Limit Distribution Theory for Maximum Likelihood Estimation of a Log-Concave Density.
Limit Distribution Theory for Maximum Likelihood Estimation of a Log-Concave Density.
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DOI：
10.1214/08-aos609
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发表时间：
2009-06-01
影响因子：
4.5
通讯作者：
Wellner JA
中科院分区：
文献类型：
--
作者：
Balabdaoui F;Rufibach K;Wellner JA
我们发现对数凹密度的非参数最大似然估计量 (MLE) 的极限分布，即 f0 = exp phi0 形式的密度，其中 phi0 是 ℝ 上的凹函数。 MLE 的存在性、形式、特征和一致收敛率由 和 给出。分布函数方面的对数凹 MLE 的表征与所研究的 [0, ∞) 上凸密度的最小二乘估计量的表征相同（直到符号）。我们使用这种联系来表明，在可比较的平滑度假设下，MLE 及其导数的极限分布与凸密度估计问题中的相同（直到符号）。特别是，通过允许一些更高的导数在感兴趣点处消失来稍微改变平滑度假设，我们发现点状极限分布取决于 Hk 0 处的二阶和三阶导数，即积分布朗运动过程的“下包络线”减去漂移项，该漂移项取决于感兴趣点处 ψ0 = log f0 的消失导数的数量。我们还建立了模式 M(f0) 的结果估计量的极限分布，并建立了一个新的局部渐近极小极大下界，该下界显示了我们的模式估计量在收敛速度和常数对总体值的依赖性方面的最优性。
We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form f0 = exp ϕ0 where ϕ0 is a concave function on ℝ. Existence, form, characterizations and uniform rates of convergence of the MLE are given by and. The characterization of the log–concave MLE in terms of distribution functions is the same (up to sign) as the characterization of the least squares estimator of a convex density on [0, ∞) as studied by. We use this connection to show that the limiting distributions of the MLE and its derivative are, under comparable smoothness assumptions, the same (up to sign) as in the convex density estimation problem. In particular, changing the smoothness assumptions of slightly by allowing some higher derivatives to vanish at the point of interest, we find that the pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0 = log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.