfrf que estfs sulem 5 g læs. ^frf estg se sulfm cfs jreauemakfs, pgr edelpcg em ec afsg he cfs 0 prklerfs jreauemakfs lemgres f 5 se sulfm4 >.2:0 + 8.160 < 5.6?5 y cf ôctklf jreauemakf se sulf agm cf que fmteaehe4 8.:80 + 2.15: < 60.;;2  He koufc jgrlf se agrrkoem cfs jreauemakfs g`servfhfs (0 + 2) < 6> y (60 + :) < 62 he jgrlf que4  Môlerg he ackemtes x ackemtes  x > 6 0 1 8 5 : ? 2

Jreauemakf g`servfhf Gk 0 2 6> 60 62 00 00 6: 60

 j(x) >.>>:? >.>11? >.>280 >.68>8 >.6?55 >.6?55 >.68:0 >.6>88 >.>:51

>.>1:1 Fpckafmhg hec test ;he dk aufhrfhf:(abk 0) temelgs4

E k agrreokhf

Gk agrreokhf

5.6?5

6>

6>.?26 6?.;:2 00.8:> 00.8:> 62.?6? 61.1:;

6> 60 62 00 00 6:

60.;;2

62

 

Ltrg. Kvæm Froóeccg

0

0

 Ϗ 

<

( 6>∐5.6?5 ) 5.6?5

0

+

( 6> ∐6>.?26) 6>.?26

0

+

( 60∐6?.;:2 ) 6?.;:2

0

+

( 62 ∐00.8:>) 00.8:>

Ec estf estfhì hìst stka kag g tkem tkemee i   ‑  p  p    ‑ 6 orfhgs he ck`ertfh, hgmhe

0

+

( 00∐00.8:> ) 62.?6?

sgm m i   sg

0

+

( 00∐62.?6? ) 61.1:;

ec mô môle lerg rg he

acfskjk acf skjkafak afakgme gmess he cfs jreauemak jreauemakfs fs em este afsg sgm ;,  p  es ec môlerg he pfræletrgs elpcefhgs, aglg se elpceñ cf lehkf emtgmaes p < 6 pgr tfmtg4 ; ‑ 6 ‑ 6 < ? orfhgs he ck`ertfh y agm um mkvec hesefhg he ζ < >.>05 ec vfcgr arktkag Ϗ 0>.>05 (?) < 6:.>602. Aglg ec vfcgr Ϗ 0afcaucfh afcaucfhg g < 6>.850 y pgr tfmtg es lemgr que 6:.>602 g `kem 6>.850 3 6:.>602 mg se reabfzf cf bkpñtesks mucf y se agmacuye que cgs hftgs se fdustfm f umf hkstrk`uakñm he ^gkssgm, agm estg se hf pgr beabg que ec fmæcksks lftelætkag pgr jkcfs  pfrf cf prgorflfakñm se puehe ccevfr f af`g agm um lìmklg lìmklg he errgr hec >.>05 < 0.5 %.

 @gmhfh he fduste fduste pfrf umf hkstrk`uakñm `kmglkfc  `kmglkfc 

Agm cf `gmhfh he fduste he Ϗ 0 (g dk aufhrfhf) es pgsk`ce tfl`këm heterlkmfr sk cf hkstrk`uakñm he hftgs prgvkeme he umf jumakñm `kmglkfc. EDEL^CG.  Agm ec prgpñsktg he pcfmefr cf agmtrftfakñm, ec hkreatgr he seceaakñm he  persgmfc he cf aglpföìf pkemsf que ec prgaesg he emtrevkstfs puehe ser fprgxklfhg pgr 

umf hkstrk`uakñm hkstrk`uakñm `kmglkfc agm  p < >.8>, es heakr, agm umf pgsk`kckhfh hec 8>% he que aufcquker afmhkhftg g`temof umf afckjkafakñm pgsktkvf em aufcqukerf he cfs emtrevkstfs. Qk ec hkreatgr hesef prg`fr estf bkpñtesks f um mkvec he skomkjkafmakf he >.0>, ·añlg he`e  prgaeher7 Afckjkafakgmes pgsktkvfs  pgsk`ces em cfs tres emtrevkstfs ( x)  x) > 6 0

 Môlerg he afmhkhftgs que g`tkemem afhf afckjkafakñm. Gk 62 8? 08

1

66 Ρ < 6>>

Qe pcfmtef cfs bkpñtesks estfhìstkafs4  Bg4 cfs jreauemakfs hec môlerg he afmhkhftgs que g`tkemem afhf afckjkafakñm prgvkemem he umf hkstrk`uakñm `kmglkfc.  Bf4 cfs jreauemakfs hec môlerg he afmhkhftgs que g`tkemem afhf afckjkafakñm MG  prgvkemem he umf hkstrk`uakñm `kmglkfc. `kmglkfc. Gtrf jgrlf semakccf es4  Bg4 cgs hftgs se fdustfm f umf umf hkstrk`uakñm `kmglkfc   Bf4 cgs hftgs MG se fdustfm f umf hkstrk`uakñm `kmglkfc  `kmglkfc 

+

(6: ∐ 6

 

Ltrg. Kvæm Froóeccg

Cf jumakñm `kmglkfc tkeme cf yf agmgakhf jgrlucf4 m ∐ x

j   (( x )< mA mAxx ∛ p ∛ q  x

Xemelgs que p < >.8, em agmseauemakf, q < 6 ‑ >.8 < >.: ^frf cg aufc se estklfm cgs vfcgres he j  he j ( x)  x) pfrf afhf afsg hec > fc 1 y se tkeme cf skoukemte tf`cf4  x > 6 0 1

j(x) >.06: >.810 >.022 >.>:8

E k 06.: 81.0 02.2 :.8

Cfs jreauemakfs jreauemakfs esperfhfs Ek se afcaucfrgm agm ec tgtfc he cfs jreauemakfs jreauemakfs g`servfhfs g`servfhfs < 6> 6>>, >, pg pgrr edel edelpc pcg4 g4 >.06 >.06: : (6 (6>> >>)) < 06 06.: .:,, pf pfrf rf cf seoum seoumhf hf >. >.81 810 0 (6 (6>> >>)) < 81 81.0 .0 y fsì suaeskvflemte. Aglg mkmoumf jreauemakf esperfhf es lemgr f –5“ se puehe fpckafr ec test he abk aufhrfhg he cf jgrlf hkreatf4 0

0

 Ϗ  <

( 62∐06.: ) 06.:

0

+

( 8? ∐81.0 ) 81.0

0

+

( 08 ∐02.2 ) 02.2

0

+

( 66∐:.8 ) :.8

.:>> + >.118 + >.2>> + 1.1>: 86

Ec vfcg vfcgrr arkt arktka kag g he Ϗ 0  pfrf cf hkstrk`uakñm `kmglkfc es i ‑ 6, skemhg i < 8, em agmseauemakf 8 ‑ 6 < 1 orfhgs he ck`ertfh. Ec vfcgr arktkag Ϗ 0>.0 (1) < 8.:86:. Aglg 5.>86 = 8.:86: se reabfzf cf bkpñtesks mucf y se agmacuye que cf hkstrk`uakñm he cfs jreauemakfs mg se hkstrk`uyem em jgrlf `kmglkfc.