Maximum likelihood estimation (MLE)

Definition

Maximum likelihood estimation (MLE) is a statistical method used for estimating the parameters of a statistical model. The technique focuses on finding the parameter values that maximize the likelihood function, which measures how well the model explains the observed data. In multiple linear regression, MLE provides a way to estimate coefficients by maximizing the probability of observing the given data under a specified model.

5 Must Know Facts For Your Next Test

1. MLE is particularly powerful in large samples, where it produces estimates that are asymptotically unbiased and efficient.
2. In multiple linear regression, MLE estimates coincide with OLS estimates when errors are normally distributed.
3. The MLE approach assumes that the data are independent and identically distributed (i.i.d.).
4. Calculating MLE often involves taking the logarithm of the likelihood function to simplify the optimization process, leading to the log-likelihood.
5. MLE can be sensitive to model misspecification, meaning that incorrect assumptions about the underlying distribution can lead to biased estimates.

Review Questions

• How does maximum likelihood estimation relate to the assumptions made in multiple linear regression?
  • Maximum likelihood estimation relies on certain assumptions about the underlying distribution of errors in a regression model. Specifically, for MLE to yield valid results in multiple linear regression, it is typically assumed that errors are normally distributed and independent. If these assumptions hold true, MLE estimates will be consistent with those obtained using ordinary least squares. However, if these assumptions are violated, it may result in inaccurate or biased parameter estimates.
• Compare maximum likelihood estimation and ordinary least squares in terms of their estimation methods for regression coefficients.
  • Maximum likelihood estimation (MLE) and ordinary least squares (OLS) both aim to estimate regression coefficients but use different approaches. OLS minimizes the sum of squared residuals between observed and predicted values, while MLE maximizes the likelihood function based on how likely it is to observe the given data under specific parameter values. When errors are normally distributed, both methods yield identical estimates. However, MLE has broader applications in various statistical models beyond just linear regression.
• Evaluate how maximum likelihood estimation impacts model selection and parameter inference in multiple linear regression.
  • Maximum likelihood estimation plays a crucial role in model selection and parameter inference by providing a systematic way to evaluate different models based on their likelihood of explaining observed data. By comparing likelihoods or log-likelihoods across competing models, researchers can determine which model best fits the data. Moreover, MLE allows for hypothesis testing regarding parameter significance through likelihood ratio tests, enhancing our understanding of which predictors are meaningful in explaining variability in the response variable.

Related terms

Likelihood Function: A function that measures the probability of observing the given data as a function of the model parameters.

Ordinary Least Squares (OLS): A method for estimating the parameters in a linear regression model by minimizing the sum of squared differences between observed and predicted values.

Parameter Estimation: The process of using sample data to infer the values of parameters in a statistical model.