* <math>R(K) = \int K(\bold{x})^2 \, d\bold{x}</math>, with {{nowrap|''R''(''K'') {{=}} (4''π'')<sup>''−d''/2</sup>}} when ''K'' is a normal kernel
<ul>
<li>* <math>R(K)\int =\bold{x} \intbold{x}^T K(\bold{x})^2 \, d\bold{x}</math>,with= {{nowrap|''R''m_2(''K'') \bold{{=I}} (4''π'')_d<sup>''−d''/2</supmath>}} when ''K'' is a normal kernel,
:with <strong>I</strong><sub>d</sub> being the ''d × d'' [[identity matrix]], with ''m''<sub>2</sub> = 1 for the normal kernel
with* D<strongsup>I2</strong><sub>d</subsup>''ƒ'' beingis the ''d × d'' [[identityHessian matrix]],withof ''m''<sub>2</sub>second =order 1partial forderivatives theof normal kernel''ƒ''
<li>* <math>\bold{\Psi}_4 = \int (\operatorname{vec} \, \operatorname{D}^2 f(\bold{x})) (\operatorname{vec}^T \operatorname{D}^2 f(\bold{x})) \, d\bold{x}</math> is a ''d''<sup>2</sup> × ''d''<sup>2</sup> matrix of integrated fourth order▼
<li>D<sup>2</sup>''ƒ'' is the ''d × d'' Hessian matrix of second order partial derivatives of ''ƒ''
▲<li><math>\bold{\Psi}_4 = \int (\operatorname{vec} \, \operatorname{D}^2 f(\bold{x})) (\operatorname{vec}^T \operatorname{D}^2 f(\bold{x})) \, d\bold{x}</math> is a ''d''<sup>2</sup> × ''d''<sup>2</sup> matrix of integrated fourth order
partial derivatives of ''ƒ''
<li>* vec is the vector operator which stacks the columns of a matrix into a single vector e.g. <math>\operatorname{vec}\begin{bmatrix}a & c \\ b & d\end{bmatrix} = \begin{bmatrix}a & b & c & d\end{bmatrix}^T.</math>
</ul>
The quality of the AMISE approximation to the MISE<ref name="WJ1995"/>{{rp|97}} is given by
