Density Estimation and Combination under Model Ambiguity

1             I ∖Kl K-x f ∖^κ(x^x- ^{- x}) ^- f Kh(^xi ^{- x}j )g(^xi )^dxj↑ f⁽ (^xi) - g(^xi)∖ -,      _f,₇,₇×

H3ni / _j an(^xi, ^xj)bn(^xi) h J                    g(x )                    y g(x )     ^d dx ⁽⁷⁷⁾

Under assumptions A1 and A2 and given that EH23ni =0, by Lemma1 in Hall(1984) we have that 2V^b23n
is asymptotically normally distributed with zero mean and variance given by:

which can be easily seen if we consider that ^f (^xi) ^g(^xi) = ^{h μ}2^g—(^x)and that
g(xi)            g(xi)

Also this term will affect the asymptotic distribution of κI1 .

To summarize all previous steps, we can rewrite the expansion of κI1 in the following way:

^κI 1 = V^zIln + Bbn ^- 2 ^½1n + V22n + ²V23nJ ^{-                       (79)}

N(^0,σ2n)+h-μ2 / g(^2)(x)dx+^o(h^2)-1 fσ^n + Op^{(n 3/2}h ¹) + ²N(0^,σ2n) ^{+ h}-μ2 [ (g  (^x))

1n    2 2                        2 n                               2n     4 2      g(x)

Once more, following Hall(1984), from the definition of V^b21n and the fact that nh → ∞, we have that the
difference between _n(n_-i) Pi Pj=i αn(xi, xj) and σ^g_l is negligible w.r.t. 2U2_n, hence the previous expression

KKI1 - (nh^1/2)^-1√2σ1N1 - (nh¹^)-1√2σ2N2 - n^-1¾2√2σ3N3 + Bbn - |cn

where N1 ,N2 and N3 are asymptotically normal N(0,1); and

It is important to notice that B^bn, which is Op (n^-1/2h²), will asymptotically cancel out with n^-1/2h² √2σ₃N₃,
since they are of the same order of magnitude.

Thus, we have the following results: asn →∞,h→ 0, nh →∞ and nh⁵ → 0

nh^1/2(jcI 1 + 2Cn) →^d √2σιNι - √2σ3N2.

29

<<   27   28   29   30   31   32   33   >>

More intriguing information