Adaptive Empirical Likelihood Procedures

• Adaptive empirical likelihood procedures are nonparametric methods that modify the likelihood construction to ensure feasibility and robust finite-sample performance.
• They incorporate data-driven techniques such as pseudo-observation adjustments, penalization for estimating-equation selection, and split-sample methods to handle high-dimensional data.
• These approaches achieve computational adaptation and automatic calibration while preserving key asymptotic properties like Wilks’ chi-square behavior.

Searching arXiv for papers on adaptive/adjusted/penalized empirical likelihood and related variants. Adaptive empirical likelihood procedure is not a single universally standardized method, but a family of empirical-likelihood-based constructions in which the likelihood surrogate, the estimating equations, the calibration device, or the computational representation is made data-driven to accommodate difficult inferential regimes. In the literature represented here, the phrase is most naturally associated with procedures that modify ordinary empirical likelihood to restore feasibility, improve finite-sample calibration, select informative estimating equations, handle heavy tails or dependence, or adapt inference to unknown asymptotic regimes (Chen et al., 2016, Gamage et al., 2016, Chang et al., 2017, Li et al., 2021, Tang et al., 2012). A crucial distinction runs through this literature: some methods are explicitly adjusted empirical likelihood procedures, some are penalized or split-sample EL procedures, and some are only adaptive in an informal sense because they are data-driven or automatically calibrated rather than “adaptive” in a formal decision-theoretic sense (Li et al., 2019, Kim et al., 2021, Chaudhuri et al., 2024).

1. Conceptual scope and terminological boundaries

Empirical likelihood is a nonparametric likelihood construction based on estimating equations rather than a fully specified parametric density. In the formulations summarized here, the common starting point is a moment or estimating-equation restriction such as $E\{g(X,\theta_0)\}=0$, followed by maximization over probability weights subject to those constraints (Chang et al., 2017, Liu et al., 2023, Vexler et al., 2018). The attraction of this framework is that it often preserves likelihood-ratio-style inference and Wilks-type chi-square calibration without requiring direct covariance estimation (Gamage et al., 2016, Liu et al., 2023).

Within this broad framework, “adaptive empirical likelihood procedure” is used most coherently as an umbrella description for methods that alter the empirical likelihood construction in response to the statistical or computational structure of the problem. The data block supports several distinct meanings.

One meaning is finite-sample or geometric adaptation. Adjusted empirical likelihood adds a pseudo-observation so that the empirical likelihood ratio is always well-defined and the convex-hull feasibility problem is repaired (Li et al., 2019, Chen et al., 2016, Gamage et al., 2016, Gamage et al., 2016). Another is estimating-equation adaptation, where penalization of the empirical-likelihood Lagrange multiplier induces selection among many candidate estimating equations, resulting in a data-driven subset of active constraints (Chang et al., 2017, Li et al., 2021). A third is computational adaptation, in which the empirical likelihood is rebuilt on blockwise summaries rather than the full sample to address massive data while preserving first-order efficiency and Wilks’ theorem (Jaeger et al., 2017, Liu et al., 2023). A fourth is automatic calibration, where empirical-likelihood statistics are calibrated by Monte Carlo or bootstrap rather than by fixed parametric covariance formulas, especially for simultaneous inference (Kim et al., 2021).

The literature also marks clear boundaries. "Likelihood based inference for current status data on a grid: A boundary phenomenon and an adaptive inference procedure" is explicitly adaptive and likelihood-based, but it is not an empirical likelihood method in the standard Owen sense (Tang et al., 2012). Conversely, several papers are not named “adaptive empirical likelihood,” yet are adaptive in an informal but technically substantive sense because they are nonparametric, data-driven, or automatically calibrated (Kim et al., 2021, Chaudhuri et al., 2024, Chaudhuri et al., 2020).

2. Feasibility repair and finite-sample adjustment

A large part of the adaptive-EL literature is driven by a basic geometric problem: empirical likelihood may be undefined when the origin is not in the convex hull of the estimating functions. In ordinary EL or JEL, the constraints are feasible only if the estimating equation can be solved with nonnegative weights summing to one (Chen et al., 2016, Gamage et al., 2016, Gamage et al., 2016). This difficulty is amplified in small samples, in tail problems where only a small number of extremes are used, and in time-series settings where frequency-domain estimating functions can be poorly behaved (Li et al., 2019, Gamage et al., 2016, Gamage et al., 2016).

The standard adjustment is to add one pseudo-observation proportional to the negative empirical mean of the estimating functions. In the heavy-tail tail-index paper, this takes the form

$$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$

with $a_n>0$, leading to the adjusted empirical likelihood ratio

$$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$

and

$$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$

with $\lambda$ solving the corresponding constraint equation (Li et al., 2019). The paper states explicitly that one can verify there always exists a probability vector satisfying the constraint, so $l_{\mathrm{AEL}(\gamma)}$ is always well-defined (Li et al., 2019).

The same repair appears in jackknife EL. "Adjusted Jackknife Empirical Likelihood" constructs jackknife pseudo-values $\widehat V_i$, then adds

$$\widehat V_{n+1} = -a_n U_n = -\frac{a_n}{n}\sum_{i=1}^n \widehat V_i$$

and defines the adjusted likelihood ratio on $n+1$ pseudo-observations (Chen et al., 2016). The paper’s main theorem is that, for one-sample and two-sample U-statistics,

$$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$0

so the adjustment preserves the Wilks-type first-order limit while guaranteeing that the statistic is well-defined for all parameter values (Chen et al., 2016).

Time-series AEL papers apply the same idea in the frequency domain. For short-memory models, "Adjusted Empirical Likelihood for Time Series Models" augments Monti’s Whittle-based empirical likelihood by

$$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$1

with

$$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$2

and proves that the adjusted statistic still has asymptotic $$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$3 calibration (Gamage et al., 2016). "Adjusted Empirical Likelihood for Long-memory Time Series Models" applies the same construction to ARFIMA models and proves

$$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$4

under the Fox and Taqqu conditions (Gamage et al., 2016).

A common misconception is that these procedures are “adaptive” because they change the target parameter or estimating equations. The papers do not support that interpretation. Their contribution is more specific: they adapt the empirical likelihood geometry so that the optimization problem exists and finite-sample coverage improves, while leaving first-order asymptotics unchanged (Li et al., 2019, Chen et al., 2016, Gamage et al., 2016, Gamage et al., 2016).

3. Adaptive selection of estimating equations and variables

The most explicit adaptive empirical-likelihood framework in the supplied material is the doubly penalized high-dimensional EL literature. "A new scope of penalized empirical likelihood with high-dimensional estimating equations" studies i.i.d. data $$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$5, a $$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$6-dimensional parameter $$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$7, and an $$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$8-dimensional estimating function $$y_{k_n+1}(\gamma) = -\frac{a_n}{k_n}\sum_{i=1}^{k_n} y_i(\gamma) = -a_n(\hat{\gamma}_n-\gamma),$$9 satisfying

$a_n>0$0

The paper’s main methodological proposal is

$a_n>0$1

where $a_n>0$2 penalizes $a_n>0$3 and $a_n>0$4 penalizes $a_n>0$5 (Chang et al., 2017).

The adaptive feature is not incidental. Penalizing $a_n>0$6 induces sparsity in the EL Lagrange multiplier, which the paper identifies as equivalent to selection among high-dimensional estimating equations (Chang et al., 2017). For fixed $a_n>0$7, if

$a_n>0$8

then the active set of estimating equations is controlled by thresholding the sample moment magnitudes relative to $a_n>0$9 (Chang et al., 2017). This is a genuine data-driven estimating-equation selection mechanism inside EL.

The asymptotic theory is correspondingly adaptive to sparsity. With active parameter set

$$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$0

Theorem 1 establishes a sparse local minimizer $$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$1 such that

$$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$2

and Theorem 2 gives asymptotic normality for the nonzero coordinates after bias correction (Chang et al., 2017). The paper emphasizes that this permits both parameter dimension and number of estimating equations to grow exponentially with sample size under sparsity and effective-selection conditions (Chang et al., 2017).

"Robust penalized empirical likelihood in high dimensional longitudinal data analysis" extends the same architecture to longitudinal marginal models, but replaces the standard QIF-type estimating functions by robust estimating equations

$$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$3

with

$$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$4

Its penalized criterion is

$$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$5

This again performs simultaneous selection of variables and estimating equations, but now with bounded influence functions for both $$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$6 and $$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$7, due to bounded $$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$8 and leverage weights $$l_{\mathrm{AEL}(\gamma)} = -2\log\Bigg( \sup\Big\{ \prod_{i=1}^{k_n+1}((k_n+1)p_i): p_i\ge 0,\ \sum_{i=1}^{k_n+1}p_i=1,\ \sum_{i=1}^{k_n+1}p_i y_i(\gamma)=0 \Big\} \Bigg)$$9 (Li et al., 2021).

This suggests that in the high-dimensional literature, “adaptive empirical likelihood procedure” is best understood not as a general slogan but as a concrete min-max mechanism: sparsity in $$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$0 adapts to the active model, and sparsity in $$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$1 adapts to the informative subset of estimating equations (Chang et al., 2017, Li et al., 2021).

4. Computational adaptation: split-sample, composite, and massive-data EL

A separate line of work adapts empirical likelihood to computational scale rather than to sparsity or robustness. "Split Sample Empirical Likelihood" proposes “a new approach that combines multiple non-parametric likelihood-type components to build a data-driven approximation of the true likelihood function” and shows that “the asymptotic behaviors of our approach are identical to those seen in empirical likelihood” while “significantly decreasing computational time” (Jaeger et al., 2017). The supplied details do not provide the exact paper text, so no more specific construction can be attributed reliably (Jaeger et al., 2017).

A more concrete massive-data construction is given in "A novel approach of empirical likelihood with massive data," which introduces split sample mean empirical likelihood (SSMEL) (Liu et al., 2023). The method partitions the sample into $$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$2 subsets of size $$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$3, computes one mean estimating function per block,

$$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$4

and then applies EL to the $$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$5 block means: $$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$6 The key identity is

$$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$7

which preserves the full-sample mean estimating equation exactly (Liu et al., 2023). The paper proves

$$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$8

and

$$l_{\mathrm{AEL}(\gamma)} = 2\sum_{i=1}^{k_n+1}\log\bigl(1+\lambda y_i(\gamma)\bigr)$$9

so SSMEL has the same first-order efficiency as centralized EL and retains Wilks’ theorem (Liu et al., 2023).

This is adaptive in a computational sense. The user chooses $\lambda$0 to trade optimization cost against information retention. The paper recommends $\lambda$1 and notes that if $\lambda$2 is too small, the method over-compresses the information; if $\lambda$3 is too close to $\lambda$4, the convex-hull geometry becomes weak (Liu et al., 2023). This suggests a notion of adaptive EL in which the empirical support is changed from individual observations to block means while preserving the inferential target.

The idea of combining multiple components also appears in "Composite Empirical Likelihood," whose abstract states that the method “combines multiple non-parametric likelihood-type components to build a data-driven approximation of the true function” by borrowing “the ability to avoid a parametric specification” from empirical likelihood and “multiple likelihood components” from composite likelihood (Jaeger et al., 2015). The supplied details explicitly say that the paper text was unavailable and that no verified formulas or theorems could be extracted, so only this abstract-level description can be used responsibly (Jaeger et al., 2015). Even at that level, it supports the view that adaptive EL can also mean modular combination of multiple likelihood-type pieces (Jaeger et al., 2015).

5. Automatic calibration, multiple testing, and adaptive likelihood-based inference

The experimental-design paper develops an empirical-likelihood framework that is not called adaptive in a formal sense, but it is strongly data-driven in its calibration (Kim et al., 2021). In blocked designs, the estimating function is

$\lambda$5

and the EL statistic is built from

$\lambda$6

with hypotheses expressed as smooth constraints $\lambda$7 and test statistics

$\lambda$8

(Kim et al., 2021).

Its main adaptive feature is calibration. The paper proposes two single-step multiple testing procedures: asymptotic Monte Carlo (AMC) and nonparametric bootstrap (NB) (Kim et al., 2021). AMC estimates the joint covariance structure from the data and simulates the limiting quadratic forms. NB resamples from null-transformed data and recomputes the entire vector of EL statistics (Kim et al., 2021). Both procedures asymptotically control the generalized family-wise error rate and construct simultaneous confidence intervals “without explicitly considering the underlying covariance structure” (Kim et al., 2021). The paper explicitly interprets this as data-driven and robust to violations of standard mixed-model assumptions (Kim et al., 2021).

A related but conceptually distinct example is "Likelihood based inference for current status data on a grid: A boundary phenomenon and an adaptive inference procedure" (Tang et al., 2012). This paper studies the current status NPMLE on a grid with spacing $\lambda$9, where asymptotics change depending on whether $l_{\mathrm{AEL}(\gamma)}$0, $l_{\mathrm{AEL}(\gamma)}$1, or $l_{\mathrm{AEL}(\gamma)}$2 (Tang et al., 2012). Its adaptive procedure works by pretending $l_{\mathrm{AEL}(\gamma)}$3, computing a surrogate $l_{\mathrm{AEL}(\gamma)}$4, and using the boundary family of limit laws to construct confidence intervals that are asymptotically valid without knowing $l_{\mathrm{AEL}(\gamma)}$5 (Tang et al., 2012). The paper is explicit that this is not an empirical likelihood method in the standard Owen sense (Tang et al., 2012). Its relevance is conceptual: it shows that adaptation can also mean avoiding regime estimation by exploiting a boundary distribution that interpolates between competing asymptotic laws (Tang et al., 2012).

This suggests a broader perspective. In empirical-likelihood-related work, “adaptive procedure” often refers less to a particular primal-dual optimization change than to an inference scheme whose calibration is learned from the data or from a unifying asymptotic boundary law (Kim et al., 2021, Tang et al., 2012).

6. Simulation-based and Bayesian empirical likelihood as adaptive surrogates

Several papers use empirical likelihood as a likelihood surrogate in Bayesian or likelihood-free inference. Their adaptivity is mostly simulation-driven and summary-driven rather than formal in the penalized-EL sense.

"Bayesian computation via empirical likelihood" defines

$l_{\mathrm{AEL}(\gamma)}$6

subject to

$l_{\mathrm{AEL}(\gamma)}$7

and then treats

$l_{\mathrm{AEL}(\gamma)}$8

as an approximate posterior (Mengersen et al., 2012). The basic sampler draws $l_{\mathrm{AEL}(\gamma)}$9 and weights by $\widehat V_i$0, while BC-AMIS adaptively updates the proposal distribution using weighted means and covariances from earlier particles (Mengersen et al., 2012). This is an explicitly adaptive Monte Carlo mechanism built around EL, even though the EL constraints themselves are not adaptively chosen (Mengersen et al., 2012).

The ABC-EL papers move further toward simulation-based adaptation. "An easy-to-use empirical likelihood ABC method" and "On a Variational Approximation based Empirical Likelihood ABC Method" define, for observed summary $\widehat V_i$1 and simulated summaries $\widehat V_i$2,

$\widehat V_i$3

and the feasible set

$\widehat V_i$4

with empirical-likelihood weights

$\widehat V_i$5

(Chaudhuri et al., 2018, Chaudhuri et al., 2020). The resulting pseudo-posterior is

$\widehat V_i$6

in the 2018 paper (Chaudhuri et al., 2018), and, in the 2020 variational version,

$\widehat V_i$7

after adding an estimated entropy term (Chaudhuri et al., 2020).

"On an Empirical Likelihood based Solution to the Approximate Bayesian Computation Problem" formalizes the same idea as ABCel, defining

$\widehat V_i$8

$\widehat V_i$9

$$\widehat V_{n+1} = -a_n U_n = -\frac{a_n}{n}\sum_{i=1}^n \widehat V_i$$0

and posterior approximation

$$\widehat V_{n+1} = -a_n U_n = -\frac{a_n}{n}\sum_{i=1}^n \widehat V_i$$1

The paper explicitly says it does not call the method “adaptive empirical likelihood,” but it is data-dependent through the observed-summary constraints, parameter-specific reweighting, and entropy estimation (Chaudhuri et al., 2024).

A plausible implication is that in the Bayesian and ABC literature, adaptive EL means the likelihood surrogate is rebuilt at each $$\widehat V_{n+1} = -a_n U_n = -\frac{a_n}{n}\sum_{i=1}^n \widehat V_i$$2 from simulation output and observed summaries, rather than from a fixed analytic estimating equation. The papers do not claim more than that, and they also emphasize limitations: summary choice remains problem-dependent, feasibility can fail, and posterior support may be non-convex (Chaudhuri et al., 2018, Chaudhuri et al., 2020, Chaudhuri et al., 2024).

7. Scope, controversies, and recurring limitations

Several recurring limitations appear across the literature.

First, adaptivity is not uniform across papers. Some methods are explicitly adjusted EL procedures and should be described that way, not as general adaptive EL frameworks (Li et al., 2019, Chen et al., 2016, Gamage et al., 2016, Gamage et al., 2016). Some are explicitly penalized EL procedures with adaptive moment selection (Chang et al., 2017, Li et al., 2021). Others are adaptive only in an informal sense because they are data-driven, automatically calibrated, or simulation-based (Kim et al., 2021, Chaudhuri et al., 2024, Chaudhuri et al., 2020).

Second, constraint choice remains central. In Bayesian computation via EL, the success of BCel depends heavily on the identifying quality of the estimating equations, and the normal example in the paper shows that adding more constraints can worsen posterior approximation (Mengersen et al., 2012). The easy-to-use EL-ABC paper likewise emphasizes that the method cannot automatically down-weight uninformative summaries, so poor summaries can cause infeasibility or underestimation of uncertainty (Chaudhuri et al., 2018).

Third, convex-hull feasibility remains a structural issue. Adjusted EL papers exist precisely because unadjusted EL may have no solution when the zero vector is outside the convex hull of the estimating functions (Li et al., 2019, Chen et al., 2016, Gamage et al., 2016, Gamage et al., 2016). ABC-EL methods inherit related support irregularities because the observed summary may lie outside the convex hull of simulated summaries at a candidate parameter value (Chaudhuri et al., 2018, Chaudhuri et al., 2024).

Fourth, different adaptive mechanisms target different problems. Robust doubly penalized EL targets outliers, heavy tails, and high-dimensional moment sets (Li et al., 2021). SSMEL targets massive-data computation (Liu et al., 2023). Adjusted EL targets existence and small-sample calibration (Li et al., 2019, Chen et al., 2016). Bootstrap- or Monte-Carlo-calibrated EL targets simultaneous inference under complex covariance structures (Kim et al., 2021). These should not be conflated.

Finally, not every adaptive likelihood-based procedure is empirical likelihood. The current-status-grid paper provides a strong example of adaptive inference through a boundary family of asymptotic distributions, but it belongs to nonparametric maximum likelihood under shape constraints, not to empirical likelihood in the Owen/Qin–Lawless sense (Tang et al., 2012).

Taken together, the literature suggests that an adaptive empirical likelihood procedure is best defined operationally rather than doctrinally. It is an EL-based method that changes the empirical likelihood construction, the active estimating equations, the support geometry, the calibration rule, or the computational representation in a data-driven way to maintain inference under nonstandard conditions. The most explicit realizations in the supplied material are pseudo-observation-based adjusted EL (Li et al., 2019, Chen et al., 2016, Gamage et al., 2016, Gamage et al., 2016), doubly penalized EL with estimating-equation selection (Chang et al., 2017, Li et al., 2021), and split-sample mean EL for massive data (Liu et al., 2023).