# Estimating Distributions with a Fixed Number of Modes

• Martin B. Mächler
Chapter
Part of the Lecture Notes in Statistics book series (LNS, volume 109)

## Abstract

A new approach for non- or semi-parametric density estimation allows to specify modes and antimodes. The new smoother is a Maximum Penalized Likelihood (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.

## Key words and phrases

Nonparametric semi-parametric density estimation smoothing roughness penalty maximum penalized likelihood inflection point boundary value problem differential equation unimodality multimodality

## AMS 1991 subject classifications

Primary 62G07 secondary 34B10 41A29 65D07 65D10

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