[Paper Review] Expansions for Quantiles and Multivariate Moments of Extremes for Distributions of Pareto Type

Saralees Nadarajah, Christopher S. Withers|ArXiv.org|Mar 25, 2009

Probability and Risk Models8 references6 citations

TL;DR

This paper develops asymptotic expansions for quantiles and multivariate moments of extreme order statistics from Pareto-type distributions, using an inversion theorem to derive series expansions in inverse powers of $ n $ and $ n^{-\beta/\alpha} $. The key contribution is a general framework applicable to distributions like Cauchy, Student $ t $, $ F $, and stable laws, enabling precise moment approximations for extreme values with explicit error bounds.

ABSTRACT

Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-α- i β}$ for $α> 0$ and $β> 0$. An inversion theorem is proved and used to derive an expansion for the quantile $F^{-1} (u)$ and powers of it. From this an expansion in powers of $(n^{-1}, n^{-β/α})$ is given for the multivariate moments of the extremes $\{X_{n, n - s_i}, 1 \leq i \leq k \}/n^{1/α}$ for fixed ${\bf s} = (s_1, ..., s_k)$, where $k \geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $α< 1$.

Motivation & Objective

• To derive asymptotic expansions for quantiles and multivariate moments of extreme order statistics from distributions with power-law tails.
• To extend existing asymptotic results for extreme value moments beyond leading-order approximations to include higher-order terms in $ n^{-1} $ and $ n^{-\beta/\alpha} $.
• To provide a general method applicable to a broad class of heavy-tailed distributions, including Cauchy, Student $ t $, $ F $, and stable laws of index $ \alpha < 1 $.
• To establish a rigorous inversion theorem linking the tail behavior of the distribution function to the quantile function expansion.

Proposed method

• An inversion theorem is derived to express the quantile function $ F^{-1}(u) $ as a power series in $ (1-u)^{\alpha_i} $, where $ \alpha_i = (i\beta - 1)/\alpha $, based on the tail expansion $ 1 - F(x) = x^{-\alpha} \sum_{i=0}^{\infty} c_i x^{-i\beta} $.
• The method uses Bell polynomials and generating functions to compute higher-order moments of normalized extremes $ Y_{n,s} = X_{n,n-s}/n^{1/\alpha} $.
• Moments of the extremes are expanded in powers of $ n^{-1} $ and $ n^{-\beta/\alpha} $, with coefficients derived from the $ c_i $ parameters of the tail expansion.
• The approach is validated through explicit expansions for $ EY_{n,s} $, $ EY_{n,s_1}Y_{n,s_2} $, and $ \text{Covar}(Y_{n,s_1}, Y_{n,s_2}) $, up to $ O(n^{-3}) $, using recursive coefficient computation.
• The framework is applied to specific distributions by identifying $ \alpha $, $ \beta $, and $ c_i $, enabling direct computation of moment expansions.

Experimental results

Research questions

• RQ1 How can higher-order asymptotic expansions for moments of extreme order statistics be derived for distributions with power-law tails?
• RQ2 What is the structure of the quantile function expansion when the tail of the distribution satisfies $ 1 - F(x) \sim x^{-\alpha} \sum_{i=0}^{\infty} c_i x^{-i\beta} $?
• RQ3 How do the moments of normalized extremes $ X_{n,n-s}/n^{1/\alpha} $ behave asymptotically in terms of $ n^{-1} $ and $ n^{-\beta/\alpha} $?
• RQ4 Can the method be systematically applied to standard heavy-tailed distributions such as Student $ t $, $ F $, and stable laws?

Key findings

• For the Cauchy distribution, the first-order expansion of $ EY_{n,s} $ is $ s^{-1} - n^{-2}\pi^2(s+1) + O(n^{-3}) $, with $ Y_{n,s} = (\pi/n)X_{n,n-s} $.
• The joint second moment $ EY_{n,s_1}Y_{n,s_2} $ for Cauchy is $ (1 - n^{-1})(s_1 - 1)^{-1}s_2^{-1} - n^{-2}\pi^2 D_{2,{\bf s}}/3 + O(n^{-3}) $, where $ D_{2,{\bf s}} = (s_2 + 1)/(s_1 + 1) + s_1/s_2 $.
• For Student $ t $ with $ N $ degrees of freedom, the expansion involves powers of $ n^{-2/N} $, with the first correction term appearing at order $ n^{-2/N} $.
• For the $ F $-distribution with $ N > 2 $, the expansion is valid up to $ O(n^{-2a_0}) $, where $ a_0 = \min(2/(N-2), 1) $, and $ Y_{n,s} = X_{n,n-s}/(n c_0)^{1/\alpha} $.
• For stable laws of index $ \alpha < 1 $, the expansion yields moments up to $ O(n^{-2}) $, with $ a = 1 $, and the first correction term appears at $ n^{-1} $.