Subsampling estimates a statistic's sampling uncertainty from smaller subsets drawn without replacement. The m-out-of-n bootstrap instead draws observations with replacement from the original observations. Both recompute the statistic on smaller samples and use an appropriate scale adjustment. They can support inference for some nonregular estimators where the ordinary bootstrap fails—an issue for a quant whose error bars depend on an extreme or a constrained estimate. Smaller samples alone do not establish validity.
The sampling rule changes the experiment
For independent, identically distributed observations, the basic distinction is:
| Method | Size of each replicate | Replacement within a replicate |
|---|---|---|
| Ordinary nonparametric bootstrap | Yes | |
| Subsampling | No | |
| m-out-of-n bootstrap | Yes |
Subsampling can enumerate all subsets of size , or approximate that collection with uniformly selected subsets. Different subsets may overlap: “without replacement” applies within a subset. The m-out-of-n bootstrap draws independently from the empirical distribution, so repeated original rows are possible. The replicate count controls numerical approximation; controls the statistical procedure. Bickel, Götze and van Zwet, §3.
Match the root before comparing errors
A common construction recomputes the estimator as on replicate and collects
This centres at the full-sample estimate, which replaces the unknown parameter. Its justification requires that this substitution be negligible on the smaller-sample scale; a typical condition is , together with an appropriate limit theorem. Studentized roots require their own denominator and justification. applies to the root- example below; it is not a universal resampling rate. Romano and Shaikh, §2.1.
Consider a hypothetical scaling example: , full-sample estimate , and one replicate of size giving . Assume the relevant rate is . Then
The corresponding offset on the full-sample estimator's scale is
The raw difference describes the smaller sample. Treating it directly as a full-sample error misses the scale conversion. These numbers describe one hypothetical replicate, not an estimated quantile, standard error or confidence interval. The chosen is an arithmetic illustration, not a recommended sample size.
Normalise at the replicate size, then translate to the full-sample scale. No distribution or interval is estimated from this single replicate.
If a valid collection of roots supplies quantiles and , inversion gives the approximate root-based confidence interval
The upper root quantile enters the lower endpoint. This follows by solving for . Appropriate quantile continuity and distributional approximation conditions are needed; raw percentiles of the smaller-sample estimates do not perform this inversion.
Where the justification can break
A common asymptotic regime is with . It lets smaller-sample statistics approach their limiting law while reducing the effect of estimating the original distribution. It is not a universal guarantee for either method: the statistic, centering, rate, limiting law and data assumptions still matter. With-replacement resampling can introduce ties that require additional conditions. When the ordinary bootstrap already works, smaller resamples may sacrifice accuracy. Bickel, Götze and van Zwet, §§1–3.
Nor does “nonsmooth” automatically mean bootstrap failure. The same paper contrasts a median under positive-density conditions with nonregular failures. Even pointwise asymptotic validity need not give reliable coverage uniformly over nearby data-generating laws; Romano and Shaikh make that distinction explicit.
For serially dependent returns, arbitrary row subsets destroy temporal structure. A time-series subsampling method may instead compute the statistic on contiguous blocks of length , with scaling and dependence conditions justified for that statistic. This differs from a block bootstrap that assembles a longer replicate from multiple resampled blocks. Lahiri, §2.8.
Simulation and Resampling develops the ordinary bootstrap workflow and separates simulation error from sampling uncertainty. Its public page is a preview; the full lesson requires membership.
Sources
- Bickel, Götze and van Zwet (1997), Resampling Fewer Than n Observations: Gains, Losses, and Remedies for Losses, §§1–3, especially §2.2 and Theorems 1–2: success and failure examples, the two smaller-sample constructions and their differing conditions.
- Romano and Shaikh (2012), On the Uniform Asymptotic Validity of Subsampling and the Bootstrap, introduction and §2.1: roots, feasible centering, quantiles and uniform coverage requirements.
- Lahiri (2003), Resampling Methods for Dependent Data, chapter excerpts, §2.8: contiguous-block subsampling and scale corrections. The numerical illustration and figure here are original.