Weighted Empirical CDFs

• Weighted empirical CDFs are nonparametric estimators that assign nonnegative weights to data points, allowing adjustment for survey biases and heterogeneous sampling.
• They support robust inference by tailoring weighting schemes for hypothesis testing, quantile estimation, and goodness-of-fit assessments.
• Advanced computational methods, such as fast summation and divide-and-conquer algorithms, enable efficient implementation in high-dimensional and machine learning applications.

A weighted empirical cumulative distribution function (weighted ECDF) is a nonparametric estimator of the cumulative distribution function in which each data point is assigned a nonnegative weight, generalizing the standard ECDF and providing a flexible tool for inference, modeling, and hypothesis testing under heterogeneous sampling, survey biases, or bespoke design. These functions feature centrally in modern statistical theory—including survey sampling, empirical process theory, robust inference, goodness-of-fit testing, and machine learning—and are tightly linked to resampling schemes, fractional entropy measures, and efficient computational methodologies.

1. Structural Definition and Properties

The classical ECDF for a sample $x_1, \ldots, x_n$ assigns uniform weight $1/n$ to each observation. The general weighted ECDF takes the form: $$\widehat{F}_n^{(w)}(x) = \frac{1}{W} \sum_{i=1}^n w_i \cdot \mathbb{I}\{x_i \le x\},$$ where $w_i \geq 0$ and $W = \sum_{i=1}^n w_i$ is the (possibly normalized) total mass. This framework admits several key specializations: inverse-probability weighting, exponential forgetting (for time series), mixture distribution deconvolution, and bias correction in sampling.

Weighted ECDFs inherit many theoretical properties from their unweighted counterparts; however, their limiting behavior is shaped by the configuration of weights. For example, in survey sampling with informative selection, the empirical CDF converges to a weighted version of the superpopulation CDF, where the weight function encodes informative selection probabilities (Bonnéry et al., 2012).

2. Motivations and Theoretical Foundations

Weighted ECDFs arise naturally in a range of modern statistical contexts:

• Informative Sampling and Design-based Inference: When sampling probabilities depend on data values (as in length-biased or cluster sampling), the standard ECDF is inconsistent for the population CDF; the limiting form is a weighted CDF with respect to the sampling weights, as established in weighted Glivenko–Cantelli-type theorems (Bonnéry et al., 2012).
• Empirical Likelihood and Divergence Minimization: Weighted ECDFs are the building blocks of empirical likelihood, generalized empirical likelihood (GEL), and Cressie–Read-type profiles, with the imposed weights ensuring moment constraints for unbiased estimating equations, robust estimation, and posterior validity (Turbatu, 2017).
• Goodness-of-Fit and Hypothesis Testing: Weighted ECDFs underpin tail-sensitive tests (Meissner, 2012), approaches based on sup-functionals of weighted empirical processes (Stepanova et al., 2014), and multiplier (weighted bootstrap) schemes for constructing approximate replications of the null-distribution (Kojadinovic et al., 2012).
• Robust Estimation and Quantile Smoothing: In robust statistics and in nonstationary time series, weighted ECDFs provide running (time-decaying) quantile, median, or other L-estimator values, and enable quantile estimation for weighted mixture distributions (Akinshin, 2023).

These diverse applications motivate a unified mathematical treatment of weighted ECDFs, including the development of weighted quantile estimators, consistency and asymptotic normality results, and demonstrations of how weight structure shapes the estimator's variance, convergence, and inferential validity.

3. Weighted ECDFs in Empirical Process Theory and Statistical Inference

From an empirical process perspective, weighted ECDFs are key objects in both classical and modern central limit theorems. For instance, for a uniform process $\{X_t: t\in E\}$ on $(0,1)$, the weighted empirical process $$U_n(t,x) = n^{-1/2}\sum_{j=1}^n w(x)[\mathbb{I}\{X_j(t) \le x\} - x]$$ converges in $\ell^\infty(E \times [0,1])$ under envelope and weighted $L$-conditions (Yang, 2014). The limiting process is Gaussian with covariance derived from a weighted Brownian bridge. This provides a rigorous foundation for weighted time-dependent likelihood and confidence band estimation.

Weighted bootstrap (multiplier) versions of the empirical process enable large-sample, computationally efficient alternatives to the parametric bootstrap for hypothesis testing. In the general setting: $$F_n''(x) = \frac{1}{\sqrt{n}}\sum_{i=1}^n (Z_i - \overline{Z})\left[\mathbb{I}\{X_i \le x\} - \psi_{\hat\theta}(X_i)^\top \dot{F}_{\hat\theta}(x)\right],$$ where $\{Z_i\}$ are i.i.d. mean-zero variance-one multipliers, $\psi_{\hat\theta}$ is a model score, and $\dot{F}_{\hat\theta}(x)$ the CDF's gradient; this structure guarantees weak convergence to the desired null distribution under suitable regularity conditions (Kojadinovic et al., 2012). Such schemes achieve dramatic computational gains with no loss in power for large $n$ and high-dimensional data.

Weighted ECDFs also facilitate the construction of self-studentizing estimators in extreme value theory. For example, estimation of the Pickands dependence function in bivariate or multivariate extremal models can be based on weighted estimating equations utilizing empirical copulas and suitable weight functions, providing more precise interval estimates and reduced mean squared error (Peng et al., 2013).

4. Weighted ECDFs in Model Assessment and Testing

Weighted versions of the ECDF embody flexible, distribution-sensitive test statistics:

• Tail-sensitive test statistics employ transformations such as $$A_{R,n} = -\frac{1}{n}\sum_{i=1}^n\ln[1 - F(x_i)^a]$$ to upweight deviations in the tails; cumulant expansions and explicit limiting distributions (including Gamma distributions for specific weighting) are derived rigorously, leading to analytic tail-aware alternatives to classical Kolmogorov–Smirnov methods (Meissner, 2012).
• Sup-functionals of weighted empirical processes generalize Kolmogorov–Smirnov's maximum deviation to suprema of standardized processes: $$T_n(q) = \sup_{0 < F_0(t) < 1}\sqrt{n}\frac{\lvert \mathbb{F}_n(t) - F_0(t) \rvert}{q(F_0(t))}$$. Optimal choice of the weight $q$ (an Erdős–Feller–Kolmogorov–Petrovski upper-class function) regularizes tail blow-up and achieves minimax detection and optimal adaptivity for sparse mixture hypothesis testing (Stepanova et al., 2014).
• Weighted L_p-based goodness-of-fit testing is implemented by constructing empirical weight functions (e.g., for the Poisson: $w^*(x) = f_n(x)/f_\lambda(x)$) and evaluating weighted L_p distances against the null. Closed-form statistics and Monte Carlo experimentation demonstrate strong power, especially for complex alternatives such as overdispersion (Kirui et al., 20 Feb 2024).

Confidence interval and band construction also benefits: weighted sup-functionals yield sharper, data-adaptive confidence bands narrower in the tails and consistent at the nominal level, outperforming classical unweighted approaches in both coverage and sensitivity (Stepanova et al., 2014).

5. Advanced Applications: Entropy, Information, and Machine Learning

Weighted empirical CDFs are foundational in advanced topics:

• Weighted Entropy and Information Generating Functions: The cumulative information generating function (CIGF), $$G_X(\alpha, \beta) = \int [F(x)]^\alpha [1-F(x)]^\beta dx$$, and its weighted extensions via mixture or distortion functions unify cumulative entropy, Gini mean semi-difference, and generalize to fractional entropies and higher dimensions (Capaldo et al., 2023).
• Fractional and Dynamic Extensions: Weighted fractional generalized cumulative past entropy (WFGCPE) and weighted cumulative residual entropy generating functions (WCREGF) are defined via functionals such as $$\mathrm{CPE}_\gamma^\psi(X) = \frac{1}{\Gamma(\gamma+1)}\int \psi(x) K(x) [-\ln K(x)]^\gamma dx$$, with empirical plug-in estimators and established consistency, stochastic ordering, and central limit properties (Kayal et al., 2021, S. et al., 9 Feb 2024). These connect to fractional integration (Riemann–Liouville) and provide unique distribution characterizations (e.g., for the Rayleigh law), supporting new entropy-based GOF tests (S. et al., 9 Feb 2024).
• Robust Machine Learning and Correction of Sampling Bias: Weighted ECDFs support distribution correction under covariate shift by substituting density ratio estimation with more stable, parameter-free CDF-based corrections in loss functions (e.g., via empirical V-matrices). These approaches yield robust, MVUE-based estimators that improve prediction under sampling bias, as seen in both synthetic and real datasets (Mazaheri et al., 2020).
• Neural Net Loss Engineering: In regression, the Weighted Empirical Stretching (WES) loss introduces tail-weighting via label density-derived penalty $g(x)$ and a scaling parameter $\beta$, improving regression accuracy and robustness, especially in the prediction of rare, high-impact events (Koo et al., 2020).

6. Computational Methods and Algorithmic Advances

The practical implementation of weighted ECDFs in high-dimensional settings has seen rapid advances:

• Fast Algorithms for Multivariate Weighted ECDFs: Two main techniques are established for efficient computation:
  • Fast summation via lexicographical sweeps over rectilinear grids yields $O(N\log N)$ (or $O(N)$) complexity for weighted ECDF evaluation, leveraging pre-sorted data and partitioned local sums.
  • Divide-and-conquer algorithms achieve $O(N \log(N)^{d-1})$ for data-aligned evaluation points, crucial in in-memory calculation and fast kernel density estimation (KDE) decompositions (Langrené et al., 2020).
• Empirical Copulas, KDE, and Survival Analysis: Expressing KDEs as weighted sums of ECDFs (e.g., for Laplacian kernels: $$\hat{f}_\text{KDE}(z)=\sum_{i=1}^N w_i e^{-|x_i-z|/h}$$ as weighted CDF/run-off sums) allows immediate application of fast ECDF algorithms to nonparametric density and regression estimation.
• Probabilistic Circuits: For computational models, probabilistic circuits—computation graphs representing PMFs/PDFs—can be adapted to compute CDFs (or weighted variants) efficiently via leaf modification and generalized inclusion–exclusion; efficient conversion between PMF and CDF representations is possible in polynomial time for binary, finite discrete (via Less-Than Encoding), and continuous (via smoothness/decomposability) variables, broadening tractable inference for weighted empirical cumulative properties (Broadrick et al., 8 Aug 2024).

7. Contemporary Directions and Open Challenges

Weighted ECDFs continue to evolve with new statistical and computational demands:

• The design of optimal weight functions for hypothesis testing, inference, or adaptive estimation under nonstationary, dependent, or high-dimensional data remains an active area.
• Theoretical work is needed on efficient weighting and bias correction under complex sampling, missingness, and time-varying environments—including connections with online learning, streaming quantile estimation, and robust inferential methodologies (Akinshin, 2023).
• Probabilistic circuit representations and their transformation properties for ECDFs over mixed, high-cardinality, or structured variable spaces offer open questions for scalable modeling and inference (Broadrick et al., 8 Aug 2024).
• Entropy-type and information-oriented functionals built from weighted ECDFs provide a rich ground for developing new stochastic orderings, uncertainty characterizations, and model diagnostics, especially as fractional and dynamic generalizations mature (Capaldo et al., 2023, S. et al., 9 Feb 2024).
• Applications in benchmarking, ranking, and reliability require rigorous uncertainty quantification for estimated functionals (e.g., standard errors of thresholds or percentiles), particularly for finite samples and extreme quantiles (Pernot et al., 2018).

Weighted empirical cumulative distribution functions thus provide a general and integrative framework underpinning much of modern nonparametric statistics, goodness-of-fit testing, robust inference, and machine learning, with ongoing methodological and theoretical advances supporting increasingly complex and data-driven applications.