12.5 Admissibility and completeness
7 min read•august 21, 2024
Admissibility and completeness are key concepts in theoretical statistics, shaping how we evaluate and choose estimators. These ideas help us understand which statistical methods perform best and why, connecting to broader themes of optimization and in data analysis.
These concepts have wide-ranging applications, from improving estimators to constructing powerful hypothesis tests. By exploring admissibility and completeness, we gain insights into the fundamental limits of statistical inference and learn to make more informed choices in our analyses.
Definition of admissibility
- Admissibility forms a crucial concept in theoretical statistics evaluating the performance of estimators
- Plays a vital role in determining optimal statistical procedures for parameter estimation and hypothesis testing
- Connects to broader themes of decision theory and statistical inference in the field of theoretical statistics
Admissible estimators
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Top images from around the web for Admissible estimators
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Maximum Likelihood Estimate and Logistic Regression simplified — Pavan Mirla View original
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- Estimators that cannot be uniformly outperformed by any other estimator for all parameter values
- Possess the property of minimizing expected loss or risk across the entire parameter space
- Often derived using methods such as maximum likelihood estimation or Bayesian inference
- Maintain their admissibility under one-to-one transformations of the parameter space
Inadmissible estimators
- Estimators that can be improved upon by another estimator for all parameter values
- Exhibit suboptimal performance in terms of bias, variance, or
- Often result from naive or simplistic approaches to estimation
- Can sometimes be improved through techniques like shrinkage or regularization
Completeness concept
- Completeness serves as a fundamental property in theoretical statistics for characterizing sufficient statistics
- Ensures the uniqueness of unbiased estimators based on sufficient statistics
- Connects to the broader theory of statistical inference and estimation efficiency
Complete statistics
- Statistics that allow only the zero function to have an expected value of zero for all parameter values
- Satisfy the condition for all θ implies almost surely
- Guarantee the existence of a unique minimum variance unbiased estimator (MVUE)
- Often found in exponential family distributions (normal, Poisson, binomial)
Incomplete statistics
- Statistics that do not satisfy the completeness property
- Allow for multiple unbiased estimators with potentially different variances
- May lead to difficulties in determining optimal estimators
- Can occur in discrete distributions or when dealing with nuisance parameters
Basu's theorem
- establishes a fundamental relationship between sufficiency, completeness, and ancillarity
- Provides insights into the independence of certain statistics in estimation problems
- Contributes to the understanding of optimal statistical procedures in theoretical statistics
Assumptions and conditions
- Requires the existence of a complete sufficient statistic T for the parameter θ
- Assumes the presence of a statistic S that is ancillary for θ
- Applies to families of distributions with well-defined likelihood functions
- Holds under regularity conditions ensuring the existence of expectations and derivatives
Applications of Basu's theorem
- Proves the independence of complete sufficient statistics and ancillary statistics
- Simplifies the construction of confidence intervals and hypothesis tests
- Aids in identifying minimal sufficient statistics in complex models
- Facilitates the derivation of uniformly most powerful tests in certain scenarios
Rao-Blackwell theorem
- provides a method for improving estimators in terms of variance reduction
- Demonstrates the importance of sufficient statistics in achieving optimal estimation
- Connects to the broader theory of minimum variance unbiased estimation in theoretical statistics
Statement of theorem
- Given an unbiased estimator θ̂ of θ, the conditional expectation E[θ̂|T], where T is a sufficient statistic, is also unbiased
- The variance of the conditional expectation is always less than or equal to the variance of the original estimator
- Mathematically expressed as for all θ
- Equality holds if and only if θ̂ is already a function of the sufficient statistic T
Implications for estimation
- Provides a systematic method for improving estimators by conditioning on sufficient statistics
- Guarantees that the best estimator is always a function of a sufficient statistic
- Leads to the concept of complete class of estimators in decision theory
- Forms the basis for constructing uniformly minimum variance unbiased estimators (UMVUE)
Lehmann-Scheffé theorem
- combines the concepts of sufficiency, completeness, and
- Provides conditions for achieving minimum variance unbiased estimation
- Represents a cornerstone result in the theory of point estimation in theoretical statistics
Conditions for efficiency
- Requires the existence of a complete sufficient statistic T for the parameter θ
- Assumes the estimator θ̂ is an unbiased function of the complete sufficient statistic T
- Applies to families of distributions satisfying regularity conditions
- Holds under the assumption of finite variance for the estimator
Relationship to completeness
- Completeness of the sufficient statistic ensures the uniqueness of the unbiased estimator
- Allows for the construction of the uniformly minimum variance unbiased estimator (UMVUE)
- Connects the concepts of sufficiency and completeness in achieving optimal estimation
- Demonstrates the importance of complete sufficient statistics in statistical inference
Minimaxity vs admissibility
- Minimaxity and admissibility represent two important criteria for evaluating estimators in decision theory
- Provide different perspectives on the optimality of statistical procedures
- Contribute to the understanding of trade-offs in estimation and hypothesis testing in theoretical statistics
Minimax estimators
- Estimators that minimize the maximum risk over the entire parameter space
- Provide a conservative approach to estimation by focusing on worst-case scenarios
- Often derived using game-theoretic principles or convex optimization techniques
- May not always coincide with admissible estimators in certain problem settings
Connections to admissibility
- All minimax estimators are admissible in finite parameter spaces
- In infinite parameter spaces, minimax estimators may or may not be admissible
- Admissible estimators can sometimes be constructed by randomizing between minimax estimators
- The relationship between minimaxity and admissibility depends on the specific problem and loss function
Completeness in exponential families
- Completeness plays a crucial role in the analysis of exponential family distributions
- Provides a powerful tool for deriving optimal estimators and test statistics
- Connects to the broader theory of sufficient statistics and Fisher information in theoretical statistics
Sufficiency and completeness
- Exponential families possess natural sufficient statistics that are often complete
- Completeness of the sufficient statistic ensures the existence of a unique UMVUE
- Allows for the application of Lehmann-Scheffé theorem in deriving optimal estimators
- Facilitates the construction of uniformly most powerful tests for exponential families
Examples in common distributions
- Normal distribution: Sample mean and variance form a complete sufficient statistic
- Poisson distribution: Sample sum is a complete sufficient statistic for the rate parameter
- Binomial distribution: Number of successes is a complete sufficient statistic for the probability parameter
- Exponential distribution: Sample sum is a complete sufficient statistic for the rate parameter
Admissibility in decision theory
- Admissibility serves as a fundamental concept in statistical decision theory
- Provides a framework for evaluating and comparing different decision rules
- Connects to broader themes of risk analysis and optimality in theoretical statistics
Loss functions
- Quantify the cost or penalty associated with estimation errors or incorrect decisions
- Common examples include squared error loss, absolute error loss, and 0-1 loss
- Choice of loss function influences the admissibility of decision rules
- Can be symmetric or asymmetric depending on the problem context
Risk and admissibility
- Risk defined as the expected loss over the sampling distribution of the data
- Admissible decision rules cannot be uniformly improved in terms of risk
- Inadmissible rules can sometimes be improved through techniques like shrinkage or regularization
- Admissibility often conflicts with other optimality criteria (minimaxity, unbiasedness)
Completeness and ancillary statistics
- Completeness and ancillarity represent important concepts in the theory of statistical inference
- Provide insights into the information content of different statistics
- Contribute to the understanding of optimal estimation and testing procedures in theoretical statistics
Ancillarity concept
- Ancillary statistics contain no information about the parameter of interest
- Distribution of ancillary statistics does not depend on the parameter being estimated
- Often used to condition on to improve estimation or testing procedures
- Examples include the range in normal samples or the order statistics in uniform distributions
Completeness vs ancillarity
- Complete statistics utilize all available information about the parameter
- Ancillary statistics provide no information about the parameter
- Basu's theorem establishes the independence of complete sufficient and ancillary statistics
- Completeness and ancillarity represent opposite ends of the information spectrum in statistics
Applications in hypothesis testing
- Admissibility and completeness concepts extend beyond estimation to hypothesis testing
- Provide criteria for evaluating and constructing optimal test procedures
- Connect to broader themes of power, size, and uniformly most powerful tests in theoretical statistics
Admissible tests
- Tests that cannot be uniformly improved in terms of power for all parameter values
- Often derived using likelihood ratio or Neyman-Pearson approaches
- May depend on the specific alternative hypothesis and significance level
- Can be characterized using complete class theorems in certain testing problems
Complete class of tests
- Set of tests that contains all admissible tests for a given problem
- Often derived using decision-theoretic principles or Bayesian methods
- Provides a framework for finding optimal tests within a restricted class
- Examples include the class of unbiased tests or invariant tests in certain problems
Limitations and criticisms
- Admissibility and completeness, while powerful concepts, have certain limitations in practice
- Understanding these limitations is crucial for appropriate application in real-world statistical problems
- Connects to broader discussions on the foundations and philosophy of statistical inference
Practical considerations
- Admissible estimators may be difficult to compute or implement in high-dimensional problems
- Complete sufficient statistics may not always exist or be easily identifiable in complex models
- Trade-offs between admissibility and other criteria (simplicity, robustness) in applied settings
- Sensitivity of admissibility results to model assumptions and prior specifications
Alternative approaches
- Robust statistics focus on procedures that perform well under departures from model assumptions
- Shrinkage methods (ridge regression, lasso) often outperform admissible estimators in high-dimensional settings
- Bayesian approaches provide an alternative framework for dealing with parameter uncertainty
- Machine learning techniques offer data-driven alternatives to traditional statistical inference
Key Terms to Review (19)
Admissible Estimator: 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.
Basu's Theorem: Basu's Theorem states that a statistic that is complete and sufficient for a family of distributions is also an admissible estimator. This theorem connects the concepts of completeness, sufficiency, and admissibility, highlighting how these properties interact in the context of statistical inference. Understanding Basu's Theorem is crucial as it helps determine the optimality of estimators, ensuring that they are not only efficient but also robust under various sampling scenarios.
Bayesian estimator: A Bayesian estimator is a statistical method used to estimate the parameters of a statistical model based on Bayes' theorem. It incorporates prior beliefs or information about the parameters, updating this knowledge with observed data to produce a posterior distribution, which provides a comprehensive view of uncertainty around the estimates. This approach allows for the blending of prior information with empirical evidence, making it particularly useful in situations with limited data or when prior knowledge is crucial.
Complete Statistic: A complete statistic is a type of statistic that captures all the information available in a sample about a parameter of interest, meaning that no unbiased estimator can improve upon it. In essence, if a statistic is complete, any unbiased function of that statistic is also unbiased, which establishes a strong relationship with sufficiency and efficiency. The concept of completeness plays an important role in determining the admissibility of estimators and understanding how they behave in the context of estimation theory.
Consistency: Consistency refers to a property of an estimator where, as the sample size increases, the estimates produced converge in probability to the true value of the parameter being estimated. This concept is crucial in statistics because it ensures that with enough data, the estimators will yield results that are close to the actual parameter value, providing reliability in statistical inference.
David Lehmann: David Lehmann is a prominent statistician known for his contributions to the field of theoretical statistics, particularly in the areas of admissibility and completeness of statistical procedures. His work has significantly influenced how statisticians understand the conditions under which statistical estimates can be considered optimal, especially regarding their properties in decision theory and inference.
Efficiency: In statistics, efficiency refers to the quality of an estimator in terms of the amount of information it utilizes from the data to produce estimates. An efficient estimator has the lowest possible variance among all unbiased estimators for a given parameter, which means it makes optimal use of available data. This concept is crucial in evaluating point estimations, maximum likelihood estimation, and properties of estimators, as it determines how well estimators can produce accurate and precise parameter estimates while maintaining desirable statistical properties.
Hans Fischer: Hans Fischer was a prominent German chemist known for his work on porphyrins and heme compounds, which are crucial in various biological systems. His contributions extend to the understanding of molecular structures and the role they play in biological processes, making significant impacts in areas like biochemistry and medicine.
Incomplete Statistic: An incomplete statistic is a function of the observed data that does not capture all the information available about a parameter of interest, leading to potential inefficiencies in statistical inference. It arises in situations where some data or information is either missing or not utilized fully, which can result in biased estimates or reduced power in hypothesis testing. Understanding incomplete statistics is crucial when considering the admissibility of estimators and the completeness of sufficient statistics in statistical inference.
Interval Estimator: An interval estimator is a type of estimator that provides a range of values within which a population parameter is expected to lie, along with a specified level of confidence. This is crucial because it not only estimates the parameter but also indicates the uncertainty associated with that estimate. Interval estimators often take the form of confidence intervals and are characterized by their properties like reliability and accuracy, influencing their admissibility and completeness.
Lehmann-Scheffé Theorem: The Lehmann-Scheffé Theorem is a fundamental result in statistics that provides conditions under which an estimator is considered admissible and also establishes that the best unbiased estimator is unique if it exists. This theorem connects the concepts of completeness and admissibility, emphasizing how a complete sufficient statistic can help identify the optimality of estimators.
Maximum Likelihood Estimator: A maximum likelihood estimator (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function, which measures how well a particular set of parameters explains the observed data. MLE is crucial for understanding sampling distributions, as it provides a way to derive estimates from sample data. This approach also ties into point estimation, as it offers a method for obtaining a single best estimate of an unknown parameter based on observed data, while its relationship with the Cramer-Rao lower bound establishes its efficiency in estimation. Additionally, discussions of admissibility and completeness often address whether MLEs are optimal under certain conditions, enhancing the understanding of their properties in decision theory and estimation theory.
Mean Squared Error: Mean Squared Error (MSE) is a measure of the average squared difference between estimated values and the actual value. It serves as a fundamental tool in assessing the quality of estimators and predictions, playing a crucial role in statistical inference, model evaluation, and decision-making processes. Understanding MSE helps in the evaluation of the efficiency of estimators, particularly in asymptotic theory, and is integral to defining loss functions and evaluating risk in Bayesian contexts.
Minimax Estimator: A minimax estimator is a statistical estimator that minimizes the maximum possible risk or loss, providing a robust solution against the worst-case scenario. This approach is particularly useful in decision theory, where it aims to achieve an optimal compromise between bias and variance while ensuring the performance remains acceptable under adverse conditions. By focusing on minimizing the worst-case loss, minimax estimators often serve as a safeguard in situations with limited information or uncertainty.
Non-admissible estimator: A non-admissible estimator is an estimation method that can be outperformed by another estimator across all possible parameter values in terms of expected loss. This means there exists at least one alternative estimator that yields a lower risk for every possible state of nature. Understanding non-admissibility helps identify estimators that are suboptimal and not suitable for certain applications.
Point estimator: A point estimator is a statistic used to provide a single best guess or estimate of an unknown population parameter. This estimate summarizes a sample's data into a single value, which is critical for making inferences about the larger population. Understanding point estimators involves evaluating their properties, such as unbiasedness, efficiency, and consistency, as well as concepts like admissibility and completeness, which help in determining their optimality and robustness.
Rao-Blackwell Theorem: The Rao-Blackwell Theorem is a fundamental result in statistical estimation that provides a method for improving an estimator by using a sufficient statistic. It states that if you have an unbiased estimator, you can create a new estimator by taking the expected value of the original estimator conditioned on a sufficient statistic, which will always yield a new estimator that is at least as good as the original one in terms of variance. This theorem connects closely with concepts like sufficiency, efficiency, and admissibility in statistical theory.
Risk function: The risk function measures the expected loss associated with a statistical decision-making procedure, reflecting how well a specific estimator or decision rule performs in terms of accuracy. It connects to the concepts of Bayes risk and admissibility, providing a framework for evaluating the effectiveness of different statistical methods in terms of their potential errors and their ability to minimize those errors under uncertainty.
Unbiasedness: Unbiasedness refers to a property of an estimator where the expected value of the estimator equals the true parameter it estimates. This characteristic ensures that, on average, the estimator neither overestimates nor underestimates the parameter, making it a desirable feature in statistical estimation. Unbiasedness is crucial for reliable inference and is often assessed alongside other properties such as consistency and efficiency.