Bootstrap-t Confidence Interval

We can calculate the student-t confidence interval using bootstrap. First, B bootstrap samples are generated and then for each sample we compute.
\[z^b = \frac{{{\hat \theta }^b} - \hat \theta}{\hat{SE^b}}\]
The $\hat\theta$ is the statistics we need to know and the $\hat\theta^b$ is the bootstrap replicate of it. The denominator $\hat{SE^b}$ is often calulated using bootstrap of the replicate (bootstrap of bootstrap) if the formular for standard error of $\hat\theta^b$ is not known.

Once we have the $z^b$, find the quantile $\alpha/2$-th and   $(1-\alpha/2)$-th needed for the endpoints of the interval.
 \[\alpha /2 = \frac{{\sharp ({z^b} \le {{\hat t}^{\alpha /2}})}}{B}\]
and get the interval

\[(\hat \theta - {z^{(1 - \alpha /2)}}S{E_{\hat \theta }},\hat \theta - {z^{(\alpha /2)}}S{E_{\hat \theta }})\]
The following code demonstrates the Bootstrap-t confidence interval of the skewness of the forearm data.

load forearm;
theta = skewness(forearm);
n=length(forearm);
B=1000;
[bootstat,bootsam]=bootstrp(B,'skewness',forearm);
sehats=zeros(size(bootstat));
for i=1:B
    x = forearm(bootsam(:,i));
    sehats(i)=std(bootstrp(25,'skewness',x));
end
zvals = (bootstat - theta)./sehats;

SE=std(bootstat);
alpha=0.05;
k=B*alpha/2;
szvals=sort(zvals);
tlo = szvals(k);
thi = szvals(B-k);
blo=theta - thi*SE;
bhi=theta - tlo*SE;
[blo,bhi]

ans =

   -0.3829    0.1730 

It is recommended that the number of bootstrap replicates, B, should be 1,000 or more.