MATLAB Source Code for Maximum Likelihood Estimation (MLE)

Maximum Likelihood Estimation (MLE) is a widely used parameter estimation method in statistics, with its core principle being to find parameter values that are most likely to generate the observed data by maximizing the likelihood function. This method can be implemented in MATLAB through various approaches. Implementing MLE in MATLAB typically involves several key steps: First, defining the probability distribution model according to the specific problem context. For common distributions like the normal distribution, MATLAB provides built-in functions. Next, constructing the likelihood function or log-likelihood function is essential. Since the product-form likelihood function may cause numerical underflow, the log-likelihood function is typically used instead for stability. MATLAB offers numerous optimization functions for finding maximum likelihood estimates, such as fminsearch (unconstrained nonlinear optimization using derivative-free method) or fminunc (unconstrained nonlinear optimization with gradient computation). These functions search for parameter values that maximize the likelihood function. When using these optimization functions, providing initial parameter guesses is crucial, as this choice significantly impacts optimization results and convergence speed. The implementation typically involves defining an objective function that returns the negative log-likelihood value. For more complex scenarios, MATLAB's Statistics and Machine Learning Toolbox provides specialized functions like the mle function. This function can directly estimate parameters for various probability distributions and offers additional options to handle special cases like censored data. The mle function supports distribution specification through distribution names or custom probability density functions. In practical applications, MLE implementation may encounter various challenges such as local optima and convergence difficulties. In such cases, different optimization algorithms can be employed, or multiple initial points can be tried to search for global optimum solutions. For small sample situations, regularization techniques or other adjustment methods might be necessary to improve estimation robustness. Code implementation should include error handling for convergence warnings and validation checks for parameter estimates.