Finite mixture models have long been used across a variety of fields in engineering and sciences. Recently there has been a great deal of interest in quantifying the convergence behavior of the mixing measure, a fundamental object that encapsulates all unknown parameters in a mixture distribution. In this paper we propose a general framework for estimating the mixing measure arising in finite mixture models, which we term minimum $\Phi$-distance estimators. We establish a general theory for the minimum $\Phi$-distance estimator, where sharp probability bounds are obtained on the estimation error for the mixing measures in terms of the suprema of the associated empirical processes for a suitably chosen function class $\Phi$. Our framework includes several existing and seemingly distinct estimation methods as special cases but also motivates new estimators. For instance, it extends the minimum Kolmogorov-Smirnov distance estimator to the multivariate setting, and it extends the method of moments to cover a broader family of probability kernels beyond the Gaussian. Moreover, it also includes methods that are applicable to complex (e.g., non-Euclidean) observation domains, using tools from reproducing kernel Hilbert spaces. It will be shown that under general conditions the methods achieve optimal rates of estimation under Wasserstein metrics in either minimax or pointwise sense of convergence; the latter case can be achieved when no upper bound on the finite number of components is given.

翻译：有限混合模型长期以来被广泛应用于工程与科学领域的多个分支。近年来，作为混合分布中封装所有未知参数的基本对象，混合度量的收敛行为引起了广泛关注。本文提出一个用于估计有限混合模型中混合度量的通用框架，我们称之为最小Φ距离估计量。我们建立了最小Φ距离估计量的一般理论，其中针对适当选择的函数类Φ，通过相关经验过程的极值得到了混合度量估计误差的精确概率界。该框架不仅涵盖了若干现有且看似不同的估计方法作为特例，还催生了新的估计量。例如，它将最小Kolmogorov-Smirnov距离估计量扩展到多元情形，并将矩估计法泛化至覆盖高斯分布之外更广泛的概率核族。此外，该框架还包含了利用再生核希尔伯特空间工具、适用于复杂（如非欧几里得）观测域的方法。将证明在一般条件下，这些方法在Wasserstein度量下能够达到极小化极大或逐点收敛意义下的最优估计速率——后者可在未对有限分量数设定上界时实现。