A Maximum Likelihood Estimator (MLE) is a statistical method used to estimate the parameters of a probability distribution based on a set of observed data. It is based on the principle of choosing the most likely values for the parameters that would have generated the observed data.
The Maximum Likelihood Estimator works by finding the values of the parameters that maximize the likelihood function, which is a measure of how likely the observed data is to occur given a specific set of parameter values. This is typically done through an iterative process, such as the Newton-Raphson method, to find the maximum point of the likelihood function.
The Maximum Likelihood Estimator assumes that the data is independent and identically distributed (i.i.d.), meaning that each data point is independent of the others and is drawn from the same underlying distribution. It also assumes that the data is continuous and follows a known probability distribution.
The Maximum Likelihood Estimator is a widely used and well-studied method for parameter estimation. It is also consistent, meaning that as the sample size increases, the estimated parameters will converge to the true values. Additionally, it can handle complex data and can be used to compare different models.
The Maximum Likelihood Estimator can be sensitive to outliers in the data and may not perform well if the underlying assumptions are not met. It also relies on the choice of the initial values for the parameters, and the estimation may not converge if the starting values are far from the true values. Additionally, it does not provide a measure of uncertainty for the estimated parameters.