content_image/294909.jpg content_image/294910.jpg content_image/294911.jpg content_image/294912.jpg
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 ) = 

<img src='content_image/294909.jpg'>

<img src='content_image/294910.jpg'>

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 

<img src='content_image/294911.jpg'>

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: 

<img src='content_image/294912.jpg'>

