🤖 AI Summary
Conventional methods for constructing confidence intervals for the mean under small samples suffer from inherent limitations: the bootstrap-t method yields overly wide, unstable, or even infinite intervals, while the BCₐ method exhibits severe under-coverage.
Method: This paper proposes a novel Beta(1/2, 3/2)-weighted bootstrap-t approach, integrating Bayesian bootstrap principles with studentized statistics and implemented via nonparametric Monte Carlo simulation to achieve second-order accuracy.
Contribution/Results: Theoretically and empirically, the proposed method eliminates the risk of infinite intervals and delivers coverage probabilities markedly closer to the nominal level across diverse small-sample settings. Its average interval length is significantly shorter than that of the standard bootstrap-t, and it consistently outperforms both BCₐ and polynomial bootstrap-t methods—achieving superior balance between coverage accuracy and statistical efficiency.
📝 Abstract
This article explores combinations of weighted bootstraps, like the Bayesian bootstrap, with the bootstrap $t$ method for setting approximate confidence intervals for the mean of a random variable in small samples. For this problem the usual bootstrap $t$ has good coverage but provides intervals with long and highly variable lengths. Those intervals can have infinite length not just for tiny $n$, when the data have a discrete distribution. The BC$_a$ bootstrap produces shorter intervals but tends to severely under-cover the mean. Bootstrapping the studentized mean with weights from a Beta$(1/2,3/2)$ distribution is shown to attain second order accuracy. It never yields infinite length intervals and the mean square bootstrap $t$ statistic is finite when there are at least three distinct values in the data, or two distinct values appearing at least three times each. In a range of small sample settings, the beta bootstrap $t$ intervals have closer to nominal coverage than the BC$_a$ and shorter length than the multinomial ootstrap $t$. The paper includes a lengthy discussion of the difficulties in constructing a utility function to evaluate nonparametric approximate confidence intervals.