$$\min \sum \frac{(f_{i+1}-f_i )^2}{( f_{i+1}+f_i )/2} = \sum \frac{u_i^2}{v_i/2} \approx I(F)$$
HuberSpline(x, p, grid, kappa = 0)An object of class density with solution \(f*\)
quantiles to be interpolated
probabilities associated with x
grid values for fitted object
width of Kolmogorov neighborhood
R. Koenker and J. Gu
$$\Leftrightarrow \quad \min \sum w_i \; \mbox{s.t.} \; u_i^2 \leq 2 v_i w_i$$
subject to interpolation of constraints \(F(x_j) = p_j, \; j=1,...,n.\) When \(\kappa > 0\), \(I(F)\) is minimized within a Kolmogorov neighborhood of the constraint points, rather than interpolating them. The generalization to Kolmogorov neighborhoods is due to Donoho and Reeves (2013).
N.B. When the grid is not equispaced, one would have to include grid spacings.
P. J. Huber. (1974) "Fisher Information and Spline Interpolation." Ann. Statist. 2 (5) 1029 - 1033,
D. L. Donoho and G. Reeves, (2013) Achieving Bayes MMSE Performance in the Sparse Signal Gaussian White Noise Model when the Noise Level is Unknown, Proc. IEEE Symposium Istanbul, Turkey.