maximumlikelihoodmetoder

Maximum likelihood methods are a class of statistical techniques used to estimate the parameters of a probability distribution. The core idea is to find the parameter values that maximize the likelihood of observing the actual data. In simpler terms, it asks: "Given the data I have, what are the most probable values for the underlying parameters of the model that generated this data?"

The process begins by defining a statistical model with unknown parameters. A likelihood function is then constructed, which represents the probability of observing the given data as a function of these parameters. Maximizing this likelihood function, often through calculus or numerical optimization, yields the maximum likelihood estimates (MLEs) for the parameters. These MLEs are considered the best estimates in the sense that they make the observed data "most likely".

Maximum likelihood estimation is widely used across various fields, including econometrics, biostatistics, and machine learning. It offers desirable statistical properties, such as asymptotic efficiency and consistency, meaning that as the sample size increases, the estimates converge to the true parameter values and achieve the lowest possible variance among unbiased estimators. However, MLEs can sometimes be biased for small sample sizes.