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, 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
The confidence intervals are constructed from a sample $$X = (x_1,\,x_2,\,\ldots x_n)$$ of $n$ i.i.d. observations drawn from a distribution $$F_\eta$$, which is completely specified by an unknown vector of parameters $&eta;$. The parameter for which confidence intervals are to be established is some function $$\theta(\eta)$$.

An estimate $$\hat{\eta}(X)$$ of the parameters is obtained from the observed data $$X$$, for example using maximum likelihood estimation. This estimate also yields an estimate $$\hat\theta(X) = \theta(\hat\eta(X))$$ for the parameter $&theta;$.

A Monte Carlo method is used to generate a number $B$ of synthetic samples $$X^\star$$, also of size $n$, from the distribution $$F_{\hat{\eta}(X)}$$, i.e. with the parameters $&eta;$ set to their estimated value. Typically, $$B \sim 2000$$. The same process used to estimate $&theta;$ from the observed sample $X$ is repeated on each synthetic sample $$X^\star$$, yielding $B$ bootstrap replicates $$\hat{\theta}(X^\star)$$. Thus the distribution of replicates is: $$ \hat{G}(t) = \mathrm{Prob}\left(\hat{\theta}(X^\star) < t\right)\,, \quad X^\star = (x^\star_1,\,x^\star_2,\,\ldots x^\star_n)\,, \quad x_i^\star\sim \hat{F}_{\hat{\eta}(X)} \quad \mathrm{i.i.d.}\,, $$ where it is important to stress that observed sample $X$ is fixed, and the random variable is the synthetic sample $$X^\star$$.