Estimating Distributions with a Fixed Number of Modes

A new approach for non- or semi-parametric density estimation allows to specify modes and antimodes. The new smoother is a Maximum PenalizedLikelihood (MPL) estimate with a novel roughness penalty. It penalizes a relative change of curvature which allows considering modes and inflection points. For a given number of modes, the score function, $$l^\prime = (\log f)^\prime$$ can be represented as $${l^\prime }\left( x \right) = \pm (x - {w_1}) \cdots (x - {w_m})\cdot{\text{ }}exp{h_l}(x)$$, a semiparametric term with parameters w_j (model order m) and nonparametric part h_l(·). The MPL variational problem is equivalent to a differential equation with boundary conditions. The exponential and normal distributions are smoothest limits of the new estimator for zero and one mode, respectively.