Density Estimation: New Spline Approaches and a Partial Review

Apart from kernel estimators, there have been quite a few different approaches of “generalized splines” for density estimation. In the present paper,Maximum Penalized Likelihood (mpl) approaches are reviewed. In conclusion, penalizing the log density seems most promising. In my “wp” approach for semi-parametric density estimation, a novel roughness penalty is introduced. It penalizes a relative change of curvature which allows considering modes and inflection points. For a given number of modes, l′ = (log f)′ can be represented as l′(x) = ±(x−w1) · · · (x−wm) · exphl(x), a semi-parametric term with parameters wj (model order m) and nonparametric part hl(·). The mpl problem is equivalently solved via a boundary value differential equation.