Bounded completeness

Definition

Bounded completeness refers to a property of a family of distributions in statistics, where the set of expectations for all bounded measurable functions is complete. This means that if every bounded function has a finite expected value under the distribution, then the family of distributions captures all relevant information. This concept is essential when considering estimators and their efficiency in statistical inference.

5 Must Know Facts For Your Next Test

1. Bounded completeness applies particularly to families of distributions that are bounded with respect to the measure they generate.
2. In practical terms, if an estimator is bounded complete, it means you can reliably use it without worrying about missing information that could affect your conclusions.
3. The concept emphasizes the importance of considering only those functions whose expectations are well-defined and finite, ensuring stability in statistical inference.
4. Bounded completeness often helps in establishing the efficiency of estimators, as it relates directly to the concept of admissibility.
5. Statistical theories involving bounded completeness often arise in the context of decision theory and optimal statistical procedures.

Review Questions

• How does bounded completeness relate to the properties of estimators in statistical inference?
  • Bounded completeness plays a significant role in understanding estimators' behavior by ensuring that estimators derived from a complete family yield reliable results. If an estimator is derived from a bounded complete family, it assures that we won't overlook essential information when estimating parameters. This property leads to more efficient estimates because it guarantees that all relevant functions are accounted for when calculating expectations.
• Discuss the implications of bounded completeness for decision theory in statistics.
  • In decision theory, bounded completeness ensures that decisions based on estimators from complete families are not only optimal but also reliable under uncertainty. When estimators maintain this property, it implies that they effectively use all available data without leaving out crucial information. This makes decisions based on such estimators less prone to errors and biases, thus enhancing overall decision-making effectiveness within statistical frameworks.
• Evaluate how bounded completeness contributes to establishing the efficiency and admissibility of estimators.
  • Bounded completeness contributes significantly to establishing both efficiency and admissibility by providing a framework within which we can assess the adequacy of estimators. An efficient estimator minimizes variance among unbiased estimators, while admissibility ensures no other estimator performs better in terms of risk. When estimators are from a bounded complete family, they tend to be both efficient and admissible because they incorporate all relevant information while avoiding redundancy, leading to better statistical practices overall.

Related terms

completeness: Completeness is a property of a statistical family where no non-trivial unbiased estimator exists for zero if the parameter is in the family.

sufficient statistic: A sufficient statistic is a statistic that captures all necessary information from the data regarding a parameter of interest, meaning no additional data can improve the estimate.

Lebesgue integrable: A function is Lebesgue integrable if it is measurable and its integral over its domain exists and is finite, which is crucial for discussing expectations in bounded completeness.