The bootstrap-t algorithm is inspired by Student’s t-test, as its name indicates. It introduces a similar parameter T and calculates its percentiles, which can then be used to establish confidence intervals for the initial parameter of interest. With that said, since the T-percentiles are unknown, it is necessary to approximate them, at which point the bootstrapping begins taking place. Using an estimation of the overall distribution and multiple bootstrap data sets drawn from it, it is possible to produce T*, a replication of the original parameter (DiCiccio & Efron, 1996). With enough different data sets produced and analyzed, with their results then ordered, it is possible to assign each percentile a value from the resulting set. The αth percentile is assigned the (B times α)th value in a set of B ordered results. With these results, one can establish a confidence interval for the data that has been found to be second-order accurate (DiCiccio & Efron, 1996). With that said, the algorithm has several weaknesses, notably the high computational intensity and the numerical instability, which can produce extremely large confidence intervals.
The BCa method also relies on bootstrap data sets that are sampled from the data. Generally, the number of such structures necessary for accurate estimations varies from several hundred (for standard error) to multiple thousands or more (for confidence intervals) (DiCiccio & Efron, 1996). Replications of parameters of interest are procured from each of these and used to estimate the confidence intervals. They are defined by a complicated formula that features two parameters that give the method its name: the bias-correction and the acceleration. The former is estimated using the entirety of the sample compared to the parameter’s estimated value, aiming to correct upward or downward biases in the sample. The latter aims to measure the speed of changes in the standard error on a normalized scale and can be convoluted to define, though, per DiCiccio and Efron (1996), it can be estimated using Fisher’s score function. BCa is also second-order accurate and correct under general circumstances, also being transformation invariant and exactly correct under the normal transformation model. However, it is highly complex and can be overly conservative and close to the non-bootstrap confidence intervals.
The final method discussed, ABC, stands for “approximate bootstrap confidence [intervals].” According to DiCiccio and Efron (1996), it is a middle ground between the two approaches discussed above, abandoning BCa’s bootstrap cumulative distribution function and introducing a nonlinearity parameter. ABC involves mapping from the natural parameter vector η to the expectation parameter µ using a predetermined function. µ can then be used to compute the standard deviation estimate, followed by the calculation of a number of numerical second derivatives to calculate the method’s three constants. The method diverges into several variations at this point, with the simplest, ABCquadratic, calculating the endpoint as a direct function of the inputs (DiCiccio & Efron, 1996). The other versions of the algorithm require slightly more effort, addressing issues such as nonlocality-related boundary violations. A significant advantage of ABC over its two counterparts is that it requires one-hundredth of the computation, but it also less automatic and demands smoothness properties for the parameter of interest (Efron & Hastie, 2016). Still, in simpler cases, using the method can confer dramatic advantages, and it is frequently used where the standard interval may have sufficed.
References
DiCiccio, T. J., & Efron, B. (1996). Bootstrap confidence intervals. Statistical Science, 11(3), 189-228.
Efron, B., & Hastie, T. (2016). Computer age statistical inference: Algorithms, evidence, and data science. Cambridge University Press.