Admissible Estimator

Definition

An admissible estimator is a statistical estimator that cannot be improved upon in terms of lower expected loss when compared to other estimators. This concept is important in decision theory and is closely related to the notions of risk functions and optimality in estimation. Admissibility highlights the idea that if there exists no other estimator with lower risk for all parameter values, then the estimator in question is deemed admissible.

5 Must Know Facts For Your Next Test

1. An estimator can be admissible even if it is not uniformly best, meaning there may be other estimators that are better for specific parameter values but worse overall.
2. In some cases, an inadmissible estimator might still be preferred due to simplicity or computational convenience, even if its risk is higher.
3. Admissibility can be assessed by comparing risk functions of different estimators across the parameter space to find any alternatives that provide lower risks.
4. Certain classes of estimators, such as Bayes estimators under specific prior distributions, are often admissible.
5. The concept of admissibility plays a crucial role in determining which estimation strategies should be used when making statistical inferences.

Review Questions

• How does the concept of admissibility relate to the performance of different estimators?
  • Admissibility relates to the performance of estimators by indicating that an admissible estimator cannot be outperformed by any other estimator in terms of lower expected loss across all parameter values. This concept allows statisticians to identify which estimators are robust and reliable choices when estimating parameters. It emphasizes the importance of understanding not just individual performance, but how estimators compare across a broader context.
• In what situations might a statistician prefer an inadmissible estimator over an admissible one?
  • A statistician might prefer an inadmissible estimator over an admissible one when factors such as simplicity, interpretability, or computational ease are prioritized over minimal risk. For example, if an estimator is easier to calculate or understand but has a slightly higher risk than another option, it may be chosen for practical reasons. This illustrates that while admissibility is important, real-world applications often require balancing theoretical ideals with practical constraints.
• Evaluate the implications of the Complete Class Theorem on the understanding of admissible estimators and their use in statistical inference.
  • The Complete Class Theorem asserts that if an estimator is admissible, it belongs to a complete class where no other estimator can outperform it in terms of risk. This has profound implications for statistical inference because it helps researchers focus on using admissible estimators without needing to consider alternatives that could lead to worse decisions. It provides assurance that by selecting an admissible estimator, one is making a theoretically sound choice based on the minimization of expected loss.

Related terms

Risk Function: A function that describes the expected loss associated with an estimator, typically used to evaluate its performance.

Minimax Estimator: An estimator that minimizes the maximum risk across all possible parameter values, offering a robust approach to estimation.

Complete Class Theorem: A theorem that states if an estimator is admissible, then it belongs to a complete class of estimators, meaning that it cannot be dominated by any other estimator.