Jump to content

User:Allais.andrea/Bootstrap BCa confidence intervals

From Wikipedia, the free encyclopedia

Bootstrap BCa confidence intervals are confidence intervals based on resampling. They can be applied to a wide class parametric and nonparametric inference problems, with minimal adaptation effort. They were first introduced by Bradley Efron in 1987[1], and were later proven to be second order accurate and second order correct. The acronym BCa stands for "bias corrected and accelerated".

Construction - Parametric model

[edit]

The confidence intervals are constructed from a sample of n i.i.d. observations drawn from a distribution , which is completely specified by an unknown vector of parameters η. The parameter for which confidence intervals are to be established is some function .

An estimate of the parameters is obtained from the observed data , for example using maximum likelihood estimation. This estimate also yields an estimate for the parameter θ.

A Monte Carlo method is used to generate a number B of synthetic samples , also of size n, from the distribution , i.e. with the parameters η set to their estimated value. Typically, . The same process used to estimate θ from the observed sample X is repeated on each synthetic sample , yielding B bootstrap replicates . Thus the distribution of replicates is: where it is important to stress that observed sample X is fixed, and the random variable is the synthetic sample .

References

[edit]
  1. ^ Efron, Bradley (1987). "Better bootstrap confidence intervals". Journal of the American Statistical Association. 82 (397): 171–185. doi:10.1080/01621459.1987.10478410.