Maximum Likelihood Estimator

Definition

A maximum likelihood estimator (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function, which measures how well a particular set of parameters explains the observed data. MLE is crucial for understanding sampling distributions, as it provides a way to derive estimates from sample data. This approach also ties into point estimation, as it offers a method for obtaining a single best estimate of an unknown parameter based on observed data, while its relationship with the Cramer-Rao lower bound establishes its efficiency in estimation. Additionally, discussions of admissibility and completeness often address whether MLEs are optimal under certain conditions, enhancing the understanding of their properties in decision theory and estimation theory.

5 Must Know Facts For Your Next Test

1. Maximum likelihood estimators are derived by maximizing the likelihood function, which quantifies how probable the observed data is for different parameter values.
2. MLEs are often preferred because they have desirable properties like consistency, asymptotic normality, and efficiency under regular conditions.
3. In large samples, the distribution of MLEs can be approximated using normal distribution due to the Central Limit Theorem.
4. The Cramer-Rao lower bound provides a theoretical limit on the variance of any unbiased estimator, showing that MLEs can achieve this bound under specific conditions.
5. Maximum likelihood estimators may not always be admissible; there can exist better (lower variance) estimators in certain scenarios, highlighting their limitations in decision theory.

Review Questions

• How does maximizing the likelihood function relate to sampling distributions and why is this important?
  • Maximizing the likelihood function helps derive estimators that best explain the observed data. This process connects directly to sampling distributions as it allows us to understand how sample data reflects population parameters. By using MLEs, we can evaluate how likely various parameter estimates are given different samples, which provides insight into the reliability and variability of those estimates.
• Discuss the efficiency of maximum likelihood estimators in relation to the Cramer-Rao lower bound.
  • Maximum likelihood estimators are often considered efficient because they can achieve the Cramer-Rao lower bound under regular conditions. This means that no unbiased estimator can have a variance lower than that bound. Understanding this relationship allows statisticians to assess whether an estimator is optimal and how close it comes to achieving the best possible precision in estimation.
• Evaluate how maximum likelihood estimators fit within the concepts of admissibility and completeness in statistical decision theory.
  • When evaluating maximum likelihood estimators in terms of admissibility and completeness, it's crucial to consider their performance compared to other potential estimators. MLEs can sometimes be non-admissible if there exist other estimators with better properties in terms of variance or bias. By analyzing these aspects within decision theory frameworks, we can determine situations where MLEs may not be the best choice and explore alternative strategies that yield more robust decision-making outcomes.

Related terms

Likelihood Function: A function that represents the probability of observed data given specific parameters; it is used in maximum likelihood estimation to find parameter values that maximize this probability.

Consistent Estimator: An estimator that converges in probability to the true value of the parameter as the sample size increases; maximum likelihood estimators are often consistent under regular conditions.

Asymptotic Normality: The property that, as sample size increases, the distribution of an estimator approaches a normal distribution, which is a key consideration in the properties of maximum likelihood estimators.