Abstract Given a large sample of observations from an unknown univariate continuous distribution, it is often of interest to empirically estimate the global mode of the underlying density. Applications include samples obtained by Monte Carlo methods with independent observations, or Markov Chain Monte Carlo methods with weakly dependent samples from the underlying stationary density. In either case, often the generating density is not available in closed form and only empirical determination of the mode is possible. Assuming that the generating density has a unique global mode, a non-parametric estimate of the density is proposed based on a sequence of mixtures of Beta densities which allows for the estimation of the mode even when the mode is possibly located on the boundary of the support of the density. Furthermore, the estimated mode is shown to be strongly universally consistent under a set of mild regularity conditions. The proposed method is compared with other empirical estimates of the mode based on popular kernel density estimates. Numerical results based on extensive simulation studies show benefits of the proposed methods in terms of empirical bias, standard errors and computation time. An R package implementing the method is also made available online.

中文翻译：

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基于弱相关样本的模态经验估计

摘要 给定来自未知单变量连续分布的大量观测样本，凭经验估计潜在密度的全局模式通常很有趣。应用包括通过具有独立观测值的 Monte Carlo 方法获得的样本，或具有来自基础平稳密度的弱相关样本的 Markov Chain Monte Carlo 方法。在任何一种情况下，生成密度通常都不是封闭形式的，只能凭经验确定模式。假设生成密度具有唯一的全局模式，基于 Beta 密度的混合序列提出了密度的非参数估计，即使模式可能位于边界上，也允许估计模式密度的支持。此外，在一组温和的规律性条件下，估计的模式显示出强烈的普遍一致性。将所提出的方法与基于流行的核密度估计的模式的其他经验估计进行比较。基于广泛模拟研究的数值结果显示了所提出方法在经验偏差、标准误差和计算时间方面的优势。实现该方法的 R 包也可在线获得。