statsmodels.nonparametric.kernel_density.KDEMultivariate.imse¶

KDEMultivariate.imse(bw)[source]¶

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

Notes

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).

References