Threshold Exceedance Estimation in Spatially Correlated Areal Data Using Maxima-Nominated Sampling

Abstract

We study estimation of the proportion of areal units in a spatially correlated domain whose success probabilities exceed a prespecified threshold. Such problems arise in health surveillance, environmental monitoring, and social policy, where the goal is to estimate the fraction of high-risk areas. We propose a DUST-MNS design that combines maxima-nominated sampling (MNS) with the probability-proportional-to-size dependent unit sequential technique (pps-DUST), thereby promoting spatial spread while mitigating the effect of spatial autocorrelation. The design forms $n$ candidate sets of size $k$ and obtains final measurements only from the area judged to be at highest risk in each set, yielding $n$ measured areas from $nk$ screened candidates. Ranking may be based on expert judgment, prior surveys, or easily obtained auxiliary covariates. We derive a closed-form estimator of the exceedance probability $\theta$ based on data from DUST-MNS design, establish its bias and variance, and show that, in the rare-to-moderate exceedance regime $\theta<\theta^\star(k)$, the proposed DUST-MNS estimator outperforms its SRS and DUST-SRS counterparts, where $\theta^\star(k)$ depends only on $k$. We also provide guidance on the choice of $k$, derive efficiency bounds under a Beta model, extend the method to imperfect ranking, and develop variance estimation and bootstrap confidence intervals. An application to county-level stroke prevalence data from CDC PLACES, using diabetes prevalence as the ranking concomitant, illustrates the proposed approach.