Admissible Estimator

Definition

An admissible estimator is a statistical estimator that cannot be improved upon in terms of its expected loss when compared to other estimators. This means that for every possible true value of the parameter being estimated, there is no other estimator that has a lower risk (expected loss) across all possible parameter values. Admissibility is an important concept in decision theory, linking it to minimax procedures which focus on minimizing the maximum risk associated with estimators.

5 Must Know Facts For Your Next Test

1. An admissible estimator is one that does not have any other estimator that dominates it in terms of lower risk for all parameter values.
2. In practice, admissibility helps statisticians select estimators that are reliable and robust, especially under uncertainty.
3. A commonly known example of an admissible estimator is the sample mean under the squared error loss, which can’t be improved upon uniformly.
4. Not all estimators are admissible; some can be dominated by others, leading to a situation where one estimator consistently outperforms another.
5. The concept of admissibility plays a significant role in the development of minimax procedures, where decisions aim to mitigate potential risks.

Review Questions

• What characteristics make an estimator admissible, and how does this concept relate to expected loss?
  • An estimator is considered admissible if there exists no other estimator that has a uniformly lower risk across all parameter values. This means that for every possible true parameter, the expected loss from using an admissible estimator cannot be improved upon by any other. Admissibility is crucial because it provides a benchmark for evaluating the effectiveness of different estimators based on their risks.
• Discuss how admissibility interacts with minimax procedures and the implications for choosing estimators.
  • Admissibility directly informs minimax procedures by identifying estimators that minimize the maximum possible risk. When choosing an estimator, statisticians look for those that are not only admissible but also align with the minimax criterion. This ensures that they can manage worst-case scenarios effectively while still maintaining reliability across a range of possible outcomes.
• Evaluate the importance of admissible estimators in statistical decision theory and their practical applications.
  • Admissible estimators play a vital role in statistical decision theory by providing a foundation for making optimal choices when estimating parameters. Their importance lies in their ability to ensure that decisions made based on these estimators lead to minimal expected loss, even in uncertain conditions. Practically, they guide statisticians in selecting methods for data analysis and interpretation, helping improve overall accuracy and reliability in diverse fields such as economics, medicine, and engineering.

Related terms

Risk Function: The risk function measures the expected loss of an estimator, reflecting how well it performs in estimating a parameter across different scenarios.

Minimax Estimator: A minimax estimator is designed to minimize the maximum possible risk, providing a robust approach in the face of uncertainty regarding the true parameter value.

Bayesian Estimator: A Bayesian estimator incorporates prior beliefs or information about parameters and updates these beliefs based on observed data, often resulting in improved estimates.