A Layered Hill Estimator for Heavy-Tailed Distributions with Missing Extreme Data

Core Concepts

This paper introduces the layered Hill estimator, a novel method for estimating the tail exponent of heavy-tailed distributions that is more robust to missing extreme data than the traditional Hill estimator.

Abstract

Kang, T., & Owada, T. (2024). Layered Hill estimator for extreme data in clusters. arXiv preprint arXiv:2411.05808v1.

This paper proposes a new estimator, called the layered Hill estimator, for the tail exponent of a heavy-tailed distribution, aiming to address the limitations of the traditional Hill estimator, particularly its sensitivity to missing extreme data.

Key Insights Distilled From

Layered Hill esimator for extreme data in clusters

by Taegyu Kang,... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.05808.pdf

Deeper Inquiries

How does the choice of the connectivity radius in the geometric graph construction affect the performance of the layered Hill estimator, and is there an optimal way to select this radius?

The choice of the connectivity radius significantly impacts the performance of the layered Hill estimator. This radius determines the scale at which we consider points "close" and thus group them into clusters for constructing higher-order layers.
Here's a breakdown of the impact:

Small radius: A small radius leads to sparsely connected graphs, potentially leaving many points isolated and excluded from higher-order layers. This can result in a loss of information and less stable estimates, especially for higher-order layered Hill estimators.
Large radius: A large radius can lead to densely connected graphs, potentially merging distinct clusters and diluting the layered structure. This can reduce the estimator's sensitivity to missing extremes, as the distinction between layers becomes less pronounced.
Optimal radius: The optimal radius depends on the data distribution, specifically the density of extreme points and the desired balance between bias and variance.
Selecting the radius:
There isn't a universally optimal way to select the connectivity radius. However, data-driven approaches can be employed:

Visual inspection of cluster structure: Visualizing the data and the resulting geometric graph for different radii can provide insights into a suitable range for the radius.
Cross-validation: One could split the data into training and validation sets. For different radii, calculate the layered Hill estimator on the training set and evaluate its performance (e.g., mean squared error) on the validation set. Choose the radius that minimizes the error.
Adaptive methods: Explore adaptive methods that adjust the connectivity radius based on the local density of extreme points. This could involve techniques like nearest neighbor graphs or density-based clustering algorithms.
Further research is needed to develop more robust and data-driven methods for selecting the optimal connectivity radius in the context of the layered Hill estimator.

While the layered Hill estimator shows robustness to missing extreme values, could it be overly sensitive to outliers or noise in the non-extreme data points?

While the layered Hill estimator demonstrates robustness to missing extreme values, it's not inherently designed to handle outliers or noise in non-extreme data points. Here's why:

Focus on extremes: The layered Hill estimator primarily focuses on the tail behavior of the distribution, utilizing the structure of extreme value clusters. Non-extreme data points have minimal influence on the estimator's calculations.
Geometric graph construction: The connectivity radius used to construct the geometric graph primarily affects the clustering of extreme points. Outliers or noise in non-extreme regions are less likely to significantly alter the graph structure or the resulting layered Hill estimates.
Potential sensitivity:
However, certain scenarios might introduce sensitivity:

Outliers near extremes: Outliers located close to the extreme value region could potentially be misclassified as extreme points, influencing the geometric graph construction and impacting the layered Hill estimates.
High noise levels: Extremely high noise levels, even in non-extreme regions, might obscure the true underlying structure of extreme value clusters, potentially affecting the estimator's performance.
Mitigation strategies:

Outlier detection and removal: Employing outlier detection techniques before applying the layered Hill estimator can help mitigate the influence of spurious extreme points.
Robust distance metrics: Exploring robust distance metrics less sensitive to outliers during the geometric graph construction could improve the estimator's resilience.
In summary, while the layered Hill estimator is inherently more robust to missing extremes, it's essential to be mindful of potential sensitivities to outliers or noise, especially in the extreme value regions. Preprocessing steps like outlier removal and the use of robust distance metrics can enhance the estimator's reliability in such scenarios.

If we consider the tail exponent as a measure of "surprise" in a dataset, how can the layered Hill estimator be applied to fields like anomaly detection or surprise minimization in reinforcement learning?

The tail exponent, as a measure of the heaviness of the tail, can indeed be interpreted as an indicator of "surprise" in a dataset. A heavier tail (smaller tail exponent) suggests a higher probability of observing extreme, unexpected values. This connection opens up interesting applications for the layered Hill estimator in fields like anomaly detection and surprise minimization in reinforcement learning.
Anomaly Detection:

Identifying unusual events: In anomaly detection, the goal is to identify data points or events that deviate significantly from the norm. The layered Hill estimator can be used to estimate the tail exponent of a data stream, providing a dynamic measure of "surprise."
Adaptive thresholds: A sudden decrease in the estimated tail exponent could signal an increased likelihood of anomalies, allowing for adaptive threshold setting in anomaly detection systems.
Robustness to missing extremes: The layered Hill estimator's robustness to missing extremes is particularly valuable in anomaly detection, where extreme events might be sparsely observed or even deliberately removed from training data.
Surprise Minimization in Reinforcement Learning:

Quantifying surprise: In reinforcement learning, surprise often relates to encountering unexpected states or rewards. The layered Hill estimator can quantify this surprise by estimating the tail exponent of the reward distribution or state transition probabilities.
Exploration vs. exploitation: Agents can leverage the estimated tail exponent to balance exploration (seeking surprising, potentially high-reward states) and exploitation (exploiting known high-reward strategies). A heavier tail might encourage more exploration.
Robust learning in uncertain environments: The layered Hill estimator's robustness to missing extremes can be beneficial in reinforcement learning scenarios with sparse rewards or sudden shifts in the environment, enabling more stable and reliable learning.
Challenges and Considerations:

Computational cost: The layered Hill estimator, especially for higher-order layers, can be computationally expensive for large datasets or high-dimensional data. Efficient implementations and approximations might be necessary.
Choice of connectivity radius: As discussed earlier, the choice of connectivity radius in the geometric graph construction is crucial and requires careful consideration or data-driven optimization.
Interpretation and domain adaptation: The interpretation of the tail exponent as "surprise" and its application to specific domains require careful adaptation and validation.
In conclusion, the layered Hill estimator, with its ability to robustly estimate the tail exponent, holds promising potential for applications in anomaly detection and surprise minimization in reinforcement learning. By providing a measure of "surprise" in datasets, it can contribute to more adaptive, robust, and efficient learning algorithms.