Maximum Likelihood Estimator

Definition

The maximum likelihood estimator (MLE) is a statistical method for estimating the parameters of a probability distribution by maximizing the likelihood function. It finds the parameter values that make the observed data most probable, providing a way to point estimate unknown parameters based on sample data. MLE is widely used because it has desirable properties, such as consistency and asymptotic normality, making it a key concept in point estimation and properties of estimators.

5 Must Know Facts For Your Next Test

1. The MLE is derived from the principle of finding the parameter values that maximize the likelihood of observing the given sample data.
2. One important property of MLEs is that they are consistent, meaning they converge to the true parameter value as sample size increases.
3. MLEs can be sensitive to model misspecification; if the assumed model is incorrect, MLE may yield biased estimates.
4. In large samples, MLEs have asymptotic properties such as being normally distributed around the true parameter value.
5. The method can be applied to various statistical models, including normal distributions, binomial distributions, and Poisson distributions.

Review Questions

• How does the maximum likelihood estimator relate to point estimation in statistics?
  • The maximum likelihood estimator is a fundamental method of point estimation that provides a way to estimate unknown parameters based on observed data. By maximizing the likelihood function, MLE identifies parameter values that are most consistent with the data. This approach aligns with the goal of point estimation, which seeks to provide single best estimates of parameters rather than intervals or ranges.
• Discuss how consistency and asymptotic normality are important properties of maximum likelihood estimators.
  • Consistency ensures that as more data is collected, the maximum likelihood estimator will converge to the true parameter value, providing reliable estimates in practice. Asymptotic normality indicates that with large sample sizes, the distribution of MLEs approaches a normal distribution. These properties are crucial because they give statisticians confidence in using MLEs for inference and hypothesis testing.
• Evaluate the potential pitfalls of using maximum likelihood estimators and how these might affect statistical analysis.
  • While maximum likelihood estimators are powerful tools, they can be sensitive to assumptions about the underlying model. If a model is incorrectly specified, it may lead to biased or inconsistent estimates, affecting conclusions drawn from statistical analyses. Additionally, in small samples, MLEs may not perform well and could result in large variance. Understanding these limitations is essential for accurate interpretation and application in real-world scenarios.

Related terms

Likelihood Function: A function that measures how well a statistical model explains the observed data for different parameter values.

Consistency: A property of an estimator indicating that as the sample size increases, the estimator converges in probability to the true parameter value.

Asymptotic Normality: A property where, as the sample size approaches infinity, the distribution of an estimator approaches a normal distribution.