MATLAB Code Implementation of Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a widely used parameter estimation method that aims to find parameter values most likely to have generated the observed data by maximizing the likelihood function. Implementing MLE in MATLAB typically involves defining the likelihood function, selecting appropriate optimization algorithms for solution, and evaluating statistical properties of the estimates, such as calculating the Cramer-Rao Lower Bound (CRLB) to measure estimation accuracy. ### Implementation Steps for Maximum Likelihood Estimation Define the Probability Model First, specify the probability distribution model for the data (e.g., Gaussian distribution, Poisson distribution) and write the corresponding likelihood function. For independent and identically distributed samples, the likelihood function is the product of probability densities at each sample point. Construct the Likelihood Function In MATLAB, the negative log-likelihood function is typically used as the objective function (since optimizers usually handle minimization problems). For example, for Gaussian-distributed data, the negative log-likelihood function may include squared error terms and variance parameters. The implementation often uses MATLAB's anonymous functions or custom function files to define the negative log-likelihood. Optimization Solution Use MATLAB's optimization tools (such as `fminsearch`, `fminunc`, or the `mle` function from the Statistics and Machine Learning Toolbox) to find parameter values that minimize the negative log-likelihood function. Proper initial value selection is crucial, especially for non-convex problems where different starting points may lead to different local minima. Cramer-Rao Lower Bound Calculation CRLB provides the lower bound for the variance of unbiased estimators, reflecting the theoretical optimal estimation accuracy. Calculation requires: Computing the second derivative of the likelihood function (Fisher Information Matrix). Taking its inverse to obtain CRLB. In MATLAB, this can be implemented using the Symbolic Math Toolbox (with `sym` variables) or numerical differentiation methods like finite differences. ### Important Considerations Model Suitability: Ensure the data distribution assumptions are appropriate, otherwise MLE results may be biased. Numerical Stability: Logarithm transformation helps avoid numerical overflow issues from multiplying likelihood functions. CRLB Verification: If the estimated variance is close to CRLB, it indicates the estimator is efficient. By combining MATLAB's numerical optimization and statistical tools, one can efficiently implement maximum likelihood estimation and perform theoretical boundary analysis. Typical implementation involves writing custom functions for likelihood calculation, using optimization options for better convergence, and validating results through statistical diagnostics.