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Maximum likelihood estimation of a log-concave density and its distribution function: basic properties and uniform consistency. (English) Zbl 1200.62030

Summary: We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to the supremum norm on a compact interval for the density and hazard rate estimator is at least \((\log (n)/n)^{1/3}\) and typically \((\log (n)/n)^{2/5}\), whereas the difference between the empirical and estimated distribution function vanishes with rate \(o_{\text p}(n^{ - 1/2})\) under certain regularity assumptions.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62N02 Estimation in survival analysis and censored data

Software:

logcondens

References:

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