Tail-Accuracy Degradation and Optimal Block Length for Block Self-Normalized Quantile Empirical Saddlepoint Inference under Dependence

Technical Summary: Tail-Accuracy Degradation and Optimal Block Length for Block Self-Normalized Quantile Empirical Saddlepoint Inference under Dependence

Problem Statement
Quantile inference is fundamental to risk measurement but faces a structural difficulty: the asymptotic variance of the sample quantile depends on the reciprocal of the density at the quantile ($1/f(q_\tau)$), which is difficult to estimate stably in small samples, heavy tails, or extreme quantiles. Existing frequentist procedures under dependence typically sacrifice either higher-order accuracy or density-freeness.

• Some methods (e.g., Hoga–Schulz) use self-normalization but rely on expanding windows and conditional density assumptions, yielding only first-order accuracy.
• Others (e.g., SNQESA) combine self-normalization with empirical saddlepoint approximation to achieve density-free, higher-order accuracy ($O(n^{-3/2})$), but only under independent sampling.
• The intersection of dependence, quantile inference, self-normalization, and density-free higher-order accuracy remains unaddressed. Specifically, it is unknown how much tail accuracy degrades when applying block self-normalized pivots to dependent data, nor is there a theoretical basis for choosing the optimal block length ($\ell$).

Methodology
The paper studies a block self-normalized quantile pivot, $T_n(t)$, constructed from a strongly $\alpha$-mixing sample.

1. Pivot Construction: For a block length $\ell$ and number of blocks $m = \lfloor n/\ell \rfloor$, the indicator score $g_i(t) = \mathbb{1}\{X_i \le t\} - \tau$ is averaged over blocks. The pivot is defined as:
   $$T_n(t) = \frac{\sum_{j=1}^m \bar{g}_j(t)}{\sqrt{\sum_{j=1}^m \bar{g}_j(t)^2}}$$
   This self-normalization uses the within-block quadratic form rather than an explicit long-run variance estimator, rendering the pivot density-free at the true quantile $q_\tau$ (Theorem 1).
2. Theoretical Framework: The analysis relies on Edgeworth expansions of the pivot's cumulants under $\alpha$-mixing. The author decomposes the relative tail error, $\text{err}(\ell) = P(T_n \ge x)/(1-\Phi(x)) - 1$, into distinct components driven by sample size ($n$), block length ($\ell$), and dependence structure ($\lambda$).
3. Lattice Analysis: The paper acknowledges that the pivot takes lattice values, making its tail function a step function. It investigates the lower bound for approximating this step function with continuous functions.

Key Contributions and Results

1. Quantification of Tail-Accuracy Degradation (Theorem 2):
   The paper establishes that under dependence, the relative tail error does not vanish as the number of blocks grows. Instead, it converges to a positive limit known as the variance-inflation floor, $E_\infty(\ell) \sim c_\Phi / \ell$.

   • The error expansion is given by: $\text{err}(\ell) = -C_{\text{eff}}^1 n^{-1/2} + D(\ell) + O(n^{-1})$.
   • The leading term ($n^{-1/2}$) arises from bias and skewness.
   • The second term, $D(\ell)$, represents the floor caused by the long-run variance of the block pivot being inflated by dependence, while the Gaussian benchmark assumes unit variance.
   • Crucially, the scale of degradation is $n^{-1/2}$, not $m^{-1/2}$ (the number of blocks), correcting a potential misconception from mean-based analyses.

2. Optimal Block Length (Theorem 3):
   By balancing the variance-inflation floor (which decreases as $\ell$ increases) against the bias/skewness transient (which is independent of $\ell$ in the limit), the paper derives a closed-form expression for the optimal block length:
   $$\ell^* \propto \sqrt{n}$$

   • The proportionality constant depends on the lag-1 correlation $\lambda$ and the quantile level $\tau$.
   • Constraints: An interior optimum exists only when $\lambda > 0$ and $\tau < 1/2$. If $\lambda \le 0$, the floor becomes a "ceiling" (negative error), and if $\tau \ge 1/2$, the transient and floor signs align, preventing cancellation.
   • This result transforms the block length from a heuristic tuning parameter into a computable quantity based on sample size and estimated dependence.

3. Uniform-Accuracy Lower Bound (Theorem 4):
   Due to the lattice structure of the pivot, any continuous approximation of the tail probability incurs a uniform error of at least $\Theta(m^{-1}\ell^{-3/2})$. The paper proves this bound is attainable via piecewise-linear construction. However, at the optimal block length ($\ell \sim \sqrt{n}$), this lattice error is $O(n^{-5/4})$, which is negligible compared to the $O(n^{-1})$ coverage error of the continuous Edgeworth expansion.

4. Simulation Performance:
   Numerical experiments on a two-state Markov chain indicator process ($\tau=0.3$) demonstrate that the proposed method (ESA-an) achieves coverage of 95% $\pm$ 0.15% across 16 configurations of dependence ($\lambda$) and block length ($\ell$).

   • In contrast, the first-order Gaussian baseline deteriorates significantly with dependence, dropping to 60.1% coverage at $\lambda=0.8$ with small block lengths.
   • The method is computationally efficient (0.0037 ms per replication), comparable to the moving block bootstrap and roughly 170 times faster than block empirical likelihood, as it requires no resampling.

Significance and Claims
The paper claims to occupy the "junction" between density-free higher-order accuracy (provided by SNQESA for independent data) and dependence handling (provided by block frameworks). Its primary contributions are:

• Decoupling Dependence from Accuracy: It demonstrates that dependence introduces a variance-inflation floor that cannot be eliminated by increasing the number of blocks; it can only be mitigated by optimizing the block length.
• Closed-Form Optimization: It provides the first analytic derivation of an optimal block length for self-normalized quantile inference under dependence, showing the scaling law $\ell^* \propto \sqrt{n}$ is independent of the dependence strength's exponent.
• Practical Utility: The method offers a computable, density-free procedure for constructing confidence intervals with second-order coverage ($O(n^{-1})$) near the optimal block length, addressing the sensitivity to bandwidth selection and density estimation that plagues standard quantile inference.

The author notes limitations: the results assume $\lambda > 0$ and $\tau < 1/2$, require stronger exponential deviation inequalities than polynomial mixing for full rigor, and do not yet cover discontinuous densities at the quantile. The extension to quantile regression is noted as a natural corollary but not the primary focus of this work.