Density Estimation and Combination under Model Ambiguity

1

n(n — 1)2

a²_n(xi,xj) = σ²_n + Op(n^-3/2h^-1),
i j,i6=j

(70)

2U^b2l =

2

n(n — 1)

i  i6=j

a_l(xj, x)a_l(xi, x)g(x)dx =

n(n — 1) XX H2ⁿ(x^i,x^j)^,
i i6=j

(71)

1 K ʌ 1 1 [Kh(x
H2n(xi,Xj)= h h₂ -----

— χj ) — ʃ Κh(χi — χj )g(χj )dχj
g(χi)

Κh(xi-Xz)-J^Κh(xi-Xz)g(xz)dXz
g(χi)

g(xi)dxi.

E Hn(^χi^,xj )] = h4 E

xi ^—

xi — xj ) — ʃ Kh (xi
g(xi)

xj) — ʃ Kh(xi — xj)g(xj)dxj ∖ KKh(xi
g⁽xi)                  ^g к

— xj)g(xj)dxj∖ μKh(xi — xz) — ʃKh(xi
^g к                  g⁽xi)

xz) — ʃ Kh(xi — xz)g(xz)dxz λ ^(x )d
g(xi)                     ^{g g i} '

— xz)g(xz)dxz             ²

------------------ I g(xi)dxi

2

''T^∙ .

x_i

g(xj)g(x_z)dxj dx

=⅛/KP^k

xi — xj )Kh(χi
g²(xi)

xz ) l ∖_rl
----g(xi)dxi

g(xj )g(xz )dXj dx_z + o( ɪ ) =

K (u)K (u + v)
----,------—-—du
g(xj + hu)

g(xj )g(xj +hu-hz)dxj hdv+o(-) = — f
hh

g²(xj)

²

K(u)K(u + v)du

g²(xj)dxjdv =

' h^-1

K K K(u)K(u + v)du dv + o(^)∙

(72)

σ²_2l ' 2n

2_h-1

K (u)K (u + v)du

dv∙

(73)

(74)

Vλ_n — 1 P h ^f (^χi)^-g(^χi) i2
22ⁿ n Z-^i      g(xi)

= l Pi b²_n(xi), which is a purely deterministic Bias-squared term, and it will

affect the mean of the asymptotic distribution∙ That is,