Estimating Maximum Likelihood Using Probability Density Functions in MATLAB

Given data and corresponding probability density functions, we can utilize MATLAB to compute maximum likelihood estimates. This process involves first calculating the likelihood function based on the provided data, then incorporating the appropriate probability density function into this calculation. MATLAB's numerical optimization tools, such as fminsearch or fminunc functions, are employed to locate the maximum value of the likelihood function. During optimization, careful selection of initial parameter values and appropriate optimization algorithms is crucial for obtaining accurate estimation results. Common approaches include using method of moments estimates as starting points or implementing multiple random initializations to avoid local optima. Furthermore, we can apply statistical hypothesis testing methods, such as likelihood ratio tests or Wald tests, to verify the significance of maximum likelihood estimates, ensuring the reliability of our obtained results. For implementation, MATLAB's mle function provides built-in support for common distributions, while custom distributions require manual likelihood function definition and optimization using Optimization Toolbox routines.