Pseudo Empirical Likelihood Approach

• Pseudo Empirical Likelihood Approach is a nonparametric method that extends classical empirical likelihood using plug-in weights and calibration to enhance robustness.
• It integrates adjustments from survey design and non-probability samples to produce reliable point estimates and range-respecting confidence intervals.
• The framework employs Lagrange multipliers and bootstrap calibration for data-driven optimization, ensuring double robustness against model misspecification.

The pseudo empirical likelihood approach refers to a suite of nonparametric inferential procedures that adapt or extend traditional empirical likelihood to accommodate situations where standard likelihood or design assumptions are violated, the sample arises from complex or unknown mechanisms (such as non-probability samples), or auxiliary information/calibration is required. Core to this framework is the replacement or augmentation of classical likelihood formulations—often via plug-in weights, model-based adjustments, or augmenting constraints—with the aim of producing estimators and confidence intervals with reliable frequentist or Bayesian properties, even in settings where the canonical empirical likelihood is not directly applicable.

1. Mathematical and Methodological Foundations

The canonical empirical likelihood (EL) constructs a nonparametric likelihood via maximizing sample probabilities $(p_i)$ under normalization and (typically) moment constraint(s):

$\ell_{\mathrm{EL}}(p) = \sum_{i=1}^n \log(p_i)$

subject to

$$\sum_{i=1}^n p_i = 1, \quad \sum_{i=1}^n p_i h(Y_i, \theta) = 0$$

for a chosen estimating function $h(\cdot,\cdot)$ encoding the model or moment restriction (0805.3203).

The pseudo empirical likelihood (PEL) generalizes this paradigm to complex survey, non-probability, or model-assisted settings by incorporating non-uniform, estimated, or data-dependent weights:

$$\ell_{\mathrm{PEL}}(p) = n \sum_{i \in S} \tilde{d}_i \log p_i$$

with normalized weights $\tilde{d}_i$ (survey design, propensity, or calibration-based), $\sum_{i \in S} \tilde{d}_i = 1$ (Chen et al., 12 Aug 2025). For non-probability samples, $$\tilde{d}_i^\mathcal{A} = (\hat{\pi}_i^\mathcal{A})^{-1} / \hat{N}^\mathcal{A}$$, where $\hat{\pi}_i^\mathcal{A}$ is an estimated propensity score.

Maximization is performed under normalization and (optionally) additional calibration constraints:

$$\sum_{i \in S} p_i = 1, \quad \sum_{i \in S} p_i m_i = \bar{m}^\mathcal{B}$$

where $m_i$ are model-fitted values (e.g., regression predictions) and $\bar{m}^\mathcal{B}$ is an external benchmark, producing a doubly robust estimator if either the propensity or outcome model is correct (Chen et al., 12 Aug 2025, Huang et al., 12 Jan 2024).

A likelihood ratio for hypothesis or interval construction is formed:

$$r_{\mathrm{PEL}}(\mu) = \ell_{\mathrm{PEL}}(\hat{p}(\mu)) - \ell_{\mathrm{PEL}}(\hat{p})$$

where $\hat{p}(\mu)$ is the maximizer under constraint $\sum_{i} p_i y_i = \mu$.

2. Inference for Non-Probability Surveys

For survey samples with unknown inclusion mechanisms, a central objective is estimation of finite population parameters (e.g., means, totals, proportions). PEL addresses this by modeling the probability of inclusion (propensity score $\pi_i^\mathcal{A})$ as a function of covariates $x_i$, typically via a chosen parametric form, and estimating parameters $\hat{\alpha}$ externally, often with auxiliary reference data or calibration targets (Chen et al., 12 Aug 2025).

The normalized pseudo-weights are constructed as:

$$\tilde{d}_i^\mathcal{A} = (\hat{\pi}_i^\mathcal{A})^{-1} / \hat{N}^\mathcal{A}, \quad \hat{N}^\mathcal{A} = \sum_{j \in S_\mathcal{A}} (\hat{\pi}_j^\mathcal{A})^{-1}$$

These weights are used in the PEL function and its maximizers yield point estimates for population means or totals that are formally equivalent to inverse probability weighted estimators.

Calibrated PEL estimators additionally incorporate outcome model predictions, leading to joint constraints involving both the observed $y_i$ and the fitted $m_i$. The constrained PEL function is then maximized over $(p_i)$ with Lagrange multipliers, yielding weights of the form:

$$\hat{p}_i = \tilde{d}_i^\mathcal{A} / (1 + \lambda (\hat{m}_i - \bar{m}^\mathcal{B}))$$

The crucial property is that the resulting estimator maintains consistency if either the propensity model or outcome model is correct—a double robustness phenomenon.

3. Confidence Intervals and Range-Respecting Inference

For binary variables (e.g., estimating a population proportion), standard construction of confidence intervals via asymptotic normality can yield intervals outside the admissible range $[0,1]$. The PEL ratio statistic, either with or without calibration, produces range-respecting, data-driven intervals.

The ratio statistic:

$$r_{\mathrm{PEL}}(\mu) = \ell_{\mathrm{PEL}}(\hat{p}(\mu)) - \ell_{\mathrm{PEL}}(\hat{p})$$

is used to construct confidence sets via an adjusted chi-squared reference:

$$\left\{ \mu : -2 r_{\mathrm{PEL}}(\mu)/a \leq \chi^2_{1,1-\alpha} \right\}$$

where $a$ is a design- or model-based scaling constant accounting for the complex weighting.

Finite-sample performance of this approach, including coverage probabilities and interval lengths, has been robustly validated in simulation studies, particularly for binary $y$, outperforming normal approximation intervals, especially in terms of keeping in-range and achieving targeted coverage (Chen et al., 12 Aug 2025).

4. Theoretical Properties and Asymptotics

Point estimators arising from PEL coincided asymptotically with standard IPW or doubly robust survey estimators when weights and constraints are correctly specified, but the PEL framework produces likelihood ratio statistics that, after appropriate scaling, converge to chi-squared distributions:

$$-2 r_{\mathrm{PEL}}(\mu) / a \xrightarrow{d} \chi^2_1$$

where $a$ accounts for the complex sampling design, propensity model, or calibration constraints (Chen et al., 12 Aug 2025). Explicit formulas for $a$ are derived in theorems, involving the structure of the weights and model variances.

Validity of this asymptotic reference supports use of PEL-based intervals and tests as pivotal tools for non-probability survey inference.

5. Robustness, Calibration, and Practical Features

A defining strength of the PEL approach is its robustness to model misspecification, inherited from the doubly robust construction. If only the propensity model or only the outcome model is correct, consistency is preserved for the parameter estimator. Failure to use model-calibrated constraints results in loss of validity if the propensity model is misspecified; with calibration, valid inference is maintained if either component is correct. This is corroborated by detailed simulation studies demonstrating PEL confidence intervals' reliability across model violation scenarios (Chen et al., 12 Aug 2025).

Another key feature is the data-driven orientation of the intervals, their invariance to transformations, and respect for the natural parameter space (especially for binary/proportional targets).

Empirically, interval coverage probabilities are close to the nominal levels across diverse designs. The impact of non-probability sample size is typically more pronounced than that of the reference sample.

6. Algorithmic Aspects and Bootstrap Calibration

Optimization of the PEL function with normalization and calibration constraints is implemented via Lagrange multipliers, and the weights are updated in closed form for given multipliers. Solving for $λ$ requires root-finding on typically low-dimensional calibration equations.

When closed-form scaling constants for the likelihood ratio distribution are hard to compute, empirical calibration (via bootstrap) is employed: resampling is conducted, and for each resample, the ratio statistic is recomputed under the parameter constraint, yielding an empirical distribution from which quantiles are taken to determine critical values and resulting interval boundaries. This procedure is computationally tractable and yields accurate nominal coverage even when theoretical adjustment factors are complex to evaluate (Chen et al., 12 Aug 2025).

7. Impact and Comparative Summary

The pseudo empirical likelihood approach represents a flexible, robust, and theoretically grounded methodology for finite-population and causal inference, especially when inclusion mechanisms are non-probabilistic or partially unknown, and auxiliary information is available for calibration. Its key advances relative to standard weighted or doubly robust estimators are (i) nonparametric, likelihood-based inference with correct coverage, (ii) confidence intervals that are both range-respecting and transformation-invariant, and (iii) a framework that unifies model-assisted and model-based survey methods under a likelihood perspective.

Simulation studies consistently indicate superiority of PEL confidence intervals for binary outcomes over normal approximations and improved robustness to weighting/model misspecification. Practical relevance is demonstrated for modern survey applications where data may be sourced partly or entirely from non-probability samples, enabling more reliable inference in official statistics and empirical social science (Chen et al., 12 Aug 2025).