Density power divergence (DPD) is designed to robustly estimate the underlying distribution of observations, in the presence of outliers. However, DPD involves an integral of the power of the parametric density models to be estimated; the explicit form of the integral term can be derived only for specific densities, such as normal and exponential densities. While we may perform a numerical integration for each iteration of the optimization algorithms, the computational complexity has hindered the practical application of DPD-based estimation to more general parametric densities. To address the issue, this study introduces a stochastic approach to minimize DPD for general parametric density models. The proposed approach can also be employed to minimize other density power-based $\gamma$-divergences, by leveraging unnormalized models. We provide \verb|R| package for implementation of the proposed approach in \url{https://github.com/oknakfm/sgdpd}.

翻译：密度功率散度（DPD）旨在存在异常值的情况下，稳健地估计观测数据的潜在分布。然而，DPD涉及对要估计的参数密度模型功率的积分；仅对特定密度（如正态分布和指数分布）才能推导出该积分项的显式形式。尽管我们可以在优化算法的每次迭代中执行数值积分，但计算复杂度阻碍了基于DPD的估计在实际中应用于更一般的参数密度。为解决这一问题，本研究引入了一种随机方法来最小化一般参数密度模型的DPD。所提出的方法还可通过利用非归一化模型，用于最小化其他基于密度功率的$\gamma$-散度。我们提供了实现所提方法的\verb|R|软件包，下载地址为\url{https://github.com/oknakfm/sgdpd}。