Maximum likelihood estimation

Maximum Likelihood Estimation (MLE) is a method used in statistics to estimate the parameters of a statistical model. The principle behind MLE is to find the parameter values that maximize the likelihood function, which measures how well the model with those parameters explains the observed data. MLE is widely used in various fields, including Econometrics, Biostatistics, and Machine Learning.

Overview

The likelihood function is a fundamental concept in statistical inference, representing the probability of observing the given data under different parameter values of a statistical model. In the context of MLE, the goal is to find the parameter values that make the observed data most probable. This approach is based on the principle of likelihood, which was introduced by Ronald A. Fisher in the early 20th century.

Mathematical Formulation

Given a set of independent and identically distributed (i.i.d.) observations \(X_1, X_2, ..., X_n\) from a probability distribution with a parameter \(\theta\), the likelihood function \(L(\theta)\) is defined as the joint probability of the observed data:

\[L(\theta) = f(X_1, X_2, ..., X_n | \theta)\]

where \(f\) is the probability density function (pdf) or probability mass function (pmf) of the observations. The maximum likelihood estimate \(\hat{\theta}\) of the parameter \(\theta\) is the value that maximizes \(L(\theta)\).

Estimation Process

The estimation process typically involves the following steps: 1. Specify the statistical model and its associated likelihood function. 2. Derive the likelihood function based on the observed data. 3. Find the parameter values that maximize the likelihood function. This is often done using calculus, by setting the derivative of the likelihood function with respect to the parameter to zero and solving for the parameter. 4. Assess the goodness-of-fit of the model and the reliability of the estimates through various diagnostic and validation techniques.

Applications

MLE is used in a wide range of applications, including: - Estimating the parameters of a normal distribution or other probability distributions. - Fitting models in regression analysis. - Estimating parameters in generalized linear models (GLMs). - Parameter estimation in Machine Learning algorithms.

Advantages and Limitations

MLE has several advantages, including consistency (the estimates converge to the true parameter values as the sample size increases) and efficiency (the estimates have the smallest possible variance among all unbiased estimators). However, MLE also has limitations, such as sensitivity to outliers and the assumption of a correctly specified model.

See Also

• Statistical Inference
• Bayesian Inference
• Estimator
• Likelihood Function
• Fisher Information