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m →Density estimation in R with a full bandwidth matrix: make H_PI render in mathjax |
m Correcting two errors: 1) the definition of AMISE contained an extra transpose on vecH, and 2) m_2 comes from the integral of xx^T K(x), not xx^t K(x)^2. Corrections from Wand and Jones reference. |
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: <math>\operatorname{AMISE} (\bold{H}) = n^{-1} |\bold{H}|^{-1/2} R(K) + \tfrac{1}{4} m_2(K)^2
(\operatorname{vec}^T \bold{H}) \bold{\Psi}_4 (\operatorname{vec}
where
* <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
* <math>\int \bold{x} \bold{x}^T K(\bold{x})
:with <strong>I</strong><sub>d</sub> being the ''d × d'' [[identity matrix]], with ''m''<sub>2</sub> = 1 for the normal kernel
* D<sup>2</sup>''ƒ'' is the ''d × d'' Hessian matrix of second order partial derivatives of ''ƒ''
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