The bootstrap is a versatile method for quantifying statistical uncertainty. Among its variants, a popular approach, the studentized bootstrap, provably achieves higher-order coverage error reduction compared to other benchmarks. However, its implementation typically requires an analytical form of the standard error, or otherwise an additional layer of resampling effort which can be computationally expensive. In this paper, we introduce what we call the studentized cheap bootstrap that achieves the same higher-order coverage accuracy as the conventional studentization, but substantially thinning the computational effort in the additional resampling layer to only very few Monte Carlo replications. Intriguingly, while conventional wisdom views "studentization" as an informal link between the bootstrap and t-distribution, we provide a first recognition that this link is in fact formal, notably with a distinct insight that the degree of freedom in the t-distribution corresponds to the Monte Carlo computation effort in the additional resampling layer, rather than the data size as in traditional thinking. Moreover, our desirable higher-order coverage accuracy builds crucially on this insight, as well as explicit calculations and geometric analyses of higher-order terms in the Edgeworth and Cornish-Fisher expansions tailored to limiting t-distributions.

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