The maximum likelihood estimator (MLE) is pivotal in statistical inference, yet its application is often hindered by the absence of closed-form solutions for many models. This poses challenges in real-time computation scenarios, particularly within embedded systems technology, where numerical methods are impractical. This study introduces a generalized form of the MLE that yields closed-form estimators under certain conditions. We derive the asymptotic properties of the proposed estimator and demonstrate that our approach retains key properties such as invariance under one-to-one transformations, strong consistency, and an asymptotic normal distribution. The effectiveness of the generalized MLE is exemplified through its application to the Gamma, Nakagami, and Beta distributions, showcasing improvements over the traditional MLE. Additionally, we extend this methodology to a bivariate gamma distribution, successfully deriving closed-form estimators. This advancement presents significant implications for real-time statistical analysis across various applications.

翻译：极大似然估计（MLE）在统计推断中具有核心地位，但其应用常受限于许多模型缺乏闭式解。这一问题在实时计算场景（尤其是嵌入式系统技术）中尤为突出，此时数值方法并不可行。本研究提出了一种广义形式的MLE，可在特定条件下生成闭式估计量。我们推导了所提估计量的渐近性质，并证明该方法保留了一一变换不变性、强相合性及渐近正态分布等关键性质。通过将该广义MLE应用于伽马分布、中川分布和贝塔分布，验证了其相较于传统MLE的改进效果。此外，我们将该方法扩展至二元伽马分布，成功推导出闭式估计量。这一进展对各类应用中的实时统计分析具有显著意义。