Variance Estimation Under Local Pivotal Methods

2026-10-04

AI Summary

This study addresses the challenge of variance estimation in local pivotal methods arising from the computational intractability of second-order inclusion probabilities. To overcome this limitation, two solutions are proposed: Monte Carlo simulation-based correction and model-assisted bootstrapping. Methodologically, Gaussian processes are introduced to construct synthetic populations, while pairwise independence tests are incorporated to enhance estimator stability. Experimental results demonstrate that, across various noise levels and sampling configurations, the model-assisted bootstrap approach exhibits superior robustness compared to conventional correction techniques. Ultimately, this work effectively improves the precision of statistical inference under spatially balanced sampling designs.

Abstract

Spatially balanced sampling designs, such as the local pivotal methods (LPM), improve representation of auxiliary-variable space, but design-based variance estimation is challenging because second-order inclusion probabilities are generally intractable and may be zero or near zero. We propose two approaches for estimating the variance of the Horvitz-Thompson mean and the finite-population variance. The first estimates joint inclusion probabilities by Monte Carlo simulation and uses pairwise independence tests to modify the resulting variance estimators. The second, a model-assisted bootstrap (MABS), uses an interpolating Gaussian process to construct a synthetic finite population and estimates variance by resampling under the original LPM design. In simulations, the modified empirical approach performs well when responses are strongly associated with auxiliary variables, but its performance for the Horvitz-Thompson mean deteriorates as this association weakens. MABS exhibits more stable performance across noise levels and sampling settings, including under LPM2 and non-Gaussian noise. For finite-population variance estimation, however, no method shows a uniform advantage. These results illustrate the complementary strengths of the proposed approaches for inference under spatially balanced sampling.

Problem

• variance estimation
• spatially balanced sampling
• local pivotal methods
• inclusion probabilities

Innovation

• Local Pivotal Methods
• Variance Estimation
• Monte Carlo Simulation
• Model-Assisted Bootstrap
• Spatially Balanced Sampling