The integrated mean square error for the conditional KDE.
Parameters: | bw: array_like :
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Returns: | CV: float :
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Notes
For more details see pp. 156-166 in [R22]. For details on how to handle the mixed variable types see [R23].
The formula for the cross-validation objective function for mixed variable types is:
CV(h,\lambda)=\frac{1}{n}\sum_{l=1}^{n} \frac{G_{-l}(X_{l})}{\left[\mu_{-l}(X_{l})\right]^{2}}- \frac{2}{n}\sum_{l=1}^{n}\frac{f_{-l}(X_{l},Y_{l})}{\mu_{-l}(X_{l})}
where
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)}
where K_{X_{i},X_{l}} is the multivariate product kernel and \mu_{-l}(X_{l}) is the leave-one-out estimator of the pdf.
K_{Y_{i},Y_{j}}^{(2)} 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
[R22] | (1, 2) Racine, J., Li, Q. Nonparametric econometrics: theory and practice. Princeton University Press. (2007) |
[R23] | (1, 2) Racine, J., Li, Q. “Nonparametric Estimation of Distributions with Categorical and Continuous Data.” Working Paper. (2000) |