  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.
