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Data collection costs can vary widely across variables in data science tasks.Two-phase designs are often employed to save data collection costs. Intwo-phase studies, inexpensive variables are collected for all subjects in thefirst phase, and expensive variables are measured for a subset of subjects inthe second phase based on a predetermined sampling rule. The estimationefficiency under two-phase designs relies heavily on the sampling rule.Existing literature primarily focuses on designing sampling rules forestimating a scalar parameter in some parametric models or some specificestimating problems. However, real-world scenarios are usually model-unknownand involve two-phase designs for model-free estimation of a scalar ormulti-dimensional parameter. This paper proposes a maximin criterion to designan optimal sampling rule based on semiparametric efficiency bounds. Theproposed method is model-free and applicable to general estimating problems.The resulting sampling rule can minimize the semiparametric efficiency boundwhen the parameter is scalar and improve the bound for every component when theparameter is multi-dimensional. Simulation studies demonstrate that theproposed designs reduce the variance of the resulting estimator in varioussettings. The implementation of the proposed design is illustrated in a realdata analysis.

A maximin optimal approach for model-free sampling designs in two-phase studies