Deriving Lehmer and Hölder Means as Maximum Weighted Likelihood Estimates for the Multivariate Exponential Family

Core Concepts

The Lehmer and Hölder families of means can be derived as maximum weighted likelihood estimates for a subset of the multivariate exponential family of probability distributions.

Abstract

The paper establishes the relationship between the weighted maximum likelihood estimator (MWLE) and the Lehmer and Hölder families of means in the context of the multivariate exponential family of probability distributions. Key highlights: The Lehmer and Hölder means are shown to be MWLEs for specific subclasses of the multivariate exponential family. The MWLE depends not only on the probability distribution, but also on the relevance of the data, as encoded by the weight function. For the Lehmer mean, the weight function is proportional to the observation raised to a power, while for the Hölder mean, the weight function is independent of the observation value. The paper provides a probabilistic interpretation of these mean families, which could potentially broaden their applicability in various domains that rely on maximum likelihood estimation. A case study using the Weibull distribution is presented to illustrate the derivation of the Lehmer and Hölder means as MWLEs. The Lehmer and Hölder means differ in their behavior - the Lehmer mean is bounded between the minimum and maximum observations, while the Hölder mean is bounded between the harmonic mean and the maximum observation.

Stats

The dataset used in the case study is the U.S. Senate statewide election results from 1976 to 2020, containing the proportion of votes for the Democratic, Republican, and Other parties.

Quotes

"The Lehmer and Hölder families of means are used quite often. Recently, their link with maximum likelihood estimators has been established only in the case of univariate probability densities. In this article, we have shown that these two families of means are maximum likelihood estimators in the case of weighted data and PDFs belonging to the multivariate exponential family."

Key Insights Distilled From

Deriving Lehmer and Hölder means as maximum weighted likelihood estimates for the multivariate exponential family

by Djemel Ziou,... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00964.pdf

Deeper Inquiries

How can the probabilistic interpretation of the Lehmer and Hölder means be leveraged to improve decision-making in real-world applications that rely on these central tendency measures

The probabilistic interpretation of the Lehmer and Hölder means offers a unique perspective that can significantly enhance decision-making in various real-world applications. By understanding these central tendency measures as Maximum Weighted Likelihood Estimates (MWLEs) for the multivariate exponential family, we can leverage this knowledge to improve statistical analyses and modeling in different fields. In fields such as finance, health, image processing, and pattern recognition, where central tendency measures play a crucial role, the probabilistic interpretation of Lehmer and Hölder means can provide valuable insights. For example, in financial analysis, understanding the probabilistic nature of these means can help in risk assessment, portfolio management, and investment strategies. By incorporating the weighted likelihood approach, decision-makers can make more informed choices based on the underlying data distribution and variability. Moreover, in healthcare applications, such as disease characterization or treatment effectiveness assessment, the probabilistic interpretation of these means can aid in identifying patterns, trends, and outliers in patient data. This can lead to more personalized and effective healthcare interventions. Similarly, in image processing and pattern recognition, leveraging the probabilistic interpretation of Lehmer and Hölder means can improve feature extraction, classification accuracy, and object detection algorithms. Overall, by integrating the probabilistic interpretation of Lehmer and Hölder means into decision-making processes, practitioners can enhance the robustness, accuracy, and reliability of their analyses, leading to more informed and data-driven outcomes in various real-world applications.

What are the potential limitations or drawbacks of using the Lehmer and Hölder means as MWLEs, and how can these be addressed or mitigated

While using Lehmer and Hölder means as MWLEs offers several advantages in statistical estimation and decision-making, there are potential limitations and drawbacks that need to be considered and addressed to ensure the validity and reliability of the results. Sensitivity to Outliers: One limitation of using MWLEs based on Lehmer and Hölder means is their sensitivity to outliers in the data. Outliers can significantly impact the estimation process, leading to biased results. To mitigate this issue, robust statistical techniques such as robust estimation methods or data preprocessing steps can be employed to reduce the influence of outliers on the central tendency measures. Assumption of Exponential Family: Another drawback is the assumption that the underlying data distribution belongs to the multivariate exponential family. If this assumption is violated, the MWLEs derived from Lehmer and Hölder means may not be optimal or accurate. Conducting model diagnostics and assessing the goodness-of-fit of the exponential family assumption can help address this limitation. Computational Complexity: Calculating MWLEs using Lehmer and Hölder means for high-dimensional data sets can be computationally intensive. Efficient algorithms and computational techniques can be implemented to handle the computational complexity and improve the scalability of the estimation process. Interpretability: While the probabilistic interpretation of Lehmer and Hölder means provides valuable insights, the interpretation of these central tendency measures may not always be intuitive or straightforward. Enhancing the interpretability of the results through visualization techniques, sensitivity analyses, and model validation can help address this limitation. By addressing these limitations through robust statistical methods, model diagnostics, computational optimizations, and enhanced interpretability, the use of Lehmer and Hölder means as MWLEs can be more effectively applied in real-world decision-making scenarios.

Given the connections between the Lehmer and Hölder means and the multivariate exponential family, what other families of means or central tendency measures might be derived using a similar approach, and what insights could they provide

The connections between the Lehmer and Hölder means and the multivariate exponential family open up possibilities for deriving other families of means or central tendency measures using a similar approach. By extending the concept of MWLEs to different probability distributions and families, we can uncover new insights and applications in various domains. One potential family of means that could be derived using a similar approach is the Generalized Mean family. Generalized means encompass a wide range of central tendency measures, including the arithmetic mean, geometric mean, harmonic mean, and others. By exploring the probabilistic interpretation of Generalized Means as MWLEs for specific distributions or data structures, we can gain a deeper understanding of their properties and applications in statistical estimation and decision-making. Furthermore, the concept of MWLEs based on central tendency measures can be extended to non-parametric estimation methods, such as kernel density estimation or quantile regression. By incorporating the probabilistic interpretation of central tendency measures into non-parametric approaches, we can enhance the flexibility and robustness of statistical modeling in complex data scenarios. Overall, by exploring the connections between central tendency measures, probability distributions, and maximum likelihood estimation, we can uncover new families of means, central tendency measures, and estimation techniques that offer valuable insights and applications in diverse fields such as finance, healthcare, image processing, and beyond.