Bias-Corrected and Accelerated Interval

This $BC_a$ interval is introduce by Efron (1987) and is said to be superior in coverage performance, range-preserving, etc and is an improvement over the percentile confidence interval.

The $BC_a$ interval adjusts the endpoint of $(1-\alpha) 100\%$ by two parameter, $\hat a$ and $\hat z_0$.

\[BC_a = (\hat \theta_B^{\alpha_1},\hat \theta_B^{\alpha_1})\]
where
\[\alpha_1=\Phi\left(\hat z_0 + \frac{\hat z_0+z^{\alpha/2}}{1-\hat a(\hat z_0+z^{\alpha/2}}\right)\]
\[\alpha_2=\Phi\left(\hat z_0 + \frac{\hat z_0+z^{(1-\alpha/2)}}{1-\hat a(\hat z_0+z^{(1-\alpha/2)}}\right)\]
The $\Phi$ denotes the standard normal cumulative distribution function. Therefore, for $\alpha/2=0.05$, we have $z^{\alpha/2}=z^{0.05}=-1.645$, because $\Phi(-1.645)=0.05$.

We can also see that when $\hat a=0$ and $\hat z_0=0$, this $BC_a$ is the same as the percentile interval.

\[\alpha_1=\Phi(z^{\alpha/2})=\alpha/2\]
The bias-correction term is $\hat z_0$, and is the proportion of bootstrap replicates $\hat\theta^b$ that are less than the statistics $\hat\theta$.
\[\hat z_0 = \Phi^{-1}\left(\frac{\sharp(\hat\theta^b<\hat\theta)}{B}\right)\]
The acceleration term $\hat a$ is
\[\hat a=\frac{-SKEW(\hat\theta^{(-i)})}{6\sqrt n}\]
where $\hat\theta^{(-i)}$ is the value of the statistics using the i-th jackknife sample. Please see the detailed definition here.

Luckily for us, Matlab already has the bootci() function and its default method is $BC_a$ !!!

load forearm;
theta = skewness(forearm);
n=length(forearm);
B=2000;
skew = @(x)(skewness(x));
ci=bootci(B,skew,forearm) 

ci =
   -0.3904
    0.1731