The integrated mean square error for the conditional KDE.

Parameters:bw (array_like) – The bandwidth parameter(s).
Returns:CV – The cross-validation objective function.
Return type:float


For more details see pp. 156-166 in [1]. For details on how to handle the mixed variable types see [2].

The formula for the cross-validation objective function for mixed variable types is:

\[CV(h,\lambda)=\frac{1}{n}\sum_{l=1}^{n} \frac{G_{-l}(X_{l})}{\left[\mu_{-l}(X_{l})\right]^{2}}- \frac{2}{n}\sum_{l=1}^{n}\frac{f_{-l}(X_{l},Y_{l})}{\mu_{-l}(X_{l})}\]


\[G_{-l}(X_{l}) = n^{-2}\sum_{i\neq l}\sum_{j\neq l} K_{X_{i},X_{l}} K_{X_{j},X_{l}}K_{Y_{i},Y_{j}}^{(2)}\]

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.


[1]Racine, J., Li, Q. Nonparametric econometrics: theory and practice. Princeton University Press. (2007)
[2]Racine, J., Li, Q. “Nonparametric Estimation of Distributions with Categorical and Continuous Data.” Working Paper. (2000)