Returns the Integrated Mean Square Error for the unconditional KDE.

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


See p. 27 in [1] for details on how to handle the multivariate estimation with mixed data types see p.6 in [2].

The formula for the cross-validation objective function is:

\[CV=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{N} \bar{K}_{h}(X_{i},X_{j})-\frac{2}{n(n-1)}\sum_{i=1}^{n} \sum_{j=1,j\neq i}^{N}K_{h}(X_{i},X_{j})\]

Where \(\bar{K}_{h}\) is the multivariate product convolution kernel (consult [2] for mixed data types).


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