The integrated mean square error for the conditional KDE.
Parameters: | bw: array_like :
The bandwidth parameter(s).
|
Returns: | CV: float :
The cross-validation objective function.
|
Notes
For more details see pp. 156-166 in [R15]. For details on how to
handle the mixed variable types see [R16].
The formula for the cross-validation objective function for mixed
variable types is:
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where
System Message: WARNING/2 (G_{-l}(X_{l}) = n^{-2}\sum_{i\neq l}\sum_{j\neq l}
K_{X_{i},X_{l}} K_{X_{j},X_{l}}K_{Y_{i},Y_{j}}^{(2)}
)
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where
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is the multivariate product kernel and
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is the leave-one-out estimator of the pdf.
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is the convolution kernel.
The value of the function is minimized by the _cv_ls method of the
GenericKDE class to return the bw estimates that minimize the
distance between the estimated and “true” probability density.
References
[R15] | (1, 2) Racine, J., Li, Q. Nonparametric econometrics: theory and
practice. Princeton University Press. (2007) |
[R16] | (1, 2) Racine, J., Li, Q. “Nonparametric Estimation of Distributions
with Categorical and Continuous Data.” Working Paper. (2000) |