Nonparametric Bootstrap and its Applications.  Vic Patrangenaru
Georgia State University

Suppose tn is an estimator of the parameter $\theta$ constructed from a random sample X1,...,Xn from a probability distribution P and $\sqrt{n}(t_n-\theta)$ converges in law to ${\mathcal{N}}(0,{\sigma}^2).$ If $\hat{\sigma}_n$ is a consistent estimator of $\sigma,$ a c.i. of nominal coverage $1-\alpha$ is given by $[t_n-z_{1-\alpha/2}\hat{\sigma}_nn^{-{1/2}},t_n+z_{1-\alpha/2}\hat{\sigma}_nn^{-{1/2}}];$ the coverage error , difference between nominal coverage and true probability coverage, may be significant especially when the sample size is not too large. One way to reduce the coverage error is to use Efron's nonparametric bootstrap, a procedure for estimating the distribution of $t_n-\theta$ or of $(t_n-\theta)/{\hat{\sigma}_n}$ based on X1,...,Xn, without making parametric assumptions on P. In the case P is continuous, R.N.Bhattacharya showed that the bootstrap estimate of $(t_n-\theta)/{\hat{\sigma}_n}$ reduces the coverage error of the normal approximation.
Nonparametric bootstrap is a key technique in statistical analysis of small size samples from huge dimensional vectors, such as in statistical image analysis. Some applications in medical imaging and pattern recognition will be presented.