Abstract

Data collection costs can vary widely across variables in data science tasks. Two-phase designs

can be employed to save data collection costs. This paper considers the two-phase studies where inexpensive variables are collected for all subjects in the first phase, and expensive variables are measured for a

subsample of subjects in the second phase based on a predetermined sampling rule. The estimation efficiency under two-phase designs relies heavily on the sampling rule. Existing literature primarily focuses on

designing sampling rules for estimating a scalar parameter in some parametric models or specific estimating problems. However, real-world scenarios are usually model-unknown and involve two-phase designs

for model-free estimation of a scalar or multi-dimensional parameter. This paper proposes a maximin criterion to design an optimal sampling rule based on semiparametric efficiency bounds. The proposed method

is model-free and applicable to general estimating problems. The resulting sampling rule can minimize the

semiparametric efficiency bound when the parameter is scalar and improve the bound for every component

when the parameter is multi-dimensional. Simulation studies demonstrate that the proposed designs reduce

the variance of the resulting estimator in various settings. The implementation of the proposed design is

illustrated in a real data analysis.

Information

Preprint No.SS-2024-0359
Manuscript IDSS-2024-0359
Complete AuthorsRuoyu Wang, Qihua Wang, Wang Miao
Corresponding AuthorsQihua Wang
Emailsqhwang@amss.ac.cn

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Acknowledgments

We would like to thank the editor, associate editor, and the anonymous reviewer for their

very insightful and helpful comments, which led to a significant improvement of our paper. Wang’s research was supported by the National Natural Science Foundation of China

(General program 12271510), and a grant from the Key Lab of Random Complex Structure

and Data Science, CAS. Miao’s research was supported by the National Key R&D Program

(2022YFA1008100) and the National Natural Science Foundation of China (Genral program

12071015).

Supplementary Materials

contains proofs of the theoretical results; a nonparametric procedure

to estimate the conditional mean and variance simultaneously; a description of the one-step

estimation method based on the data collected in the two phases and the pilot sample; and

additional simulation results.

Supplementary materials are available for download.