logcondens: Estimate a Log-Concave Probability Density from iid Observations

Given independent and identically distributed observations X(1), ..., X(n), this package allows to compute a concave, piecewise linear function phi on [X(1), X(n)] with knots only in {X(1), X(2), ..., X(n)} such that L(phi) = sum_{i=1}^n W(i)*phi(X(i)) - int_{X(1)}^{X(n)} exp(phi(x)) dx is maximal, for some weights W(1), ..., W(n) s.t. sum_{i=1}^n W(i) = 1. According to the results in Duembgen and Rufibach (2009), this function phi maximizes the ordinary log-likelihood sum_{i=1}^n W(i)*phi(X(i)) under the constraint that phi is concave. The corresponding function exp(phi) is a log-concave probability density. Two algorithms are offered to compute the estimator: An active set algorithm and one based on the pool-adjacent-violaters algorithm. In addition, we provide functions to compute (1) the value of the density and distribution function estimate at a given point (2) a smoothed log-concave density estimator (3) the characterizing functions of the estimator and (4) to sample from the estimated distribution. Finally, two datasets that have been used to illustrate log-concave density estimation are made available.

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