Expressions for C_1, C_2, C_3 implied by Theorem 3
In[]:=
C1=
κ
2
^(1/2);​​C2=-1/6*D3/D2*γ*
κ
2
^(-1)+
μ
2,c
*
κ
2
^(-1)/2;​​C3=
κ
2
^(-3/2)*
μ
3,c
6-12*D3D2*
μ
2,b
+D33
κ
2
D2*γ*
μ
2,c
+
μ
4
*(D3^2/8/D2^2-1/24*D4/D2)+
μ
2,a
2-D3^218D2^2
κ
2
*γ^2-12
κ
2
*
μ
2,c
^2-D3^28D2^2*
κ
2
^2;
Moments of A_n, A_n^3\hat{C}_2/\hat{C}_1, and A_n^5\hat{C}_2/\hat{C}_1 as computed in Sections C.1.1 - C.1.5
EA11=-
κ
2
^(-3/2)/2*γ-
κ
2
^(-3/2)*
μ
2,c
+1/2*
κ
2
^(-1/2)
μ
2,d
;​​EA1=n^(-1/2)*EA11;​​​​EA21=-1/2/
κ
2
*
μ
2,2
+1/4*
μ
2,d
^2
κ
2
-γ/(
κ
2
^2)*
μ
2,d
-4/(
κ
2
^2)*
μ
2,b
-1/
κ
2
*
μ
1,2,d
-2/(
κ
2
^2)*
μ
2,c
*
μ
2,d
+2/(
κ
2
^3)*γ^2+8*γ/(
κ
2
)^3*
μ
2,c
+8/(
κ
2
^3)*
μ
2,c
^2-2/
κ
2
^2*
μ
2,a
-2/(
κ
2
^2)*
μ
3,c
+3;​​EA2=1+n^(-1)*EA21;​​​​EA31=
κ
2
^(-3/2)*-72*γ-6*
μ
2,c
+32*
κ
2
*
μ
2,d
;​​EA3=n^(-1/2)*EA31;​​​​EA41=
κ
2
^(-2)*
μ
4
-3
κ
2
^2+6*
κ
2
*
μ
1,2,d
+2*γ*
μ
2,d
+12*
μ
2,b
+3*
κ
2
*
μ
2,2
+32*
κ
2
*
μ
2,d
^2+6*
μ
2,d
*
μ
2,c
+12
μ
2,a
+6*
κ
2
*
μ
1,3,d
+4
μ
3,c
-12*(
μ
4
-
κ
2
^2)-8
κ
2
*γ^2-12*γ*
μ
2,d
-24*
μ
2,b
-24*γ
κ
2
*
μ
2,c
-12*
κ
2
*
μ
1,2,d
-72
μ
2,b
-16
κ
2
*γ*
μ
2,c
-24*
μ
2,c
*
μ
2,d
-48*
μ
2,a
-48
κ
2
*
μ
2,c
^2+9*(
μ
4
-
κ
2
^2)+36
κ
2
*γ^2+36*(
μ
2,b
)+144
κ
2
*γ*
μ
2,c
+36*(
μ
2,a
)+144
κ
2
*
μ
2,c
^2-6*
κ
2
*
μ
2,2
-24*
μ
2,a
-6*
κ
2
*
μ
1,3,d
-24*
μ
3,c
+30*
κ
2
^2;​​EA4=3+1/n*EA41;​​​​EA3C=EA3*C2/C1+3/2*n^(-1/2)*
κ
2
^(-2)*-D3D2*(
μ
4
-3
κ
2
^2)3+12
κ
2
*D3D2*(γ^2+2γ*
μ
2,c
)-3
κ
2
*
μ
2,c
*(γ+2
μ
2,c
)2+(2-D3/D2)*
μ
2,b
+
μ
3,c
+2
μ
2,a
;​​​​EA5C=(10*EA3-15*EA1)*C2/C1+5*3/2*n^(-1/2)*
κ
2
^(-2)*-D3D2*(
μ
4
-3
κ
2
^2)3+12
κ
2
*D3D2*(γ^2+2γ*
μ
2,c
)-3
κ
2
*
μ
2,c
*(γ+2
μ
2,c
)2+(2-D3/D2)*
μ
2,b
+
μ
3,c
+2
μ
2,a
;​​
Moments of W_n
In[]:=
M1=EA1+n^(-1/2)*1*C2/C1;​​M2=EA2+2*n^(-1/2)*EA3C+2*n^(-1)*EA4C+n^(-1)*3*(C2/C1)^2;​​M3=EA3+3*n^(-1/2)*3*C2/C1;​​M4=EA4+4*n^(-1/2)*EA5C+4*n^(-1)*EA4C*5+6*n^(-1)*15*(C2/C1)^2;
Cumulants of W_n
In[]:=
K12=M1*n^(1/2);​​K22=n*(M2-1-M1^2);​​K31=n^(1/2)*(M3-3M1);​​K41=n*(M4-3-4M3*M1-6*(M2-1)+12*M1^2);​​
Expression of A(x^2)
In[]:=
A=-2*x*(1/2*(K22+K12^2)+1/24*(K41+4*K12*K31)*(x^2-3)+1/72*K31^2*(x^4-10*x^2+15));
In[]:=
Collect[FullSimplify[A],x]
Out[]=
-
2
(2D2+D3)
5
x
2
γ
36
2
D2
3
κ
2
-
1
36
2
D2
3
κ
2
3
x
4(D2+D3)(2D2+D3)
2
γ
+9
2
(2D2+D3)
3
κ
2
+3(-2
2
D2
-4D2D3-3
2
D3
+D2D4)
κ
2
μ
4
+6D2(2D2+D3)γ
μ
2,c
-6D2(2D2+D3)γ
κ
2
μ
2,d
-
1
36
2
D2
3
κ
2
x-12
2
D2
2
γ
+18
2
D2
κ
2
μ
4
+36
2
D2
κ
2
(
μ
2,a
+2
μ
2,b
)-36
2
D2
γ
μ
2,c
-9
2
D2
2
μ
2,c
-18
2
D2
κ
2
μ
2,c
μ
2,d
+9
2
D2
2
κ
2
-2
μ
2,2
+
2
μ
2,d
-4
μ
1,2,d

The expression of A(x^2)if we choose \phi according to the reversed KL divergence as in Diciccio et al.
In[]:=
Collect[FullSimplify[A],x]/.{D21,D3-2,D46}
Out[]=
-
1
36
3
κ
2
x-12
2
γ
+18
κ
2
μ
4
+36
κ
2
(
μ
2,a
+2
μ
2,b
)-36γ
μ
2,c
-9
2
μ
2,c
-18
κ
2
μ
2,c
μ
2,d
+9
2
κ
2
-2
μ
2,2
+
2
μ
2,d
-4
μ
1,2,d

​