Asymptotic theory for nonlinear functionals and normality in continuous two-phase sampling

Establish an asymptotic theory for two-phase sampling with post-stratification over continuous populations that extends the treatment of nonlinear functionals and derives asymptotic normality for the resulting estimators.

Background

The paper develops design-based consistency, variance-estimation, and asymptotic-normality results for two-stage sampling over a continuous spatial territory. Its main results include linearization-based inference for smooth scale-invariant functions of spatial totals, but the analysis does not cover two-phase sampling with post-stratification.

The conclusion identifies two unresolved extensions in the environmental-survey setting: extending existing continuous-population convergence results to nonlinear functionals and deriving asymptotic normality in that broader continuous-population framework. Resolving these issues would provide a unified theory for designs combining multistage, multiphase, and spatial sampling.