96
7 S c a le E st im a tio n
7.1
M -e st im a to rs o f S c a le
aTheScaleModel
X=σZ,Z∼F1
F1:Distributionwithscaleparameter1.Definesσ.
LogScaleTransformtheobservationandtheparameter!
logh|X|i=eµ+logh|Z|i,eµ=loghσi,logh|Z|i∼ eF0Locationmodel. eF0long-tailedtotheleft.Density: ef0hzi=2cexphe zie z,c=(2π) −1/2.
97
z
Density
−7−6−5−4−3−2−101
0.0 0.1 0.2 0.3 0.4 0.5
98.1 cM-Estimators(backtoscale,unlogged)Max.li.forσ,F1=Φ=Nh0,1i:X
i (xi/bσ) 2−1 =0bσ 2= 1n X
i x 2i
Cutlargeterms!X
i ψhxi/bσi 2−κ =0
ψ:Huber’sψ.General:X
i χhxi/bσi=0, Rχhx/bσhFiidFhxi=0
κ?
99 WewantFisherconsistency!bσhFσi=σRχhx/bσidFσhxi= RχhzidF1hzi=0Rψ 2hzi−κ dF1hzi=0
−→κ= Rψ 2hzidF1hzi...usuallyforF1=Φ.
100.1
dInfluenceFunctionGeneralresultforM-estimators:
IFhx;Fi= 1c ψhx,θic=− Z∂∂θ ψhx,θifθ hxidx ψhx,θi=χhx/σi−→ ∂∂θ ψhx,θi=−χ ′hx/σix σ 2
IFhz;F1i= 1c χhzic= Rχ ′hzif1hzidz IFhx;Fσi=σIFhx/σ;F1i
1017.1
eWeightedScaleNew!
χhzi=ω 2hziz 2−κ κsuchthatRω 2hziz 2−κ dF1hzi=0
Advantage:Allowsforeasydefinitionofredescenders.
102.1
fLogScaleAgain:Transformtheobservationandtheparameter!
logh|X|i=µ+logh|Z|i,µ=loghσi,logh|Z|i∼ eF0
Locationmodel. eF0long-tailedtotheleft.
gMax.li.for eΦ?
OptimalEstimators?−→Exercise!
103
7.2
O th e r S c a le E st im a to rs
aInter-QuartileRangeIQR=γ(q0.75−q0.25)=γ(F −1h0.75i−F −1h0.25i) γtomakeitFisher-consistent,γ −1=Φ −1h0.75i−Φ −1h0.25i=1.35.
bMADMedianAbsoluteDeviation
MAD=γmedianh|X−medianhXi|i
γtomakeitFisher-consistent,γ −1=Φ −1h0.75i=0.6745.γ=1.483.
104 .2 cQnRousseeuw&Croux(1993).Basicidea:“Elementalsubsets”.Minimalsetofobs.thatallowsto“estimate”ascale:Pairsofobservations.bσij=|xj−xi|.Askfor“robustconsensus”.
bσ=γmediani<jh|xj−xi|i
Breakdownpoint=0.29.−→Usealowerquantiletogetbreakdownpoint0.5!Leth=[n/2]+1,
Qn=γ(|xj−xi|)[k] ,k= h2 ≈ n2 /4 γ −1= √2Φ −1h5/8i,γ=2.222.
105
7.3
L o c a tio n a n d S c a le
aThemodel
X=µ+σZ,Z∼F0
Fhx;µ,σi=Fh(x−µ)/σ;0,1i=F0h(x−µ)/σi fhx;µ,σi= 1σ fh(x−µ)/σ;0,1i= 1σ f0h(x−µ)/σi
= 1cσ exph−ρh(x−µ)/σii
NormaldistributionF0=Φ:ρhzi=z 2/2
106
Twoparameters−→M-estimators:2equations,simultaneous
•Max.li.forlong-taileddistribution:MinimizeX
i (−loghσi−ρh(xi−µ)/σi)orsolve2scoresequations.Letψ=ρ ′.
ψhx;µ,σi=sµhx;µ,σi= 1cσ ψhziz=(x−µ)/σ χhx;µ,σi=sσhx;µ,σi= 1cσ ψhzi·z− 1σSolve Pi ψh(xi−µ)/σi=0, Pi χh(xi−µ)/σi=0
107
Ifψismonotone,ψhzi·z→∞forz→∞−→notrobustIfψhzi·zbounded−→ψ“redescending”−→mayleadtonon-uniquesolutionsorinefficientest.ofµ
108
•Proposal2(Huber,1964)X
i ψh(xi−µ)/σi=0X
i ψ 2h(xi−µ)/σi=n Rψ 2hzidΦhzi
NolongerMax.li.foranydistribution.−→“ψstyle”ofM-estimatorsismoreflexible!
•Useweightedscale!χhzi=ω 2hzi(z 2−κ)
1097.3
cExperienceofPrincetonsimulationstudy(1970):Needforscaleestimationcanseriouslyaffecttherobustnesspropertiesofthelocationestimator
−→Iflocationisofprimaryinterest,andscaleisa“nuisance”parameter,thenestimatescaleasrobustlyaspossible−→MAD
Whenitcomestotesting,e.g.µ=µ0,ascaleest.isneeded.Shouldbeefficientforachievingagoodpower!−→Betterscaleestimatorneeded.Weightedscale.−→later!(whenwetreatregression)
