statsmodels.nonparametric.kernel_density.KDEMultivariateConditional.imse¶

KDEMultivariateConditional.imse(bw)[source]¶

The integrated mean square error for the conditional KDE.

Parameters:¶

bwarray_like

The bandwidth parameter(s).

Returns:¶

CVfloat

The cross-validation objective function.

Notes

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:

$$CV(h,\lambda )=\frac{1}{n}\sum _{l=1}^n\frac{G_{-l}(X_l)}{{[{\mu }_{-l}(X_l)]}^2}-\frac{2}{n}\sum _{l=1}^n\frac{f_{-l}(X_l,Y_l)}{{\mu }_{-l}(X_l)}$$

where

$$G_{-l}(X_l)=n^{-2}\sum _{i\ne l}\sum _{j\ne l}{K}_{X_i,X_l}{K}_{X_j,X_l}{K}_{Y_i,Y_j}^{(2)}$$

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

[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)