Finite-Sample Distribution Approximations

• Finite-sample distribution approximations are methods that provide non-asymptotic corrections for estimators computed from limited data, enhancing accuracy beyond classical theory.
• Techniques such as Edgeworth expansions, Berry–Esseen bounds, and Stein’s method refine normal approximations and control tail behaviors in finite-sample regimes.
• These approaches are crucial for robust estimation and nonparametric inference, offering practical tools for high-dimensional analysis and model-agnostic prediction.

Finite-sample distribution approximations refer to the derivation and analysis of precise, non-asymptotic approximations for the distribution of statistics or estimators computed from finite samples. These approximations quantify, correct, and bound the discrepancy between the empirical finite-sample distribution and its limiting—often normal or otherwise idealized—counterpart. This paradigm is central in modern statistics, nonparametric inference, high-dimensional analysis, and robust estimation, both from the theoretical and the algorithmic perspectives.

1. Edgeworth Expansions, Normal Approximation, and Higher-order Corrections

The classical approach to finite-sample distribution approximation for statistics such as sums or means—particularly under sampling without replacement from finite populations—is based on augmenting the standard central limit theorem (CLT) normal approximation by incorporating higher-order terms that account for skewness, kurtosis, and additional cumulants. The Edgeworth expansion is the canonical tool in this setting. For the normalized sample sum $S_{n,n}$, the characteristic function can be written as: $$\varphi_n(t) = \exp(-t^2/2) \left[ 1 + \sum_{m=1}^k \frac{1}{(\sigma_n \sqrt{n})^m} G_{m,n}(t) \right] + R_{k,n}(t)$$ where $G_{m,n}(t)$ are polynomials in $t$ whose coefficients depend on cumulants of the underlying population, and $R_{k,n}(t)$ is a remainder term (Mohamed et al., 2013).

Accounting for dependence induced by sampling without replacement necessitates the use of modified cumulants reflecting the finite population structure. The normal approximation is improved by including a finite number of these correction terms, yielding higher fidelity in $p$-value calculation and confidence interval construction, especially in small to moderate sample regimes.

2. Non-asymptotic Error Bounds: Berry–Esseen and Large Deviations

Finite-sample distribution approximation is fundamentally concerned with quantifying the error of the limiting approximation. The Berry–Esseen bound provides an explicit rate for the uniform deviation between the empirical distribution function and the standard normal. In the finite population case, such a bound takes the form

$$\sup_x \left| \mathbb{P} \left\{ \frac{S_{n,n}-n\gamma}{\sigma_n} < x \right\} - \Phi(x) \right| \leq C \cdot \beta_{3,n}$$

where $\beta_{3,n}$ is a Lyapunov ratio reflecting higher moment contributions and design effects from finite populations (Mohamed et al., 2013).

Beyond the central part of the distribution, Cramér-type large deviation results are essential for controlling tail events: $$1 - P_n(x) = \exp \left\{ -\frac{x^2}{2N} \left[ 1 + O\left( \frac{x+1}{N} \right) \right] \right\}$$ valid in a specific range of $x$ relative to $N$. Such estimates are critical in nonparametric testing, quality control, or risk management, where rare-event control cannot rely solely on central limit approximations.

3. Discrete and Functional Approximations: Stein’s Method and Fourier Techniques

For statistics such as the number of maxima in discrete data or the behavior of robust scale estimators, Stein’s method is widely employed to obtain total variation distance bounds between the finite-sample distribution and a discrete limit law, such as logarithmic or Poisson: $d_{TV}(K_n, L) \leq \cdots$ with $K_n$ the count statistic and $L$ the target logarithmic law (Daly, 9 May 2025).

In continuous settings where the exact density of a statistic is unknown, Fourier-based series (e.g., Fourier cosine expansions) offer a constructive approximation route. For a statistic $T_n$ with bounded, symmetric support, the density can be expanded as

$$f_n(x) \approx \sum_{k=0}^K a_{n,k} \cos \left( \frac{k\pi}{A_n} x \right)$$

where coefficients $a_{n,k}$ are constructed either by integration when $f_n$ is explicit or via moment expansion if only the moments are available (Nakagawa et al., 2021).

4. Bootstrap, Distribution-Free, and Model-Agnostic Methods

Finite-sample inference in the presence of complex (possibly high-dimensional or non-Euclidean) data and "black-box" estimators increasingly relies on resampling-based techniques and distribution-free methods. The nonparametric bootstrap provides a universally applicable approach for approximating sampling distributions, especially when asymptotics may be unreliable or higher order remainder terms are non-negligible (e.g., for HAL-TMLE or robust statistics) (Laan, 2017).

Split conformal prediction and ART (Aggregation based on Ranks of Transformed sequences) framework exemplify exact, model-agnostic approaches yielding finite-sample coverage or type-I error control independently of distributional assumptions (Hulsman, 2022, Cui et al., 8 Jan 2025). These methods utilize explicit ranking, symmetric scoring, and order statistic properties to calibrate inference without relying on parametric forms or asymptotic approximations.

Summary Table: Main Methodologies for Finite-Sample Approximation

Approach	Target Regime	Key Principle
Edgeworth Expansions	Sums, smooth functionals	Higher-order correction
Stein's Method	Discrete/extreme value statistics	Operator equations, TV bounds
Berry–Esseen, Cramér	Rate/tail bounds for sums	Uniform/tail deviation rates
Fourier Series	Densities with bounded support	Series expansion, moments
Bootstrap	Broad, model-agnostic	Empirical distribution
Split Conformal/ART	Prediction, changepoint detection	Distribution-free, ranking

5. Robust Estimation, Non-Euclidean Geometry, and Finite-sample Pathologies

Finite-sample distribution approximations are especially critical in contexts exhibiting non-Euclidean structure or strong robustness requirements (e.g., Rousseeuw–Croux scale estimators (Akinshin, 2022), sample Fréchet means on spheres (Tran et al., 2021, Eltzner et al., 2021)). In these settings, classical asymptotic regimes (e.g., normality via tangent space projections) can break down, leading to phenomena such as finite sample smeariness (FSS) which manifests as inflated dispersion of sample means, causing standard quantile-based tests to lose nominal control.

This inflation is quantified by the modulation: $m_n = \frac{n V_n}{V}$ where $V_n$ is the expected sample mean variance, and $V$ the population variance. Under FSS, $m_n > 1$, invalidating standard normal-based inference; bootstrap-based inference can restore correct calibration. On spheres and other compact positively curved spaces, FSS is ubiquitous, especially for rotationally symmetric laws, and must be accounted for in all nonparametric inference.

6. Model Complexity and Computational Perspectives

Finite-sample approximation methods must often balance accuracy with computational feasibility. Recent work introduces deterministic or pseudo-random sequence decompositions, which, by matching low-order moments via linear constraints and the quantile function of candidate distributions, can reproduce finite-sample behavior of estimators while dramatically reducing computational burden compared to classical Monte Carlo (Tuobang, 26 Feb 2024). Such approaches facilitate efficient approximation of finite-sample biases, variances, and other properties across a range of estimators and statistical functionals.

7. Applications and Implications in Modern Statistics

The developments in finite-sample distribution approximation provide concrete, actionable techniques for

• Improving the fairness and reliability of nonparametric hypothesis tests and confidence intervals by accounting for sample-dependent corrections.
• Designing robust estimators and understanding their efficiency and bias as a function of sample size (Akinshin, 2022).
• Calibrating model-agnostic inference procedures (split conformal, ART) in machine learning and changepoint detection, especially in high-dimensional or distributionally ambiguous settings (Hulsman, 2022, Cui et al., 8 Jan 2025).
• Extending probabilistic modeling and inference to non-Euclidean and non-asymptotic settings where classical statistical machinery does not readily apply (Tran et al., 2021, Eltzner et al., 2021).
• Guiding the selection of sequence transformations or constraints to build more efficient and accurate non-asymptotic approximations in robust and computationally intensive estimation (Tuobang, 26 Feb 2024).

Collectively, these contributions form the modern toolkit for quantifying, correcting, and controlling the finite-sample behavior of statistical procedures in both classical and contemporary data analysis, ensuring the validity and interpretability of inference across a wide spectrum of finite-sample regimes.