Maximum likelihood estimation

Definition

Maximum likelihood estimation (MLE) is a statistical method for estimating the parameters of a probability distribution or a statistical model by maximizing the likelihood function. It connects to the concept of fitting models to data by finding the parameter values that make the observed data most probable under the assumed model.

5 Must Know Facts For Your Next Test

1. MLE is widely used because it has desirable properties such as consistency, asymptotic normality, and efficiency under certain conditions.
2. In many cases, MLE can be computed using numerical optimization techniques when an analytical solution is difficult to obtain.
3. Maximum likelihood estimates are sensitive to model specification, meaning that incorrect model assumptions can lead to biased estimates.
4. When dealing with large samples, MLE often approximates the true parameter values closely, making it a popular choice in econometrics.
5. In the context of regression models, MLE can be used to derive estimators for parameters in logistic regression or other models where traditional methods may not apply.

Review Questions

• How does maximum likelihood estimation relate to the concept of likelihood functions in statistical modeling?
  • Maximum likelihood estimation relies on the likelihood function, which quantifies how likely the observed data is for different parameter values. By maximizing this function, MLE identifies the parameter values that make the observed data most probable under a specified model. This process is essential for fitting statistical models accurately and understanding their underlying structures.
• Discuss how maximum likelihood estimation is utilized in multiple linear regression and its impact on parameter estimation.
  • In multiple linear regression, maximum likelihood estimation provides a method for estimating the coefficients that best fit the relationship between dependent and independent variables. By maximizing the likelihood of observing the given data under the linear model assumptions, MLE leads to efficient and consistent estimates. This approach is particularly useful when traditional methods like ordinary least squares may not fully capture the complexities of the data.
• Evaluate the implications of sample selection bias in the context of maximum likelihood estimation and how it can affect model outcomes.
  • Sample selection bias can significantly distort maximum likelihood estimation results if not properly addressed. When certain observations are systematically excluded from analysis, the estimated parameters may reflect only a biased subset of the population. This leads to inaccurate conclusions about relationships within the data. Techniques like the Heckman selection model can help mitigate this bias by adjusting for sample selection issues, allowing for more reliable parameter estimates and improved model validity.

Related terms

Likelihood function: A function that measures the probability of observing the given data under various parameter values in a statistical model.

Estimator:

A rule or formula that provides an estimate of a population parameter based on sample data.

Convergence: The process by which an iterative method approaches a final value or solution, often used in relation to optimization techniques like MLE.