CHAPTER 7. X and 2 = X
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1 CHATR 7 Sco d r usd smors o. Th vrcs r d ; comr h S vrc hs cs / / S S Θ Θ Sc oh smors r usd mo o h vrcs would coclud h s h r smor wh h smllr vrc. 7-. [ ] Θ [ ] Θ ] [ 7 6 Boh d r usd sms o sc h cd vlus o hs sscs r Θ Θ quvl o h ru m Θ 7 Θ Θ Θ Sc oh smors r usd h vrcs c comrd o dcd o slc h r smor. Th vrc o s smllr h h o s h r smor. Θ Θ Θ 7-. Sc oh d r usd h vrcs o h smors c md o drm h r smor. Th vrc o s smllr h h o hus my h r smor. Θ Θ Rlv ccy 5. 4 Θ Θ Θ Θ S S 7-
2 7-4. Sc oh smors r usd: S Θ Θ Rlv ccy S Θ Θ S Θ Θ S Θ Θ Θ Θ / Bs Θ Θ Θ 4 or usdss us Θ sc s h oly usd smor. As or mmum vrc d ccy w hv: Θ Bs Rlv ccy whr s or s. Θ Bs Thus 4 Rlv ccy 6 4 I h rlv ccy s lss h or qul o Us Θ wh / 7 / or Θ s h r smor. I < < h us Θ. or usdss us Θ. or ccy us Θ wh or d us Θ wh < < Θ o s Θ S Θ Θ o s Θ S Θ Θ Bs S Θ 6 [o h hs cluds s ] To comr h hr smors clcul h rlv ccs: S Θ. sc rl.. > us Θ s h smor or S Θ S Θ S Θ 6 sc rl.. > us Θ s h smor or S Θ.8 sc rl.. > us Θ s h smor or S Θ 6 Cocluso: s h mos c smor wh s u s sd. Θ s h s usd smor. Θ 7-
3 Show h S s usd: S S S S S S S S s usd smor o Show h s sd smor o : s sd smor o. Bs c Bs dcrss s crss. 7- Show h s sd smor o. Usg [ ] Thror s sd smor o. Bs c Bs dcrss s crss. 7- Th vrg o h 6 osrvos rovdd c usd s smor o h m ull orc cus w kow s usd. Ths vlu s ouds. Th md o h sml c usd s sm o h o h dvds h oulo o wk d srog hl. Ths sm s 75. ouds. 7-
4 c Our sm o h oulo vrc s h sml vrc or.78 squr ouds. Smlrly our sm o h oulo sdrd dvo s h sml sdrd dvo or.655 ouds. d Th smd sdrd rror o h m ull orc s.655/6 /.5. Ths vlu s h sdrd dvo o o h ull orc u o h m ull orc o h sml. Oly o cocor h sml hs ull orc msurm udr 7 ouds. Our o sm or h rooro rqusd s h / Dscrv Sscs rl d Tr SDv S Od Thckss Th m od hckss s smd y rom h sml s 4. Agsroms. Sdrd dvo or h oulo c smd y h sml sdrd dvo or 9.8 Agsroms. c Th sdrd rror o h m s.85 Agsroms. d Our sm or h md s 44 Agsroms. Sv o h msurms cd 4 Agsroms so our sm o h rooro rqusd s 7/ o hs rcs ws chgd rg o h ook. Th rcs h orgl rgs volvd cocs o covrd sucly h ook h dsruo o h sml mmum d mmum. Th ollowg soluo s or rg. d d d h sml vrg o h osrvos h rdom sml. W kow h h m o h dsruo. Howvr h m o h dsruo s / so s usd smor o. 7.4 W kow h h vrc o s so s sdrd rror mus sm hs rmr w would susu our sm o o.. To
5 .. CO s Ths sdrd rror could smd y usg h sms or h sdrd dvos o oulos d. 7-6 [ ] [ ] S S S S S s c Th vlu o lh h mmzs h sdrd rror s: d Wh 4 d h vlu o lh o choos s 8/9. Th rrry vlu o.5 s oo smll d wll rsul lrgr sdrd rror. Wh 8/9 h sdrd rror s / 9 8 /.. s I.5 h sdrd rror s s 7-8 c A sm o h sdrd rror could od susug or d 7-5
6 or h quo show. d Our sm o h drc rooros s. Th smd sdrd rror s.4 7-6
7 Sco 7-!!! 7-9. d d l! l l l l 7-. or d d / l l l h or co smd usg quos sc d d. Thror s smd usg. s mmzd m d m ml: Cosdr rc low d l h m h hs lsd w o cr ssg d o d h s h h cr gs o ss h o cosdrd m hdwy. Ths hdwy c modld y h shd ol dsruo. 7-7
8 ml Rlly: Cosdr rocss whr lurs r o rs. Suos h u s u o oro u o lurs wll occur ul rod o oro. lurs wll occur oly r h m. 7-. l l l l l l l l l l... l l l l 7-8
9 7-. [ ] l l l l l l l l Uo sg l qul o zro w o / d Uo sg l qul o zro d susug or w o l l l l l l l l d l l c umrcl ro s rqurd. 7-4 Th m o s clculd rom grl o h dsy uco o Solv hs or d susu o o 7-5 hror: Th cd vlu o hs sm s h ru rmr so mus usd. Ths sm s rsol o ss cus s usd. Howvr hr r ovous rolms. Cosdr h sml d. ow 4.7 d Ths s ursol sm o cus clrly. 7-9
10 7-6 co usd sc wll lwys lss h. s. c y d Yy y y. Thus y s s gv. Thus sy or y > > so h vrc o â s lss h h o â. I s hs ss h h scod smor s r h h rs. 7-8 l l l l Sg h ls quo qul o zro d solvg or h ylds 7-9 so Θ l l l l Sg h ls quo qul o zro h mmum lklhood sm s Θ d hs s h sm rsul od r 7-
11 c l.5.5 / d W c sm h md y susug our sm or h o h quo or. 7- c c c d c so h h cos c should qul.5 c d l l l l By sco h vlu o h mmzs h lklhood s m 7- Usg h rsuls rom ml 7- w o h h sm o h m s 4. d h sm o h vrc s
12 - log Sd. Dv Th uco sms o hv rdg d s curvur s o oo rooucd. Th mmum vlu or sd dvo s 9.8 lhough s dcul o s o h grh. 7- Wh s crsd o 4 h grh looks h sm lhough h curvur s mor rooucd. As crss s sr o drm h whr h mmum vlu or h sdrd dvo s o h grh. Sco / 9 /./ ~ 5 5 [.8 ] / s
13 / s rducd y.99 s Assumg orml dsruo / s 5 6 sdrd rror o ~ 7-
14 7-4 Y 6 Y Y Y ~ 5 44 romly usg h crl lm horm / 6 / 6 z / Usg h crl lm horm:. <.5..5 < < < / / < <.674 <.674 < mus mus ms Usg h crl lm horm s romly ormlly dsrud. 8. < < < < < < <.4.4 < 8.4 < c < 6 < <
15 > 9 ~ 7 ~ 8 ~ ~ > > I B A h B A s romly orml wh m d vrc B A Th.5.5 > > > B A.48 Th roly h B cds A y.5 or mor s o h uusul wh B d A r qul. Thror hr s o srog vdc h B s grr h A Assum rom orml dsruos. hgh hgh low ~ 6 55 low ~ Sulml rcss π or > >... > or ~ 5 5 ~
16 ~ / / o cus Crl m Thorm ss h wh lrg smls s romly ormlly dsrud. 7-5 Assum s romly ormlly dsrud / > /6 5 5 / s z > z ~. Th rsuls r vry uusul Boml wh qul o h rooro o dcv chs d ] [ 7-6
17 7-58 l l l l kg h ls quo qul o zro d solvg or h w o: Θ s h mmum lklhood sm l l l l l mkg h ls quo qul o zro d solvg or h w o h mmum lklhood sm. l Θ 7-6 l l l l l mkg h ls quo qul o zro d solvg or h rmr o rs w o h mmum lklhood sm. l Θ 7-7
18 ] [l l l l l d l u l d d dv h d l 7-8
19 d-dg rcss 7-6 / Bcus s o qul o r o dd. d 7-6 / / ] / [ Γ Γ c Wh c.8. Wh 5 c.5. So S s ry good smor or h sdrd dvo v wh rlvly smll sml szs r usd. 7-9
20 7-6 Ths rcs ws chgd rg d hs soluo s or h ls rcs. Y ; so s. *. Th lklhood uco s * z * * π * π / z * Th log-lklhood s l[ * ] π * * dg h mmum lklhood smor: Bu * z * dl[ * ] 4 z d * * * * so h s z y y y z Y Y Y Y [ Y Y ] [ ] 7-
21 So h smor s usd c c rom Chyshv's quly. Th c c <. Gv ε d c c chos sucly lrg h h ls roly s r d c s qul o ε or ] [... or ] [... > > Th ] [ ] [ ] [ 7-66 ] [ cus -. ] [ 7-67 [ ] Φ. rom rcs 7-65 [ ] { } [ ] { } π Φ π Φ rom rcs 7-65 ] [ ] [ 7-
22 7-69 rom rcs 7-65 or I Y h y y y. Th Y I Y h y y. Th Y. Y y dy rom rcs 7-65 or. Y y y dy whr gro y rs s usd. Thror [ ] d [ ] 7-7 k [ ] k k Thror k 7-7 Th rdol sm o h sdrd dvo S s.6. Th m o h sml s.4 so h vlus o corrsodg o h gv osrvos r d.57. Th md o hs w qus s.57 so h w sm o h sdrd dvo s.8; slghly lrgr h h vlu od wh h rdol smor. kg h rs osrvo h orgl sml qul o 5 roducs h ollowg rsuls. Th rdol smor S s qul o.9. Th w smor rms uchgd. 7-7 T r... r... r r... r r Bcus s h mmum lm o ms. Th s h mmum lm o - ms rom h mmorylss rory o h ol d. Smlrly k k. Th k r r T T r... d r r r T / / r r r s rld o h vrc o h rlg dsruo r /. Thy r rld y h vlu /r. Th csord vrc s /r ms h ucsord vrc. 7-
23 Sco 7.. o CD S7- rom ml S7- h osror dsruo or s orml wh m / / d vrc / / /. Th Bys smor or gos o h s crss. Ths ollows sc / gos o d h smor rochs h s ccl. Thus h lm. S7- Bcus π d or h jo dsruo s π or - < < d. Th π d d hs grl s rcogzd s orml roly. Thror [ Φ Φ ] whr Φ s h sdrd orml cumulv dsruo uco. Th [ ] π Φ Φ Th Bys smor s [ ] Φ Φ d π ~. v -. Th dv - d d [ ] [ ] [ ] [ ] Φ Φ Φ Φ Φ Φ Φ Φ dv v dv v v v π π ~ w v. Th dv dv dw v v ] [ ] [ d [ ] Φ Φ Φ Φ π π w dw ~ 7-
24 S7- or d! Th m m m m Γ m! m m m m or >. Γ m m Ths ls dsy s rcogzd o gmm dsy s uco o. Thror h osror dsruo o s gmm dsruo wh rmrs m d m. Th m o h osror dsruo c od rom h rsuls or h gmm dsruo o m m m m [ ] 9 S7-4 rom ml S7- h Bys sm s ~ Th Bys sm rs o udrsm h m.. 5 S7-5. rom ml S7- ~ Th Bys sm s vry clos o h o h m.. S7-6 d.. Th As uco o hs s rcogzd s gmm dsy wh rmrs d.. Thror h osror ~ m or s..... Usg h Bys sm or <. d
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