-2013qO?-u xOIK 3K OCopula{8 z 11/ 36◀◀▶▶ n ◀▶2013c5 10F←↩↪→ |'4
OutlineIK• 3K 3K OCopula{8 •a Oz 12/ 36•u Copula{◀◀▶▶•8 ◀▶←↩↪→ |'4
1 3K∙ 3 "IK[1].37K x+ 3K O[2].A ! 5! "Copula{8 z[3]. A (" ∙kJ9{13/ 36
◀◀▶▶[1].AzzaliniCapitanio J
skew-normal(SN),◀▶skew-t(ST);[2].2 /CqJscalemixture←↩↪→skewnormal(SMSN) [3].MaGenton J
flexibleskewsymmetricdistri-|bution.'4
∙ IK[1].?
a# A5
O 3K { OCopula{8 z[2].u copula?E, '5( ? \:14/ 36[3].?
{3/xN ◀◀▶▶ {A^"◀▶←↩↪→ |'4
2a O∙~^ 1Skewnormal(SN)distribution:IK(( 3K Ox−)ξφΦx−)ξp(x∣ξ,σ2,λ)=2λCopula{σσσ8 z φ(⋅)Φ(⋅)OIO ξσλ"15/ 36PX∼SN(ξ,σ2,λ)◀◀▶▶2Skewstudentt(ST)distribution:◀▶(())x−ξν+11/2←↩↪→p(x∣ξ,σ2,λ)=2t(x,ξ,σ2;ν)Tλ;νσν+Q t(⋅;ν)T(⋅;ν)Ogdν t |Q=(x−ξ)2/σ2"aqSTN"'4
3Scalemixtureskewnormal(SMSN)distribution:CU
LX=ξ+κ1/2(U)ZIK 3KZ∼SN(0,σ2,λ),κ(⋅) \ Uk OHCopula{(⋅;ν) Cν{"8 zP X∼SMSN(ξ,σ2,λ,H)16/ 364Flexibleskew-symmetricdistribution:◀◀▶▶p(x;ξ,σ,λ)=2σ−1f0(z)G{PK(z)}◀▶f0(⋅)G(⋅)Ovf0(x)=f0(−x),G(−x)=1−←↩↪→G(x)
PK(x)=λ1x+λ3x3+⋅⋅⋅λ2K−1x2K−1 K g ,z=(x−ξ)/σ"∙ f0GO tΦFlexibleskew-t-normal|distribution{PFSTN'4
∙Aa 3A x ' SMSNaFSSaIKSNSTSTNFSNFSTN 3K √√√√√ OCopula{×√√×√8 z×××√√ `17/ 36◀◀▶▶1! STFSTN ν<10' w x A ◀▶2!3FSNFSTN K≥2N A Lλ5?1N!←↩↪→ 3!L$^~ u kP duO\ |.E,Uy . /O05"'4
IK 3K OCopula{8 z 18/ 36◀◀▶▶◀▶←↩↪→ =|α(1,0);mα=(1,1);eα=(1,−1);meα=(−1,1)'4
S~f/(insuranceclaimdatasetofDanishfirelossesfrom1980to1990)IK 3K OCopula{8 z 19/ 36◀◀▶▶◀▶←↩↪→ k2167^ | '4
S~f/k^c? ?1[[(JLIK. 3K[ OCopula{FSTNFSNSTN8 zξ σ 36λ(,)(,)(,0)ν–◀◀▶▶◀▶lAICwFSTN[ JgFSN←↩↪→ STN" |'4
[ IK 3K OCopula{8 z 111/ 36◀◀▶▶◀▶←↩↪→ |^FSTN[ (J'4
S~f/2^k SN?1[lgroup=1 group=6A .AICIK 3Kgroup8123456 OCopula{ zlLwgroup=4[ Jg group=3 group=1Lk A112/ 365" ◀◀▶▶group=4 .A O◀▶component1234w←↩↪→ξ σ|λ'4
S~f/N [IK 3K OCopula{8 z 113/ 36◀◀▶▶◀▶←↩↪→ |'4
3u Copula{37K xcopula E ' k"IK 3Kcopular>S 5 O {X
>Sm k''Copula{X"8 zl`^ copula;K"114/ 36^5[@
^u] ~ ◀◀▶▶ pd "◀▶d'u 37K x '5"←↩↪→ |'4
Copula{∙Copula eCUj,j=1,...,rlmIK[0,1] ! u=(u1,...,ur)K 3K OCopula{C(u1,...,ur)=Pr(U1≤u1,...,Ur≤ur)8 z copula"115/ 36◀◀▶▶∙Sklarn? ) >S'( (copula)"AO/XJ>S◀▶Y A copula "←↩↪→ |'4
∙Copulab X=(X1,...,Xr)kY F(x;φ)z>SO Fj(xj;φ),j=1,...,rKLIKC(u;φ)=F(F−11(u1;φ),...,F−1 3Kr(ur;φ);φ) OCopula{copula"φ"8 z ∙CopulaXJX f(x;φ)116/ 36>Sfj(xj;φ),j=1,...,r@ocopula3[0,1r] ◀◀▶▶copula◀▶c(u;φ)=f(x;φ)∏r←↩↪→j=1fj(xj;φ) xj=F−1(ujj;φ)"|'4
skew-tCopula{∙ skew-tb XQlXe IKt()(()()) 3KX2D OCopula{Q∼0Γ+Dt2r,Ω=0D,νI8 zD= diag(λ1,...,λr)
Γ(r×r)
"117/ 36e
skew-t◀◀▶▶()2−1/2◀▶fST(x;Γ,D,ν)=2rf)x;νPr(V>0det(Γ+D2)1/2t(Γ+D;x)←↩↪→ft(x;ν) t V∼Γ+D2)−1x+νtr(D(Γ+D2−1x′()x,r+(I−D1(Γ+D2−)D),r+ν)|ν'4
∙ skew-tcopulab skew-t FIKST(x;φ)K t copula 3K OCST(u;φ)=FST(x;φ)Copula{8 z 118/ 36cST(u;φ)=fST(x;φ)∏rj=1fST,j(xj;λj,ν)◀◀▶▶◀▶←↩↪→ |'4
S~f /x| N N 8 IK
) x| 3K O ACopula{∙
)xi3 <8 ∙xi!x J/ !%n9 z ∙
)n %ndCAc n gA n 119/ 36◀◀▶▶ U? O{◀▶∙N I NCD .∙8O" Gr'~?yx ?←↩↪→J" |'4
IK 3K OCopula{8 z 120/ 36◀◀▶▶◀▶←↩↪→ |'4
∙ {k /6IU3:5:2 '~ p ! !.|+^ + {IKg3N <
> < g 3K ON 5 ! g 5^ Copula{ {"8 z+N Xe + p | <121/ 36+ | <◀◀▶▶+n .|| <◀▶←↩↪→ |'4
1+<6 2U | <?1N "2+ ^ 3/y/ IK2 | <?1N " 3K OCopula{8 3du.| lz|zL{ { |T| <?1N "122/ 36◀◀▶▶◀▶←↩↪→ |'4
IK 3K OCopula{8 z 123/ 36◀◀▶▶◀▶←↩↪→ J|'4
IK 3K OCopula{8 z 124/ 36◀◀▶▶◀▶←↩↪→ |'4
IK 3K OCopula{8 z 125/ 36◀◀▶▶◀▶←↩↪→ |'4
IK 3K OCopula{8 z 126/ 36◀◀▶▶◀▶←↩↪→ xi '~|'4
IK 3K OCopula{8 z 127/ 36◀◀▶▶◀▶←↩↪→N\xJ |'4
∙ Gg ZIP .~ OPoissonK a' x G3" GO A5 g (over-IK 3Kdispersion)":)(zero-inflation)" O
CAc n gA n Copula{93J xi '58 z^ZIP(zero-inflationP oisson) .>S, |^ t copula ''X"128/ 36∙ZIP .◀◀▶▶{◀▶Pr(Y=y∣p,λ)=p+(1−p)e−λ,y=0(1−pλy)e−λ,y=1,2,...y!←↩↪→b Y GgClog(λ)=β,logit(p)=γβγ ."|'4
∙l{du GglC |^copula3(J"d\dCX\)"kbIK 3KH Oj(yij;θj)ZIP . 1ji >SCopula{yijL1ji 1i Gg"gb lskew-t dC8 xzijv Hj(y−;θ129/ 36ijj)≤FST,j(xij;λj,ν)<Hj(yij;θj)H◀◀▶▶j(y−;θijj)=Hj(yij−1;θj) |^X Y'uX ^"◀▶←↩↪→ |'4
xiuZIP . >S O xiβγE(Yi)Pr(Yi=0)y¯i#(yi=0)IK 3KS {< < ◀◀▶▶`L1 !n 8X O1o!◀▶ . O " G'~ * "uy . [ "←↩↪→ |'4
u t copula[ Gg Gg* t OCopula{ z 36uy [ J' "◀◀▶▶T~ 3 K◀▶1 xi G >Scopula2N\x Gg >S←↩↪→u t copula3c ximJ= |copula'4
48 IK•k C 8 3K OCopula{8 •|^Copula'5 Bayesian{z 132/ 36•u 8 . O K◀◀▶▶•u 8 . CJ◀▶←↩↪→ |'4
5z[1].Azzalini,A.,Genton,.,,[2].Azzalini,A.,. ,171-178. O[3].Azzalini,A.,Capitaino,A.,{ ,367-389.z [4].Go´mez,.,Venegas,O.,Bolfarine,H.,-ronmetrics133/ 3618,395-407.[5].Azzalini,A.,Dalla-Valle,A.,-normaldistri-◀◀▶▶,83,715-726.[6].Azzalini,A.,Capitanio,A.,-◀▶,579-602.[7].Ma,Y.,Genton,.,flexibleclassofskew-symmetricdistribu-←↩↪→,459-468. [8].Zeller,.,Lachos,.,Vilca-Labra,.,fl.|,343-368.'4
z(Y)[9].Cancho,V.,Dey,.,Lachos,.,Andrade,M.,-linearregressionmodelswithscalemixuturesofskew-normaldistributions:estimationandIKcaseinfl,588-602. 3K O[10].Xie,F.,Wei,B.,Lin,J.,-normalCopula{, z[11].Vanegas,L.,Cysneiros,F.,, 36◀◀▶▶◀▶←↩↪→ |'4
6 IK•x'OXA 3K OCopula{8 •OXz 135/ 36◀◀▶▶◀▶←↩↪→ |'4
IK 3K OCopula{ThankYou!8 z 136/ 36◀◀▶▶◀▶←↩↪→ |'4
