and the expression for the bias for ψ ∈ D$_{l}$$_{+}$$_{k}$ and kernel K of order l : more specifically for any ψ the bias functional provides ( E ̂ f, ψ ) − ( f, ψ ) = where R ( h ) = o ( h $^{l}$); if ψ ∈ D$_{l}$$_{+}$$_{k}$$_{+1}$ then R ( h ) = O ( h l $^{+1}$) . Note that ( f, ψ ) = E ( ψ ) where expectation is with respect to the measure given by F. Denote the expression by ( B ( h, K ) , ψ ) as it represents the value of a linear continuous functional B ( h, K ) applied to ψ. The B ( h, K ) is the leading term in the generalized bias function for the kernel estimator: E ̂ f − f ≈ O ( h $^{l}$) , ( − 1) l ∑ Σ m$_{i}$ = $_{l}$∫ k ∏ i =1 h m$_{i}$ i m$_{i}$ ! F (˜ x ) ∂ l + $^{k}$ψ ∂x m$_{1}$$_{i}$ +1 1 ...∂x m$_{ki}$ +1 k (˜ x ) d ˜ x ∫ K ( w ) w m 1 1 ...w m$_{ki}$ k dw (7) + R ( h ) , ( − 1) l ∑ Σ m$_{i}$ = $_{l}$∫ k ∏ i =1 ( h$_{i}$/ ¯ h ) m$_{i}$ m$_{i}$ ! F (˜ x ) ∂ l + $^{k}$ψ ∂x m$_{1}$$_{i}$ +1 1 ...∂x m$_{ki}$ +1 k (˜ x ) d ˜ x ∫ K ( w ) w m 1 1 ...w m$_{ki}$ k dw Bias ( ˆ f ) = E ˆ f − f = h $^{l}$B ( h, K ) + o ( h $^{l}$); (8) where for any ψ ∈ D$_{l}$$_{+}$$_{k}$$_{+1}$ ( E ̂ f, ψ ) − ( f, ψ ) = h $^{l}$( B ( h, K ) , ψ ) + o ( h $^{l}$) . 14