Maximum Likelihood Estimator (MLE)

Definition

The maximum likelihood estimator (MLE) is a method for estimating the parameters of a statistical model by maximizing the likelihood function. This approach identifies the parameter values that make the observed data most probable under the assumed model. MLE is widely used due to its desirable properties, such as consistency and asymptotic normality, making it a powerful tool in statistical inference.

5 Must Know Facts For Your Next Test

1. The likelihood function is constructed based on the joint probability of observing the data given specific parameter values.
2. To find the MLE, one often takes the natural logarithm of the likelihood function, which simplifies calculations and helps in optimization.
3. In cases where it's difficult to derive the MLE analytically, numerical methods like gradient ascent or Newton-Raphson can be employed.
4. MLE estimators are particularly useful because they are often asymptotically efficient, meaning they achieve the lowest possible variance among unbiased estimators as sample size increases.
5. The MLE can be sensitive to model specifications and outliers, so careful consideration of model assumptions is essential.

Review Questions

• How does the likelihood function relate to finding the maximum likelihood estimator?
  • The likelihood function quantifies how likely it is to observe the data given different parameter values in a statistical model. To find the maximum likelihood estimator, you maximize this function by identifying which parameter values result in the highest probability of obtaining the observed data. Essentially, MLE uses this relationship to pinpoint the parameters that best explain the data within the framework of the chosen model.
• Discuss why maximum likelihood estimators are considered asymptotically efficient and what that means for practical applications.
  • Maximum likelihood estimators are deemed asymptotically efficient because, as sample sizes grow larger, they achieve the lowest possible variance among all unbiased estimators. This efficiency implies that MLEs yield precise estimates when working with large datasets, which is often a common scenario in practice. Consequently, practitioners prefer MLEs because they provide reliable estimates that improve as more data becomes available.
• Evaluate the impact of model selection on maximum likelihood estimation and how it can affect results.
  • Model selection plays a crucial role in maximum likelihood estimation because choosing an inappropriate model can lead to biased or inconsistent estimates. If the assumed model does not accurately represent the underlying data-generating process, even a maximized likelihood can yield poor or misleading parameter estimates. Therefore, careful consideration of model assumptions and diagnostics is essential for ensuring valid conclusions drawn from MLE results, making robust model selection critical in statistical inference.

Related terms

Likelihood Function:

A function that measures the probability of observing the given data under various parameter values, providing the foundation for maximum likelihood estimation.

Consistent Estimator:

An estimator that converges in probability to the true parameter value as the sample size increases, ensuring accurate estimation with larger samples.

Asymptotic Normality:

The property that, as the sample size approaches infinity, the distribution of the estimator approaches a normal distribution, which aids in making statistical inferences.