[Paper Review] Several new tail index estimators

Vygantas Paulauskas, Marijus Vaiÿciulis|arXiv (Cornell University)|Jan 5, 2015

Financial Risk and Volatility Modeling17 references3 citations

TL;DR

This paper introduces a new class of non-monotone, power-logarithmic functions $ g_{r,u}(x) = x^r \ln^u(x) $ to construct improved tail index estimators for heavy-tailed distributions. By forming estimators based on ratios of extreme order statistics using these functions, the authors achieve better asymptotic performance than the Hill and other classical estimators across the full range of second-order parameters, with asymptotic normality and lower asymptotic mean squared error established theoretically and supported by simulations.

ABSTRACT

In the paper we propose some new class of functions which is used to construct tail index estimators. Functions from this new class is non-monotone in general, but presents a product of two monotone functions: the power function and the logarithmic function, which plays essential role in the classical Hill estimator. Introduced new estimators have better asymptotic performance comparing with the Hill estimator and other popular estimators over all range of the parameters present in the second order regular variation condition. Asymptotic normality of the introduced estimators is proved, and comparison (using asymptotic mean square error) with other estimators of the tail index is provided. Some preliminary simulation results are presented.

Motivation & Objective

• To develop a new class of tail index estimators that outperform existing methods across the full range of second-order regular variation parameters.
• To address the limitations of monotonicity in classical estimators like Hill and Pickands by introducing non-monotone functions that retain logarithmic structure.
• To establish asymptotic normality and derive asymptotic mean squared error (AMSE) for the new estimators to enable theoretical comparison with existing methods.
• To provide a theoretically grounded and empirically validated alternative to the Hill estimator with improved finite-sample performance.

Proposed method

• Propose a parametric family of functions $ g_{r,u}(x) = x^r \ln^u(x) $ for $ x \geq 1 $, where $ r \in \mathbb{R} $, $ u \in \mathbb{R} $, with constraints $ \gamma r < 1 $, $ u > -1 $, and focus on integer $ u $.
• Define a new statistic $ G_n(k,r,u) = \frac{1}{k} \sum_{i=0}^{k-1} g_{r,u}\left( \frac{X_{n-i,n}}{X_{n-k,n}} \right) $ based on ratios of extreme order statistics.
• Construct three new estimators $ \hat{\gamma}^{(1)}_n, \hat{\gamma}^{(2)}_n, \hat{\gamma}^{(3)}_n $ using $ G_n(k,r,u) $, with closed-form expressions involving $ G_n(k,r,0) $, $ G_n(k,r,1) $, and $ G_n(k,0,2) $.
• Prove asymptotic normality of the estimators using a multivariate central limit theorem and continuous mapping theorem, avoiding complex functional delta methods.
• Derive asymptotic mean squared error (AMSE) expressions and compare them with those of the Hill, moment, and moment ratio estimators.
• Use simulation studies to provide preliminary empirical validation of the theoretical findings.

Experimental results

Research questions

• RQ1 Can a new class of non-monotone functions incorporating power and logarithmic components improve tail index estimation beyond the Hill estimator?
• RQ2 How do the asymptotic bias and variance of the new estimators compare to those of the Hill, moment, and moment ratio estimators across different second-order parameters?
• RQ3 Does the proposed estimator family maintain good finite-sample performance across the full range of tail index values and second-order parameters?
• RQ4 Can asymptotic normality and AMSE expressions be rigorously derived for the new estimators without relying on complex functional central limit theorems?

Key findings

• The new estimators exhibit superior asymptotic performance compared to the Hill estimator and other popular estimators across the entire range of second-order parameters.
• Asymptotic normality of the proposed estimators is rigorously established via a multivariate central limit theorem and continuous mapping theorem.
• The asymptotic mean squared error (AMSE) of the new estimators is uniformly lower than that of the Hill estimator and other classical estimators for all considered parameter configurations.
• The estimator $ \hat{\gamma}^{(3)}_n $ is shown to be asymptotically unbiased (bias $ B^{(2)} = 0 $) for $ r \neq 0 $, while the bias diverges for $ r = 0 $, indicating a structural limitation at the logarithmic limit.
• Theoretical analysis confirms that the new estimators are consistent and converge at the standard $ \sqrt{k} $ rate, with convergence rates matching or improving upon existing methods.
• Preliminary simulation results support the theoretical findings, indicating better finite-sample performance of the new estimators in various tail index and second-order parameter settings.