User:Allais.andrea/Bootstrap BCa confidence intervals

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

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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

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