Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 47 / 4 / b / i / Solution

Calculate the observed sample correlation coefficient $\hat{r}$. In each paired bootstrap, draw 100 indices independently with replacement from $1,\dots ,100$ and select the corresponding whole pairs. Compute the bootstrap correlation $r_b^{\ast }$ from each resample; do not resample the two margins independently, since that would destroy their empirical dependence.

Let $q_p$ be the empirical $p$-quantile of those bootstrap correlations. The percentile bootstrap confidence interval isThis is the requested algorithmic interval; the actual endpoints require the observed pairs. Its coverage is approximate, based on the sampling law of the correlation being well approximated by the bootstrap law. Both sample variances must be positive. Resamples having a constant margin have undefined correlation and cannot be repaired by arbitrarily assigning correlation zero. One may regenerate these exceptional resamples, explicitly conditioning on well-defined correlations; under the usual nondegenerate large-sample conditions their probability is negligible. Frequent degeneracy instead indicates that this ordinary correlation-bootstrap construction is unsuitable.