[Paper Review] Bootstrapping Two-phase Sampling

Takumi Saegusa|arXiv (Cornell University)|Jun 21, 2014

Statistical Methods and Inference30 references3 citations

TL;DR

This paper proposes a novel nonparametric bootstrap procedure for two-phase stratified sampling without replacement, addressing variance estimation in semiparametric models where asymptotic variances lack closed forms. By combining phase-specific and stratum-specific bootstrap weights, the method establishes conditional weak convergence of inverse probability weighted empirical processes, enabling valid inference for weighted likelihood estimators and M/Z-estimators under complex sampling designs.

ABSTRACT

We propose a nonparametric bootstrap procedure for two-phase stratified sampling without replacement. In this design, a weighted likelihood estimator is known to have smaller asymptotic variance than under the convenient assumption of independence often made in practice. Variance estimation, however, has not been well studied for semiparametric models where variance may not have a closed form. Motivated by semiparametric inference, we establish conditional weak convergence of bootstrap inverse probability weighted empirical processes with several variants of calibration. Two main obstacles to applying existing bootstrap empirical process theory are the dependent and biased sample due to sampling design, and the complex limiting processes of the linear combinations of Brownian bridge processes. To address these issues, the proposed bootstrap weights take the form of the product of two weights corresponding to randomness from each phase and stratum. We apply our bootstrap to weighted likelihood estimation and establish two Z-theorems for a general semiparametric model where a nuisance parameter can be estimated either at a regular or a non-regular rate. We show different bootstrap calibration methods proposed in the survey sampling literature yield different bootstrap asymptotic distributions.

Motivation & Objective

• Address the lack of reliable variance estimation methods for weighted likelihood estimators in two-phase stratified sampling without replacement.
• Overcome the challenges of dependent, biased samples and complex limiting distributions involving linear combinations of Brownian bridge processes.
• Extend bootstrap empirical process theory to sampling designs with non-i.i.d. observations and non-exchangeable bootstrap weights.
• Provide a theoretically grounded bootstrap procedure applicable to general semiparametric models, including M- and Z-estimators.
• Demonstrate that different calibration methods in survey sampling yield distinct bootstrap asymptotic distributions, enhancing methodological flexibility.

Proposed method

• Propose a bootstrap weight structure as the product of i.i.d. weights (for phase I) and stratified sampling weights (for phase II), capturing randomness from both sampling phases and strata.
• Decompose the bootstrap IPW empirical process into phase I and phase II components to handle dependence and non-exchangeability.
• Establish conditional weak convergence of the bootstrap IPW empirical process by proving weak convergence of the phase II process given phase I weights.
• Use multiplier inequalities and symmetrization techniques to bound empirical process norms, leveraging results from empirical process theory (e.g., Lemma 2.9.1 of van der Vaart and Wellner).
• Apply the exchangeably weighted bootstrap framework to handle sampling without replacement, adapting tools from [30] and [37] to complex survey designs.
• Introduce a bottom-up proof strategy that sequentially handles phase I and phase II sampling randomness, ensuring convergence under minimal moment conditions.

Experimental results

Research questions

• RQ1 Can a bootstrap procedure be developed for two-phase stratified sampling without replacement that correctly estimates variance when asymptotic variances lack closed forms?
• RQ2 How can bootstrap theory be extended to handle dependent, non-i.i.d. samples arising from sampling without replacement in finite population surveys?
• RQ3 What role do different calibration methods in survey sampling play in shaping the asymptotic distribution of the bootstrap estimator?
• RQ4 Does the proposed bootstrap method yield valid inference for general semiparametric models, including M- and Z-estimators, under complex sampling designs?
• RQ5 Can conditional weak convergence of the bootstrap IPW empirical process be established despite non-exchangeable bootstrap weights and complex limiting processes?

Key findings

• The proposed bootstrap procedure successfully establishes conditional weak convergence of the inverse probability weighted empirical process under two-phase stratified sampling without replacement.
• The bootstrap weights, formed as the product of phase I and phase II stratum-specific weights, capture the dual sources of randomness and enable correct variance estimation.
• Different calibration methods in survey sampling lead to different asymptotic distributions of the bootstrap, demonstrating methodological sensitivity to calibration choice.
• The method applies to general semiparametric models, including estimators with regular or non-regular convergence rates, extending beyond weighted likelihood estimation.
• Theoretical results show that the bootstrap estimator achieves the correct asymptotic distribution, avoiding overestimation issues seen in standard nonparametric bootstrap under sampling without replacement.
• The proof strategy, involving decomposition and sequential convergence, overcomes the main obstacles of non-i.i.d. data and non-exchangeable weights, validating the method under minimal regularity conditions.