Theory of Statistics - George Mason University

588 8 Nonparametric and Robust Inference
E(pK(y)) ≈
IRd
κ(u) p(y) − (V u) T ∇p(y) + 1
2 (V u)T
Hp(y)V u du
= p(y) − 0 + 1
2 trace V T Hp(y)V . (8.63)
To second order in the elements of V (that is, to terms in O(|V | 2 )), the bias
at the point y is therefore
1
2 trace V V T Hp(y) . (8.64)
Using the same kinds of expansions and approximations as in equations
(8.62) and (8.63) to evaluate E (pK(y)) 2 to get an expression of order
O(|V |/n), and subtracting the square of the expectation in equation (8.63),
we get the approximate variance at y as
or
V(pK(y)) ≈ p(y)
n|V |
IR d
Integrating this, because p is a density, we have
(κ(u)) 2 du,
V(pK(y)) ≈ p(y)
S(κ). (8.65)
n|V |
AIV pK
S(κ)
= , (8.66)
n|V |
and integrating the square of the asymptotic bias in expression (8.64), we have
AISB
1
pK =
4
T
trace V Hp(y)V 2
dy. (8.67)
IR d
These expressions are much simpler in the univariate case, where the
smoothing matrix V is the smoothing parameter or window width h. We have
a simpler approximation for E(pK(y)) than that given in equation (8.63),
E(pK(y)) ≈ p(y) + 1
2 h2p ′′
(y) u 2 κ(u)du,
and from this we get a simpler expression for the AISB. After likewise simplifying
the AIV, we have
IR
AMISE S(κ) 1
pK = +
nh 4 σ4 κh4R(p), (8.68)
where we have left the kernel unscaled (that is, u 2 κ(u)du = σ 2 K ).
Minimizing this with respect to h, we have the optimal value of the smoothing
parameter
Theory of Statistics c○2000–2013 James E. Gentle