Empirical Precision Extrapolation Method

• The paper presents the Adjusted Empirical Likelihood method, which uses pseudo-observations to achieve chi-square accuracy up to O(n⁻²).
• Operator embedding and curvature-based techniques leverage convexity in operator norms to derive sharp extrapolation bounds in functional and complex geometric settings.
• Precision-specific operations and adaptive model reduction demonstrate that managing floating-point arithmetic and surrogate errors significantly improves numerical reliability.

Empirical Precision Extrapolation Method refers to a class of numerical and statistical methodologies aiming to improve the accuracy and reliability of extrapolated results across a variety of domains—statistics, scientific computing, numerical analysis, and model reduction—by employing empirically motivated modifications, explicit error controls, and/or rigorous design principles to mitigate inherent uncertainties, ill-conditioning, or coverage errors. The term encompasses methods such as adjusted empirical likelihood with high-order precision, extrapolation via polynomial and operator families, high-precision Taylor extrapolation, algorithmic treatments for floating-point arithmetic, and adaptive algorithms for reduced order modeling. These techniques combine empirical performance analysis, theoretical expansion, and simulation studies to guide practical choices of model correction or extrapolation strategy.

1. High-Order Correction and Adjusted Empirical Likelihood

Central to the empirical precision extrapolation paradigm is the Adjusted Empirical Likelihood (AEL) methodology for statistical inference (Liu et al., 2010). Standard empirical likelihood (EL) approaches may suffer from low precision in chi-square approximations and nonexistence of solutions to the estimating equations, particularly with small sample sizes or high-dimensional settings. AEL remedies these issues by augmenting the data set with a pseudo-observation, ensuring the existence of a solution by guaranteeing that the zero vector falls within the convex hull of estimating functions.

The technical construction is as follows:

• Given $g(x_i; \theta)$, the original estimating function, AEL appends $g_{n+1} = -a_n \, \bar{g}$, where $\bar{g}$ is the mean of the $g(x_i;\theta)$ and $a_n>0$ is an adjustment parameter.
• The likelihood ratio statistic is then defined over $n+1$ observations.
• By selecting $a_n = b/2$, where $b = \frac12 \alpha_4 - \frac13 \alpha_3^2$ (with $\alpha_3,\alpha_4$ being central moments), the AEL ratio statistic matches the chi-square distribution up to $O(n^{-2})$, incorporating a Bartlett-type correction.

Simulation evidence demonstrates significantly improved coverage probabilities of confidence regions—notably better than those produced by Bartlett-corrected EL or conventional methods—both in univariate and multivariate settings, including linear regression and asset-pricing contexts.

2. Operator Embedding and Curvature-Based Extrapolation

In functional analysis and complex geometry, empirical precision extrapolation is structurally realized through the embedding of operators into parameterized families with curvature-controlled properties (Lempert, 2015). An operator $A$ to be estimated is extended to a family $\{A_t\}$ acting between normed spaces $(E_t, F_t)$, indexed by a parameter $t \in (\sigma_0, \infty)$.

• The operator norm $|A_t|$ is shown to be convex, and often monotonic, in $t$ due to underlying plurisubharmonicity (e.g., $\log|A|(t)$ convexity).
• Precise estimates for $|A_t|$ are then propagated from limiting regimes ($t \to \sigma_0$, $t \to \infty$) to intermediate values by convexity, providing sharp extrapolated bounds.
• Applications include holomorphic section extension problems where L² estimates are extrapolated from weighted limits to the original problem, achieving sharp constants in complex geometry.

The empirical aspect lies in the robustness of these bounds across families and the demonstration of optimality in both abstract harmonic analytic and geometric settings.

3. Precision-Specific Operations in Numerical Computation

In floating-point arithmetic, empirical precision extrapolation techniques address operations whose accuracy degrades under naive increases in precision (Wang et al., 2015). The paper classifies such operations as "precision-specific," e.g., rounding implementations that propagate and magnify error in higher-precision arithmetic due to constant-based bit manipulations.

A lightweight detection algorithm identifies these patterns by monitoring relative errors across large input ensembles. An automatic processing method is introduced:

• Instrumented code inserts reducePrec() and resumePrec() functions to locally revert to original precision for problematic instructions and then convert back for subsequent computations.
• Experimental results on GLIBC show that high-precision computation can backfire unless these corrections are applied; the modified approach yields error reductions and higher alignment with multi-precision reference results.

The implication for general extrapolation methodologies is that increasing numerical precision is not universally beneficial and must be managed in operation-specific contexts.

4. Polynomial, Analytic, and Asymptotic Extrapolation Schemes

Empirical precision extrapolation is prominently realized in analytic function continuation from discrete data and in the prediction of convergence rates and errors:

• Least squares polynomial approximants, constructed from noisy $N+1$ equally spaced samples on $[-1,1]$ and degree $M^*$ chosen via $$M^* = \min\{\frac12 \sqrt N, \frac{\log(Q/\epsilon)}{\log \rho}\}$$, yield stability in extrapolation for analytic functions in Bernstein ellipses (Demanet et al., 2016). The error at extrapolation point $x$ is $\mathcal O(\epsilon^{-\log r(x)/\log\rho})$ up to logarithmic factors, optimally balanced between approximation and noise amplification.
• High-precision Taylor polynomial approximation, using hundreds of digits in arithmetic operations, overcomes numerical instability and the Runge phenomenon (Bakas, 2019). The method provides a-priori prediction of extrapolation spans via the calculated radius of convergence $r = 1 / \limsup_{n\to\infty} |a_n|^{1/n}$ and sharply bounded errors.

These approaches are robustified by empirical error analysis and allow methodologically sound extrapolation in scientific and engineering problems requiring stringent numerical precision.

5. Model Reduction, Error Estimation, and Adaptive Extrapolation

Goal-oriented reduced order modeling employs empirical precision extrapolation for both error prediction and adaptive surrogate construction (Stefanescu et al., 2019):

• A-posteriori error estimators approximate loss of precision in reduced models.
• Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM) form the surrogate space, while dual weighted residuals drive adaptive refinement.
• The adaptive DEIM algorithm leverages singular vectors of dual weighted residual matrices to focus on quantities of interest, and efficient implementations support both explicit and implicit Euler schemes.

Numerical results for Burgers and Shallow Water equations validate theoretical bounds and demonstrate practical error estimation suitable for precision-aware extrapolation and model selection.

6. Statistical Inference and Extrapolation Beyond Support

Extensions to nonparametric statistical inference pose particular challenges when extrapolating the conditional expectation or quantiles outside the support of the conditioning variable (Pfister et al., 15 Feb 2024). The proposed extrapolation-aware method imposes that the minimal and maximal directional derivatives (up to order $q$) on observed support dominate global behavior. Taylor expansions anchored at data points, with derivative extrema from observed regions, define robust upper and lower extrapolation bounds for target points.

• For an unknown function $f$, bounds at $x$ involve $B_{q,f,D}^{lo}(x)$ and $B_{q,f,D}^{up}(x)$ by optimizing Taylor expansions anchored at all data points in $D$.
• Plug-in estimation procedures combine pilot models (e.g., random forests, local polynomials) and derivative estimates, ensuring consistent extrapolation bounds as data become dense in $D$.
• Prediction and uncertainty quantification outside support are then constructed as intervals or midpoints of these bounds, achieving extrapolation-aware inference that adapts to data sparsity.

Simulations and data applications confirm effectiveness, highlighting the necessity of explicitly managing extrapolation uncertainty in modern nonparametric inference.

7. Uncertainty Quantification in Extrapolation via Random Walks

In quantum chemistry, extrapolation to complete basis set (CBS) limits requires explicit uncertainty estimation due to absent results at higher cardinalities (Lang et al., 12 Mar 2025). The proposed method simulates all possible extrapolation outcomes by constructing an ensemble of random walks:

• Given extrapolated values $e_X, e_{X-1}$, the next value $e_{X+1}$ is constrained within $e_X \pm |e_X - e_{X-1}|$ and sampled uniformly; the process iterates recursively.
• Millions of random walks yield a sample distribution of CBS limit predictions $ẽ_\infty$, from which confidence intervals are computed (e.g., $1\sigma$, $2\sigma$) as statistical error bars.
• The method is parameter-free, compatible with any extrapolation formula, and validated empirically against reference results, where error bounds remain tight yet conservative.

This approach provides rigorous uncertainty quantification for extrapolated results in settings lacking higher-order data and is broadly applicable wherever error propagation in extrapolation is a concern.

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In summary, the empirical precision extrapolation method spans statistical, numerical, and modeling domains, synthesizing empirical error correction and theoretical expansion to yield high-order accuracy, error estimates, and uncertainty awareness. Across methodologies—adjusted likelihoods, operator embeddings, high-precision computation, polynomial and Taylor extrapolation, adaptive reduction, and statistical interval estimation—the common thread is the empirical validation of extrapolation strategy, robustness to operational or sample limitations, and quantification of associated uncertainties to enable reliable inference or computation beyond the observed data or immediate numerical regime.