Density Estimation and Combination under Model Ambiguity



applying a change of variable from (xi,xj) = (xi,u) where u = xj-xi we get the following expression

1 C KK2(u) + £hRK(u)g(xi + hu)du2 — 2K(u) £hRK(u)g(xi + hu)du

= 4hJ J ---------------------------g2χ)---------------------------g(Xi)g(xi + hu)du '

' 41h K K2(u)du + o( 1) = O(1).
4h             h h

(61)


Similarly we can show that

E [Jn(xi,xj)Jn(xj ,xi)]


' 4-h( K2(u)du + o(h) = O(h).
4h             h h


(62)


Then it follows that

and


E H12n (xi


xj) = E £2Jn (xi,xj)+ Jn (xi,xj)Jn (xj ,xi


)1 = h/


K 2(u)du + o(ɪ) = θ(ɪ),
hh


(63)


21
σ2n = n2h J K2(u)du + o(h).


(64)


The second term in () is the expected value of a Bias term, that is


Bbn 1 X bn(xi) ' h-μ2 I g(2x')dx + o(h2),
n2

i


(65)


where g(2)(x) is the second derivative of the p.d.f. Hence Bbn = Op (n 1/2h2^ Thus, what we obtain is


^ ^ ^

V1n = V11n + Bn


σ1nN(0, 1) + y μ2


g(2)(x)dx+o(h2).


(66)


1

V2n = —
n
i


"p"" /   ∖      P/ ∖

fn(xi) f(xi)


g(xi)


—                "I 2

+ f (xi) - g(xi)
g(χi)


“ --—~~ ___

fn(xi) — f (xi)

g(xi)

f (xi) g(χi)
g(χ
i)

f fn(xi) — f (xi)

∖     g(xi)


(67)


= Vb21n + Vb22n + Vb23n.                                         (68)


V21n = 1 X
n

i


2

—1— X an(xi,Xj) I
n—1

j,i6=j


1

n(n — 1)2


ΣΣаП (xi,xj )+ (π∑∑αn(xi>Xj n(xi,xz ).
n(n — 1)

i j,i6=j                              i j6=i z6=j


(69)


The first part is a variance term and it will affect the mean of the asymptotic distribution. The second
term equals a twice centered degenerate U-statistic
U2n, which is of the same order of magnitude of V11n
and it also affects the asymptotic distribution of KI.


27




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