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Limit Distribution Theory for Maximum Likelihood Estimation of a Log-Concave Density
Fadoua Balabdaoui1, Kaspar Rufibach, Jon A Wellner
1Centre de Recherche en Mathematiques de la Decision, Universite Paris-Dauphine, Paris, France,
Annals of Statistics
|November 3, 2009
Summary
We determine the limiting distributions for the nonparametric maximum likelihood estimator (MLE) of log-concave densities. This research connects log-concave density estimation to convex density estimation, yielding optimal mode estimators.
Area of Science:
- Statistical Theory
- Nonparametric Statistics
- Density Estimation
Background:
- The nonparametric maximum likelihood estimator (MLE) for log-concave densities is well-defined, with its properties studied by Rufibach (2006) and Dümbgen & Rufibach (2007).
- A key characterization of the log-concave MLE, in terms of distribution functions, mirrors that of the least squares estimator for convex densities on [0, infinity) (Groeneboom et al., 2001b).
Purpose of the Study:
- To establish the limiting distributions of the MLE and its derivative for log-concave densities.
- To leverage the connection with convex density estimation to derive these distributions.
- To analyze the impact of varying smoothness assumptions on the limiting distributions and establish optimality for mode estimators.
Main Methods:
- Utilizing the established connection between the characterizations of log-concave MLE and convex density least squares estimators.
- Adapting smoothness assumptions from convex density estimation to the log-concave setting.
- Analyzing the behavior of an integrated Brownian motion process and its relation to vanishing derivatives of the log-density.
Main Results:
- The limiting distributions of the MLE and its derivative for log-concave densities are shown to be analogous (up to sign) to those in convex density estimation.
- Pointwise limiting distributions are found to depend on specific derivatives of a transformed integrated Brownian motion process, influenced by vanishing derivatives of the log-concave function.
- The limiting distribution of the mode estimator is derived, and a new local asymptotic minimax lower bound demonstrates its optimality.
Conclusions:
- The study provides a theoretical framework for understanding the asymptotic behavior of log-concave density estimators.
- The established connection facilitates the transfer of results from convex density estimation to the log-concave setting.
- The derived mode estimator is shown to be asymptotically optimal, offering valuable insights for statistical inference.
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