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statsmodels.nonparametric.kernel_density.KDEMultivariateConditional.imse

KDEMultivariateConditional.imse(bw)[source]

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:

System Message: WARNING/2 (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})} )

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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)} )

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9k> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

where

System Message: WARNING/2 (K_{X_{i},X_{l}})

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9k> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
is the multivariate product kernel and

System Message: WARNING/2 (\mu_{-l}(X_{l}))

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is the leave-one-out estimator of the pdf.

System Message: WARNING/2 (K_{Y_{i},Y_{j}}^{(2)})

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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)

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