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R
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(F(z)−G(z))2dz+
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R
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G(z)(1−G(z))dz
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A.6 Dawid-Sebastianiscore
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ForanyF,G∈P2(R), theexpectationoftheDawid-Sebastianiscore(9) is:
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EG[DSS(F,Y)]= (µF−µG)2
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σF2
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+σG2
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σF2
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+2logσF.
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Proof.
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EG[DSS(F,Y)]=EG
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(Y−µF)2
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σF2
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+2logσF
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=EG (Y−µF)2
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σF2
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+2logσF
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NoticingthatEG (Y−µF)2 =EG[SE(F,Y)],
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EG[DSS(F,Y)]= (µF−µG)2+σG2
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σF2
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+2logσF.
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A.7 Error-spreadscore
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ForanyF,G∈P4(R), theexpectationoftheerror-spreadscore(10) is:
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EG[ESS(F,Y)]= (σG2−σF2)+(µG−µF)2−σFγF(µG−µF) 2
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+σG2[2(µG−µF)+(σGγG−σFγF)]2
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+σG4(βG−γG2−1),
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38
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whereµF,σ2
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F,γFarethemean,thevarianceandtheskewnessoftheprobabilisticforecastF. Similarly,µG,
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σ2
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G,γGandβGarethefirstfourcenteredmomentsofthedistributionG.Theproof isavailableinAppendix
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BofChristensenetal. (2014).
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A.8 Logarithmicscore
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ForanyF,G∈P(R)suchthatFandGhaveprobabilitydensityfunctionsintheclassL1(R),theexpectation
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ofthelogarithmicscore(11) is:
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EG[LogS(F,Y)]=DKL(G||F)+H(F),
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whereDKL(G||F) istheKullback-LeiblerdivergencefromFtoGandH(F) istheShannonentropyofF.
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Theproof isstraightforwardgiventhattheKullback-LeiblerdivergenceandShannonentropyaredefinedas
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DKL(G||F)=
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R
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g(y)log g(y)
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f(y) dy;
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H(F)=
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R
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f(y)log(f(y))dy.
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A.9 Hyvärinenscore
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ForF,Gsuchthattheirdensitiesf exist,aretwicecontinuouslydifferentiableandsatisfyf′(x)/f(x)→0
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as|x|→∞andg′(x)/g(x)→0as|x|→∞, theexpectationoftheHyvärinenscoreis:
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EG[HS(F,Y)]=
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R
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f′(y)2
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f(y)2
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−2f′(y)g′(y)
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f(y)g(y) g(y)dy
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=
|
R
|
f′(y)
|
f(y) −g′(y)
|
g(y)
|
2
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g(y)dy−
|
R
|
g′(y)2
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g(y)2
|
g(y)dy
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wherethelastformulashowstheentropyoftheHyvärinenscore(secondtermontheright-handside).
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Proof.
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EG[HS(F,Y)]=E 2f′′(Y)
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f(Y) −f′(Y)2
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f(y)2
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=2
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R
|
f′′(y)
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f(y) g(y)dy−
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R
|
f′(y)2
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f(y)2
|
g(y)dy
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Integratingbyparttheintegralofthefirsttermontheright-handsideleadsto:
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R
|
f′′(y)
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f(y) g(y)dy= f′(y)
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f(y) g(y)
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+∞
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−∞
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−
|
R
|
f′(y) g′(y)f(y)−g(y)f′(y)
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f(y)2
|
dy
|
=−
|
R
|
f′(y)g′(y)
|
f(y)g(y) g(y)dy+
|
R
|
f′(y)2
|
f(y)2
|
g(y)dy
|
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