Theoretical guarantees are established for a standard estimator in a semi-parametric finite mixture model, where each component density is modeled as a product of univariate densities under a conditional independence assumption. The focus is on the estimator that maximizes a smoothed log-likelihood function, which can be efficiently computed using a majorization-minimization algorithm. This smoothed likelihood applies a nonlinear regularization operator defined as the exponential of a kernel convolution on the logarithm of each component density. Consistency of the estimators is demonstrated by leveraging classical M-estimation frameworks under mild regularity conditions. Subsequently, convergence rates for both finite- and infinite-dimensional parameters are derived by exploiting structural properties of the smoothed likelihood, the behavior of the iterative optimization algorithm, and a thorough study of the profile smoothed likelihood. This work provides the first rigorous theoretical guarantees for this estimation approach, bridging the gap between practical algorithms and statistical theory in semi-parametric mixture modeling.

翻译：本文为半参数有限混合模型中的一种标准估计量建立了理论保证，其中每个分量密度在条件独立性假设下被建模为一元密度之积。研究聚焦于最大化平滑对数似然函数的估计量，该估计量可通过主优化-最小化算法高效计算。该平滑似然函数应用了一个非线性正则化算子，该算子定义为各分量密度对数经核卷积后的指数函数。在温和的正则性条件下，通过经典M估计框架证明了估计量的一致性。随后，利用平滑似然的结构特性、迭代优化算法的行为以及对轮廓平滑似然的深入研究，推导了有限维与无限维参数的收敛速率。本研究首次为该估计方法提供了严格的理论保证，弥合了半参数混合建模中实用算法与统计理论之间的鸿沟。