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Theboundarytermisnullsincef′(x)/f(x)→0as|x|→∞andgisaprobabilitydensityfunction.
Thus,
EG[HS(F,Y)]=−2
R
f′(y)g′(y)
f(y)g(y) g(y)dy+2
R
f′(y)2
f(y)2
g(y)dy−
R
f′(y)2
f(y)2
g(y)dy
=−2
R
f′(y)g′(y)
f(y)g(y) g(y)dy+
R
f′(y)2
f(y)2
g(y)dy
=
R
f′(y)2
f(y)2
−2f′(y)g′(y)
f(y)g(y) g(y)dy
39
A.10 Quadratic score
For any F,G ∈ L2(R), the expectation of the quadratic score is :
EG[QuadS(F,Y )] = ∥f∥2
2 −2⟨f,g⟩,
where ⟨f,g⟩ = R
f(y)g(y)dy.
A.11 Pseudospherical score
For any F,G ∈ Lα(R), the expectation of the quadratic score is :
EG[PseudoS(F,Y )] = −⟨fα−1,g⟩
where ⟨fα−1,g⟩ = R
f(y)α−1g(y)dy.
∥f∥α−1
α
B Expected multivariate scoring rules
B.1 Squared error
For any F,G ∈ P2(Rd), the expectation of the squared error (12) is :
,
EG[SE(F,Y )] = ∥µF −µG∥2
2 +tr(ΣG),
where µF is the mean vector of the distribution F and µG and ΣG2 are the mean vector and the covariance
matrix of the distribution G.
Proof. Let Ti denote the projection on the i-th margin.
EG[SE(F,Y )] = EG[∥µF −Y ∥2
2]
d
=EG
d
=
=
i=1
d
i=1
(µTi(F) − Ti(Y ))2
i=1
ETi(G) [SE(Ti(F),Y )]
(µTi(F) − µTi(G))2 + σ2
Ti(G)
=∥µF −µG∥2
2 +tr(ΣG)
B.2 Dawid-Sebastiani score
For any F,G ∈ P2(Rd), the expectation of the Dawid-Sebastiani score is :
EG[DSS(F,Y )] = log(detΣF)+(µF −µG)TΣ−1
F (µF −µG)+tr(ΣGΣ−1
The proof is available in the original article (Dawid and Sebastiani, 1999).
F ).
40
B.3 Energy score
In a general setting, the expected energy score does not simplify. For any F,G ∈ Pβ(Rd), the expected
energy score (13) is :
EG[ESβ(F,Y )] = EF,G∥X −Y ∥β
2−1
2EF∥X −X′∥β
2.
B.4 Variogram score
For any F,G ∈ P(Rd) such that the 2p-th moments of all their univariate margins are finite, the expected
variogram score of order p (14) is :
d
EG[VSp(F,Y )] =
Proof.
i,j=1
EG[VSp(F,Y )] = EG
=EG
d
=
wij EF [|Xi −Xj|p]2 −2EF [|Xi −Xj|p]EG[|Yi −Yj|p]+EG[|Yi −Yj|2p] .
d