Using scipy gaussian kernel density estimation to calculate CDF inverse

You can use some python tricks for fast and memory-effective estimation of the CDF (based on this answer):

    from scipy.special import ndtr
    cdf = tuple(ndtr(np.ravel(item - kde.dataset) / kde.factor).mean()
                for item in x)

It works as fast as this answer, but has linear (len(kde.dataset)) space complexity instead of the quadratic (actually, len(kde.dataset) * len(x)) one.

All you have to do next is to use inverse approximation, for instance, from statsmodels.

The method integrate_box_1d can be used to compute the CDF, but it is not vectorized; you'll need to loop over points. If memory is not an issue, rewriting its source code (which is essentially just a call to special.ndtr) in vector form may speed things up.

from scipy.special import ndtr
stdev = np.sqrt(kde.covariance)[0, 0]
pde_cdf = ndtr(np.subtract.outer(x, n)).mean(axis=1)
plot(x, pde_cdf)

The plot of the inverse function would be plot(pde_cdf, x). If the goal is to compute the inverse function at a specific point, consider using the inverse of interpolating spline, interpolating the computed values of the CDF.