Maximum likelihood estimation (mle)

Definition

Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function. In simpler terms, MLE finds the parameter values that make the observed data most probable, which helps in deriving efficient estimators. It connects closely to concepts like consistency, as MLE estimators typically converge to the true parameter values as the sample size increases.

5 Must Know Facts For Your Next Test

1. MLE provides estimates that are asymptotically unbiased, meaning they become accurate as sample size increases.
2. The method relies on the assumption that data are independent and identically distributed (i.i.d.).
3. MLE estimators are efficient, achieving the lowest possible variance among all consistent estimators under certain regularity conditions.
4. For MLE to be valid, certain regularity conditions must hold, such as differentiability of the likelihood function.
5. In large samples, MLE follows a normal distribution due to the Central Limit Theorem, allowing for confidence interval construction around parameter estimates.

Review Questions

• How does maximum likelihood estimation ensure consistency in parameter estimation?
  • Maximum likelihood estimation ensures consistency by relying on the likelihood function, which is constructed from the observed data and parameter values. As the sample size increases, the MLE converges to the true parameter value because it maximizes the probability of observing the given data. This means that with larger samples, MLE estimators are more likely to reflect the true underlying parameters, illustrating how consistency is achieved in practice.
• What are the implications of asymptotic normality for maximum likelihood estimators and how does it relate to consistency?
  • Asymptotic normality implies that as the sample size grows, maximum likelihood estimators will distribute normally around the true parameter value. This characteristic allows us to construct confidence intervals and conduct hypothesis tests using normal distribution properties. As MLE is consistent, it becomes more accurate with larger samples, reinforcing that not only do these estimators converge to the true values, but they also exhibit predictable statistical behavior in large samples.
• Evaluate how maximum likelihood estimation can be applied in real-world scenarios and its advantages over other estimation methods.
  • Maximum likelihood estimation can be applied in various fields such as economics, biology, and engineering where model parameters need estimation from observed data. Its advantages include producing efficient and consistent estimators, particularly useful in large datasets. Compared to methods like method of moments or least squares, MLE often provides better statistical properties under certain conditions, making it a preferred choice when dealing with complex models or distributions.

Related terms

Likelihood Function: A function that measures the probability of observing the given data under different parameter values.

Asymptotic Normality:

A property of estimators indicating that as the sample size grows, the distribution of the estimator approaches a normal distribution.

Consistency:

The property of an estimator to converge in probability to the true value of the parameter being estimated as the sample size increases.