Quantile Contribution Statistic

• Quantile Contribution Statistic is a measure that quantifies the share of a total quantity contributed by observations above a given quantile threshold.
• It is applied in various fields such as income inequality analysis, risk management, and variable selection, using sample or population quantiles.
• Key aspects include its finite-sample properties, asymptotic approximations, and methods to correct downward bias for robust analysis in heavy-tailed distributions.

The quantile contribution statistic is a broad class of distributional measures quantifying the proportion of a total quantity (typically sum, income, risk, or other positive measures) attributed to observations above or below a given quantile threshold. It appears in numerous domains—statistical estimation, income inequality, risk management, high-dimensional variable selection—where tail phenomena are decisive. The precise form of the statistic varies by application, but its unifying feature is computation via sample or population quantiles.

1. Formal Definition and Variants

Given a set of i.i.d. real random variables $X_1,\dots,X_n$ with cumulative distribution function $F$, the basic quantile contribution statistic for level $p\in(0,1)$ is

$$\Lambda_n(p) = \frac{\sum_{i=\lceil np\rceil}^n X_{(i)}}{\sum_{i=1}^n X_{(i)}}$$

where $X_{(i)}$ are the order statistics. This measures the share of the total sum due to the largest $1-p$ fraction. In population terms, Taleb–Douady (Taleb et al., 2014) define for support $[x_{\min},\infty)$ and $q\in(0,1)$: $$\kappa_q = \frac{\mathbb{E}[X\,\mathbf{1}_{\{X>h(q)\}}]}{\mathbb{E}[X]} = q\,\frac{\mathbb{E}[X|X>h(q)]}{\mathbb{E}[X]}$$ where $h(q)$ is the threshold for the upper $q$-fraction.

In the quantile-functional framework (Jurečková et al., 3 Apr 2024): $$C(\alpha) = \frac{\int_{1-\alpha}^1 Q_Z(u)\,du}{\int_0^1 Q_Z(u)\,du}$$ for an unobserved error term $Z$ in a regression model; analogous functionals are used in robust inequality measures (Prendergast et al., 2015).

Risk management extends this to the context of risk contributions, using quantile-based risk measures $\rho_\Lambda$ and their differential allocations (Ince et al., 2021).

2. Exact Finite-Sample and Population Properties

The finite-sample distribution of $\Lambda_n(p)$ possesses an exact expression involving multiple integrals over the joint order statistics (Almani, 6 Nov 2025): $F_{\Lambda_n(p)}(\lambda) = 1 - n!\idotsint_{0<u_2<...<u_n<1} F\left(\frac{1-\lambda}{\lambda}\sum_{i=\lceil np\rceil}^n F^{-1}(u_i) - \sum_{i=2}^{\lceil np\rceil-1}F^{-1}(u_i)\right) du_2\dots du_n$ for $\mathbb{E}[X]>0$.

Population forms admit concise formulas: $\kappa_q = \frac{1}{\mu}\int_{1-q}^1 F^{-1}(u)\,du$ with $\mu = \mathbb{E}[X]$. For Pareto tails, this specializes to $\kappa_q = q^{(\alpha-1)/\alpha}$, reflecting tail dominance (Taleb et al., 2014).

A general theme is that exact finite-sample distribution is algebraically complex and rarely used for large $n$; large-sample approximations (see below) are greatly preferred in practice.

3. Asymptotic Theory and Approximation

A central result is almost-sure convergence: $$\Lambda_n(p) \to \frac{\mathbb{E}[X_1\,1_{\{X_1 \ge q_p\}}]}{\mathbb{E}[X_1]}$$ with $q_p=F^{-1}(p)$, by law of large numbers.

The numerator and denominator in quantile contributions both admit asymptotic normal expansions; via the Hinkley/Geary theory for correlated normals, one obtains an explicit density for the large-sample ratio (Almani, 6 Nov 2025): $$\Lambda_n(p) \approx \frac{\mu_n}{\mu}\exp\left(\mathcal N(0,\tfrac{1}{n}V_{\rm eff})\right)$$ where $$V_{\rm eff} = \frac{\sigma^2}{\mu^2} + \frac{\sigma_n^2}{\mu_n^2} - \frac{2c_n}{\mu\mu_n}$$. Simulation confirms this log-normal approximation is accurate for moderate $n$ ($< 0.08$ total area discrepancy in diverse distributions).

For quantile functionals in regression models, the plug-in estimators converge at root-n rate, with explicit asymptotic variance based on the quantile densities of the error term (Jurečková et al., 3 Apr 2024).

4. Estimation, Bias, and Consistency

The naive empirical estimator

$$\widehat{\kappa}_q = \frac{\sum_{i=1}^n X_i\,\mathbf{1}_{\{X_i>\widehat{h}(q)\}}}{\sum_{i=1}^n X_i}$$

suffers from severe downward bias for heavy-tailed data (Taleb et al., 2014). This bias is concavity-driven (Jensen's inequality): $\mathbb{E}[\widehat{\kappa}_q] \le \kappa_q$ with error decaying only as $n^{-(\alpha-1)}$ for Pareto exponents $\alpha>1$—potentially extremely slowly for $\alpha\approx1$.

Aggregation and mixing exacerbate these issues. The global quantile-contribution is superadditive: $$\mathbb{E}[\kappa''] \ge \frac{m}{m+n}\mathbb{E}[\kappa] + \frac{n}{m+n}\mathbb{E}[\kappa']$$ implying pooled data yield strictly higher expected concentration than weighted sub-means. Mixing distributions with different tail exponents likewise raises the expected contribution over the mean-exponent approach.

Practical recommendation is to fit tail models (e.g., Pareto via Hill estimator), compute theoretical $\kappa_q(\hat{\alpha},\hat{\lambda})$, and average over posterior uncertainty where feasible.

5. Robust Quantile Extensions for Inequality and Tail Analysis

Income inequality analysis uses quantile-based Lorenz curves (Prendergast et al., 2015), replacing means by quantiles to gain robustness. Explicit quantile contribution statistics include: $$L_1(F;p) = p\,\frac{Q(F;p/2)}{Q(F;1/2)},\quad L_2(F;p) = p\,\frac{Q(F;p/2)}{Q(F;1-p/2)},\quad L_3(F;p) = 2p\,\frac{Q(F;p/2)}{Q(F;p/2) + Q(F;1-p/2)}$$ Estimates are distribution-free; asymptotic variances are nearly independent of $F$, enabling direct sample-size planning. Bounded influence functions provide high outlier-resistance, contrasting with the classical Lorenz and Gini statistics.

Decomposition techniques (Quantile Ratio Index, (Prendergast et al., 2017)) partition inequality into quantile bands and yield weighted conditional contributions, facilitating analysis of which parts of the distribution drive overall concentration.

6. Advanced Applications: Variable Selection and Risk Allocation

In high-dimensional classification, the CR-statistic ("quantile contribution statistic" in the sense of orthogonal quantile-based comparison density expansion (Mukhopadhyay et al., 2011)) quantifies differences between class-conditional distributions: $CR = \sum_{k=1}^M \hat{\theta}_k^2$ with each $\hat{\theta}_k$ a rank-correlation coefficient between mid-rank score function $S_k(U)$ and class label $Y$. Under the null, $CR$ is asymptotically chi-squared distributed, enabling direct p-value computation. Robust FDR-thresholding is achieved via the CDfdr algorithm, operating on the comparison density.

In quantitative risk management, the quantile contribution appears as the derivative allocation for generalized quantile risk measures (Ince et al., 2021). For lambda-quantiles, the contribution formula is: $$\frac{\partial \rho_\Lambda(u)}{\partial u_i} = -\eta_\Lambda,Y(-\rho_\Lambda(u))\,\mathbb{E}[X_i|Y=-\rho_\Lambda(u)]$$ with sensitivity adjustment $\eta_\Lambda,Y$ depending on the tail behavior of the risk measure. Euler-type allocation generalizes to non-homogeneous risks and is validated numerically on normal portfolios.

7. Limitations, Controversies, and Best Practices

The quantile contribution statistic is not additive under naive aggregation and is downward biased for highly skewed data, especially with unknown/uncertain tail index. Sample measures can spuriously suggest rising concentration due to pooling or larger sample sizes (Taleb et al., 2014). Empirical quantile contributions are unreliable for structural change detection unless properly corrected for sample-size and parameter uncertainty.

Practitioners are advised to:

• Favor model-based estimation or extrapolative risk measures (CVaR, Expected Shortfall) over direct sample ratios in heavy-tailed regimes.
• Use robust quantile inequality measures for outlier-prone distributions.
• Combine quantile decomposition approaches to attribute concentration, track distributional change, and identify the sources of inequality or risk.

A plausible implication is that quantile contribution statistics, while essential for tail diagnosis and inequality quantification, require careful theoretical underpinning and cautious interpretation in applied settings with heavy-tailed or complex mixture data.