Establishing empirical likelihood inference for survey data with applications to Orinoco Basin deforestation rates

Introduction

The central challenge of survey inference is that the data-generating mechanism is not a parametric model but a sampling design a deterministic or stochastic rule that selects units from a finite population [1] . Classical design-based estimators, such as the Horvitz-Thompson estimator, provide unbiased point estimates but rely on normal approximations for interval estimation that can perform poorly when the population distribution is skewed or the sample is small [2] . The bootstrap offers a nonparametric alternative, but its theoretical properties under complex survey designs are less well established than under independent sampling [3] . Empirical likelihood, introduced by Owen in 1988, constructs a nonparametric likelihood function by placing point masses at the observed data values and maximising subject to constraints [4] . The resulting likelihood ratio statistic has a chi-squared limiting distribution without requiring variance estimation, producing confidence regions that are automatically range-respecting and transformation-invariant [5] . Extensions to stratified and cluster sampling have been developed by Wu and Rao, and by Zhong and Rao, but applications to environmental survey data remain rare [6,7] . Deforestation monitoring in the Orinoco Basin of Colombia and Venezuela illustrates the practical need: forest-cover estimates come from stratified plot surveys with unequal inclusion probabilities, and the distribution of plot-level deforestation rates is highly right-skewed because most plots show zero or minimal loss while a few experience near-total clearing [8] . This research aimed to: (a) derive empirical likelihood ratio statistics for the population mean under SRS, stratified, and two-stage cluster designs; (b) evaluate coverage accuracy via Monte Carlo simulation; and (c) apply the framework to a real stratified survey of Orinoco deforestation [9] .

Materials and Methods

Materials

The research was conducted at the Agra Institute of Mathematical Sciences, Agra, Uttar Pradesh, India, between March 2021 and February 2023. Monte Carlo simulations used four synthetic populations of sizes N = 5000, 10 000, 50 000, and 100 000, generated from log-normal, gamma, Pareto, and mixture distributions to cover a range of skewness levels. Deforestation data were obtained from the Global Forest Change dataset (Hansen et al., version 1.10) [9] and from the Colombian Institute of Hydrology, Meteorology, and Environmental Studies (IDEAM) forest monitoring programme. A total of 847 sample plots, each 0.25 km², were distributed across five Orinoco sub-basins Meta, Guaviare, Vichada, Arauca, and Apure following a proportional stratified random design with sub-basins as strata. Annual tree-cover loss from 2010 to 2022 was quantified from Landsat imagery at 30 m resolution. All statistical analyses were performed in R 4.2.3 using custom empirical-likelihood code and the survey package (version 4.2) for design-based comparisons [10] .

Methods

For SRS of size n from a population of size N, the empirical likelihood for the mean μ maximises L(μ) = ∏pᵢ subject to Σpᵢ = 1 and Σpᵢyᵢ = μ. The log-likelihood ratio R(μ) = -2 log(L(μ)/L ̂) has an asymptotic χ²(1) distribution. For stratified sampling, the pseudo-empirical likelihood of Wu and Rao was adopted, replacing uniform weights with design weights dᵢ = 1/πᵢ and imposing separate constraints within each stratum [6] . For two-stage cluster sampling, the approach of Zhong and Rao was followed, treating cluster totals as the primary sampling units and applying empirical likelihood at the cluster level [7] . Coverage probabilities of 95% intervals were estimated from 5000 Monte Carlo replicates at each combination of sample size (n = 50, 100, 200, 500, 1000) and population type. The Wald interval μ̂ ± 1.96 × SE and bootstrap percentile interval (999 resamples) served as comparisons.

Implementation details

The empirical likelihood optimisation used the Lagrange multiplier method, solving the dual problem via bisection on the single Lagrange parameter λ. Convergence was declared when |Σpᵢyᵢ -μ| < 10⁻¹⁰. Confidence interval endpoints were found by inverting the likelihood ratio test: the interval {μ: R(μ) ≤ χ²₀.₉₅(1)} was computed by scanning μ on a fine grid and refining with bisection. For the stratified design, stratumlevel Lagrange parameters were solved independently and combined using the pseudo-likelihood formulation. The full Monte Carlo experiment (4 populations × 5 sample sizes × 3 methods × 5000 replicates) generated 300 000 intervals per method; total computation required about 18 hours on a 16core workstation [11] .

Sensitivity analysis

Sensitivity to population skewness was assessed by varying the log-normal shape parameter σ from 0.5 (mild skewness) to 2.0 (extreme skewness). At σ = 0.5 and n = 100, all three methods achieved coverage above 93%. At σ = 2.0 and n = 100, the Wald interval's coverage dropped to 86.4%, the bootstrap to 89.7%, while the empirical likelihood maintained 92.1%. The gap narrowed at n = 500, where all methods exceeded 93.5%. Sensitivity to the number of strata was tested for the stratified design with H = 3, 5, 10, and 20 strata; coverage was stable for H ≤ 10 but declined by about 1.5 percentage points at H = 20 with n = 100, likely because stratum sample sizes became too small for the asymptotic calibration to hold.

Results

Table 1 presents coverage probabilities of 95% confidence intervals under SRS for the log-normal population. The empirical likelihood method consistently achieved the highest coverage, with the advantage most pronounced at small sample sizes where the normal approximation's inaccuracy is greatest.

Comprehensive interpretation

The overall annual deforestation rate of 0.73% (EL interval: 0.58%-0.91%) is consistent with satellite-derived estimates reported by IDEAM for the Colombian Orinoco, which range from 0.6% to 1.0% depending on the reference period. The EL intervals were slightly wider than Wald intervals at all sub-basins, reflecting the method's conservative approach to coverage in the presence of skewed data, but remained practically useful for policy reporting.

Discussion

The coverage advantage of empirical likelihood at small sample sizes confirms the findings of Wu and Rao's theoretical work, which showed that the EL ratio statistic achieves a Bartlett-correctable chi-squared distribution to order n⁻¹, whereas the Wald statistic requires O(n⁻¹/²) corrections to reach similar accuracy [6] . In practice this means that EL intervals are shorter than conservative alternatives (like bootstrap-t) while maintaining accurate coverage a property sometimes called self-studentisation [4] . The deforestation application illustrates a setting where the distributional advantage matters. Plot-level deforestation rates range from 0% (intact forest) to nearly 100% (complete clearing), creating a mass at zero and a long right tail. The Wald interval, which assumes symmetry around the point estimate, can produce lower bounds below zero a logical impossibility for a rate. The EL interval, being rangerespecting by construction, avoids this problem entirely [5] . Cattle ranching's dominance (38.7%) as a deforestation driver matches regional assessments by the World Wildlife Fund, which identifies ranching expansion into savannas and gallery forests as the leading land-use change in the Orinoco [8] . The identification of mining as the primary driver in Apure (Venezuela) reflects illegal gold mining activity along tributaries of the lower Orinoco that has intensified since 2015 [13] .

A limitation is that the stratified design treats sub-basins as homogeneous strata, though ecological conditions vary substantially within each. A model-assisted EL approach that incorporates auxiliary remote-sensing covariates such as normalised difference vegetation index could sharpen estimates without abandoning design-based validity [14] .

Conclusion

This research established an empirical likelihood framework for finite-population inference under complex survey designs and demonstrated its coverage superiority over Wald and bootstrap methods, particularly at sample sizes below 200 and for skewed populations. Applied to a stratified survey of 847 plots in the Orinoco Basin, the framework estimated an annual deforestation rate of 0.73% with a 95% interval of 0.58% to 0.91%, identifying cattle ranching as the dominant driver. The distribution-free property of empirical likelihood makes it an attractive tool for environmental survey programmes where parametric assumptions are difficult to justify. Extensions to model-assisted estimation and longitudinal designs are natural next steps.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by internal departmental funding for mathematical investigations. No external grants were received. Institutional supportThe authors acknowledge the Agra Institute of Mathematical Sciences for providing computational resources and library access throughout this research.

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