Semiparametric Empirical Likelihood

• Semiparametric empirical likelihood is a framework that maximizes a nonparametric likelihood under estimating-equation constraints to infer finite-dimensional parameters without a full parametric density.
• It integrates parametric model components with nonparametrically determined weights, yielding likelihood-ratio tests and confidence regions that adhere to asymptotic chi-square (Wilks) behavior, while allowing adjustments to improve efficiency.
• The approach is versatile, with applications in robust regression, missing data, time series, and high-dimensional models, enabling techniques such as doubly robust estimation and penalized inference.

to=arxiv_search.search 天天中彩票能json {"query":"semiparametric empirical likelihood arXiv empirical likelihood estimating equations semiparametric", "max_results": 10, "sort_by":"relevance"} to=arxiv_search.search ിച്ചിട്ടുണ്ട്ഞ്ജसन {"query":"(Özdemir et al., 2018) empirical likelihood robust estimation linear regression", "max_results": 5, "sort_by":"relevance"} Semiparametric empirical likelihood (EL) is a likelihood-based inferential framework in which a finite-dimensional parameter is specified through estimating equations or structural moment restrictions, while the underlying distribution is left unspecified and represented by unknown probability masses on the observed sample points (Liu et al., 2010, Özdemir et al., 2018, Wang et al., 2023). Rather than postulating a full parametric density, semiparametric EL maximizes a nonparametric likelihood over weights subject to simplex and estimating-equation constraints, then profiles out those weights with Lagrange multipliers to obtain estimators, likelihood-ratio statistics, confidence regions, and hypothesis tests (Liu et al., 2010, Yuan et al., 2021). Across recent work, the same core construction has been adapted to robust regression, missing data, density ratio models, structural equation models, weakly dependent time series, stratified metric spaces, capture–recapture models, spatial hierarchies, and ensemble learners.

1. Core construction and semiparametric viewpoint

In the standard estimating-equation formulation, the parameter $\theta$ is defined by

$E\{g(X;\theta)\}=0,$

and empirical likelihood maximizes

$$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$

The empirical log-likelihood ratio is

$R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$

and, when the constraints are feasible, the maximizing weights satisfy the usual dual form

$$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$

with $\lambda$ determined by the multiplier equation

$$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$

(Liu et al., 2010).

The semiparametric character is explicit in regression formulations. For the linear model

$Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$

the regression structure is specified parametrically through $\boldsymbol\beta$, while the distribution of the observations is left unspecified and represented nonparametrically by unknown masses $p_i$. The EL criterion is

$E\{g(X;\theta)\}=0,$0

under

$E\{g(X;\theta)\}=0,$1

which is the EL analogue of the OLS normal equations, with equal weights $E\{g(X;\theta)\}=0,$2 replaced by unknown $E\{g(X;\theta)\}=0,$3 (Özdemir et al., 2018).

The same logic extends beyond single-sample moment models. In the two-sample density ratio model, the baseline distribution $E\{g(X;\theta)\}=0,$4 is represented nonparametrically on the pooled observed support by masses $E\{g(X;\theta)\}=0,$5, while the second sample is linked through

$E\{g(X;\theta)\}=0,$6

and auxiliary information is imposed by

$E\{g(X;\theta)\}=0,$7

(Yuan et al., 2021). This pattern—parametric low-dimensional structure plus nonparametric masses under unbiased constraints—is the defining architecture of semiparametric EL.

2. Likelihood-ratio inference, Wilks phenomena, and high-order refinement

Under standard regularity conditions, empirical likelihood has the central likelihood-ratio property

$E\{g(X;\theta)\}=0,$8

which yields asymptotically valid confidence regions of the form

$E\{g(X;\theta)\}=0,$9

(Liu et al., 2010). This Wilks-type behavior is one reason EL is routinely described as likelihood-style inference without a fully specified parametric model.

A practical obstruction is the convex-hull condition: ordinary EL is undefined if $$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$0 does not lie in the convex hull of the estimating functions. Adjusted empirical likelihood (AEL) resolves this by adding a pseudo-observation

$$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$1

and maximizing the augmented likelihood over $$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$2 points. With a specific adjustment level,

$$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$3

where $$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$4 is the Bartlett correction factor for ordinary EL, AEL achieves the same $$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$5 chi-square accuracy as Bartlett-corrected EL while also guaranteeing existence of the adjusted estimating equations (Liu et al., 2010). In the scalar case $$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$6, the Bartlett factor is

$$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$7

under $$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$8, and the paper also discusses the two-pseudo-observation construction for $$L_n(\theta)= \sup \Biggl\{ \prod_{i=1}^n p_i : p_i\ge 0,\; \sum_{i=1}^n p_i=1,\; \sum_{i=1}^n p_i g(x_i;\theta)=0 \Biggr\}.$$9.

Jackknife empirical likelihood addresses nonlinear estimands, especially U-statistics, by replacing the original problem with a mean problem on pseudo-values

$R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$0

Adjusted jackknife empirical likelihood (AJEL) adds

$R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$1

with $R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$2 suggested and $R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$3 sufficient, so that the empirical likelihood ratio is well-defined for all $R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$4. The resulting statistic still satisfies

$R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$5

for one-sample and two-sample U-statistics (Chen et al., 2016).

Higher-order accuracy also appears in non-Euclidean EL. For Fréchet means on open books, bootstrap calibration improves the coverage error of EL confidence regions from

$R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$6

and the paper explicitly notes that EL is Bartlett correctable asymptotically (Bharath et al., 2024). This places semiparametric EL among the few nonparametric likelihood methods that can be pushed beyond first-order chi-square calibration.

3. Robust, local, and globally consistent variants

Classical regression EL inherits the sensitivity of OLS-like estimating equations. In the linear model, standard EL uses the weighted normal equations, so merely replacing equal weights by unknown $R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$7 does not sufficiently protect against outliers. A direct robustification replaces the residual moment condition by the M-estimation score equation

$R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$8

or, in EL form,

$R_n(\theta)=-2\log\bigl(n^n L_n(\theta)\bigr),$9

Here $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$0, and bounded or redescending $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$1 functions such as Huber’s and Tukey’s bisquare downweight large residuals (Özdemir et al., 2018). In the contaminated normal simulation $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$2, with $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$3 and $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$4, the reported MSEs were $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$5 for EL, $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$6 for EL-Huber, $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$7 for EL-Tukey, and $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$8 for OLS, with Tukey’s bisquare usually the best under contamination (Özdemir et al., 2018).

A different line of development treats EL as a local likelihood experiment. A local representation paper shows that for local perturbations $$p_i=\frac{1}{n}\frac{1}{1+\lambda^\top g(x_i;\theta)},$$9, the implied EL measures admit the approximation

$\lambda$0

and derives the local estimator

$\lambda$1

The paper proves consistency, local asymptotic normality, and asymptotic optimality, and emphasizes that the consistency theorem does not require differentiability of $\lambda$2 (Gao, 2014).

Global behavior is a separate issue. Standard EL theory is local: it shows that a local maximizer near the truth is consistent, but it does not imply that the global maximizer is the correct one when the EL surface has multiple local or global maxima. A global-consistency treatment establishes strong consistency of the global empirical likelihood maximizer under conditions C1–C5, including identifiability, finite moments, local Lipschitz continuity, a closed parameter space, and a growth/nondegeneracy condition at infinity (Liang et al., 2023). The same paper proposes a global maximum test based on the value of the profile EL ratio at a candidate maximizer and a remedy that expands the estimating-function vector so that the EL criterion becomes globally identifying.

4. Efficiency gains, side information, and doubly robust estimation

A major attraction of semiparametric EL is that valid side information can be encoded as constraints on the weights and converted directly into efficiency gains. For a linear functional

$\lambda$3

with side information $\lambda$4, the EL weights satisfy

$\lambda$5

and the EL-weighted estimator becomes

$\lambda$6

Under fixed finite constraints,

$\lambda$7

with

$\lambda$8

so the asymptotic variance is the variance of the residual after projection onto the span of the constraint functions (Wang et al., 2023). The same paper allows estimated constraints $\lambda$9 and a growing number of constraints $$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$0, provided growth and moment conditions hold.

In density ratio models, auxiliary information is incorporated by unbiased estimating equations

$$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$1

on top of the semiparametric relation

$$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$2

The maximum empirical likelihood estimators $$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$3 are asymptotically normal, the ELR statistic for hypotheses $$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$4 converges to $$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$5, and the paper proves a monotonicity statement: if $$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$6, dropping one valid estimating equation cannot reduce the asymptotic variance; if $$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$7, including the just-identified valid estimating equations does not change the asymptotic variance relative to the DRM-only estimator (Yuan et al., 2021).

Missing-data problems make the semiparametric efficiency role of EL especially explicit. For parameters defined by general estimating equations under missing at random,

$$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$8

an efficient doubly robust EL approach constructs stacked estimating functions combining a propensity model and a working model for $$\sum_{i=1}^{n}\frac{g(x_i;\theta)}{1+\lambda^\top g(x_i;\theta)}=0$$9. The resulting estimator is consistent if either the propensity model is correct or the regression model is correct, and when both are correctly specified it achieves the full semiparametric efficiency bound (Liu et al., 2016). In randomized trials, two-sample EL weighting for the average treatment effect,

$Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$0

uses arm-specific empirical likelihood weights with covariate-balancing constraints, is semiparametric efficient when the working outcome regressions are correct, and extends under MAR missingness to estimators with double robustness and multiple robustness (Tan et al., 2020).

5. Dependence, nuisance functions, and singular geometry

Semiparametric EL is not confined to i.i.d. settings. For stationary, strongly mixing time series, a conditional heteroscedastic partially linear single-index model is rewritten as a finite set of unconditional moments,

$Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$1

where the nuisance vector $Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$2 contains nonparametric objects such as $Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$3, $Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$4, conditional expectations, and the index density. These components are estimated by kernel smoothing, yet the empirical log-likelihood ratio

$Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$5

has the same first-order limit when $Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$6 is replaced by $Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$7. The resulting Wilks theorem is

$Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$8

(Chaumaray et al., 2020).

Recent work on Fréchet means shows that the asymptotic law of the EL ratio can depend on the local topology of the parameter space. On an open book $Y_i=\mathbf X_i^{T}\boldsymbol\beta+\varepsilon_i,$9, EL is defined off the spine by a folding map $\boldsymbol\beta$0 and on the spine by projection $\boldsymbol\beta$1 plus page-direction inequalities. If the population Fréchet mean lies in the interior of a page,

$\boldsymbol\beta$2

If the mean lies on the spine and is sticky,

$\boldsymbol\beta$3

while in the half-sticky case

$\boldsymbol\beta$4

The paper’s central point is that the EL limit is governed not just by ambient dimension but by the stratified geometry near the mean (Bharath et al., 2024). This suggests that semiparametric EL can remain viable even when the parameter of interest lives on singular spaces, but the Wilks law must then be read geometrically.

6. Specialized implementations and contemporary extensions

The flexibility of semiparametric EL is visible in the range of model classes to which the same constrained-weighting mechanism has been adapted.

Setting	Distinctive EL construction	Paper
Linear regression with AR($\boldsymbol\beta$5) errors	Transformed regression moments jointly constrain $\boldsymbol\beta$6, $\boldsymbol\beta$7, and $\boldsymbol\beta$8; simulation shows smaller MSE and bias than CML in almost all configurations	(Özdemir et al., 2020)
Linear SEMs with dependent, non-Gaussian errors	Profiles out $\boldsymbol\beta$9, adds AEL and EEL, and reports runtime up to 40 times faster for profiled EL	(Wang et al., 2017)
Bayesian spatial hierarchical models	Places EL at the data stage and a spatial prior at the process stage in SHEL models; all three data examples report lower MSPE than standard parametric analyses	(Porter et al., 2014)
Capture–recapture abundance	EL ratio for abundance has limiting $p_i$0; extensions handle one-inflation and MAR covariates by semiparametric profiling and score testing	(Liu et al., 14 Jul 2025, Liu et al., 14 Jul 2025)
High-dimensional moment systems	Penalizes the Lagrange multipliers in Bayesian penalized EL and uses Metropolis–Hastings or MAMIS rather than direct optimization	(Chang et al., 2024)
Random forests and ensembles	Recasts predictions as incomplete generalized $p_i$1-statistics and uses modified EL to restore $p_i$2 pivotality under sparse subsampling	(Chiang et al., 17 Nov 2025)

These implementations share the same statistical grammar. Unknown masses or weights are optimized under model-implied constraints; nuisance structure is either profiled out, estimated in a first stage, or absorbed into auxiliary estimating equations; and the inferential target is typically a profile EL ratio with asymptotic chi-square calibration or a modified version thereof. What changes across domains is the constraint geometry: autoregressive residual structure in time series regression, structural zeros in mixed-graph SEMs, spatial latent processes in SHEL models, one-inflated count mechanisms in capture–recapture, sparse multiplier support in penalized EL, or jackknife pseudo-values for incomplete $p_i$3-statistics.

Taken together, these developments portray semiparametric empirical likelihood as a general inferential technology for models that are too structured to be purely nonparametric and too distribution-sensitive to be handled comfortably by full parametric likelihood. The recurring advantages are likelihood-ratio inference from moment restrictions, compatibility with nuisance estimation and side information, and a capacity for refinement through adjustment, robustification, bootstrap calibration, or penalization when the basic EL geometry becomes fragile.