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

For a bootstrap-t confidence interval, estimate the original standard error $\hat{s}$ of $\hat{r}$, for example by the standard deviation of the outer paired bootstrap correlations. For each outer resample $b$, estimate its own standard error $s_b^{\ast }$ by drawing many inner paired resamples from that outer sample and taking the standard deviation of their correlations. The resulting studentized statistics areLet $t_p^{\ast }$ be their empirical quantiles. Approximating the distribution of $(\hat{r}-r)/\hat{s}$ by that of $T^{\ast }$ and solving the two inequalities for $r$ givesThe reversal of quantiles is essential. The pivot requires positive standard errors and nondegenerate correlations; its claimed 95% coverage is an approximation under bootstrap regularity, not an exact finite-sample guarantee. One may intersect the final interval with the parameter space $[-1,1]$.

The same percentile and studentized procedures can be applied to the sample mean of $X$, and then only the $X$ observations need be resampled. They are useful when a normal approximation is doubtful. However the mean already has the simple standard-error estimate $S_X/10$ here. For normal observations, the classical exact interval isand the same form is an asymptotic approximation for independent finite-variance observations under appropriate large-sample conditions. Thus bootstrap intervals are valid alternatives, but are often unnecessary for a mean with a reliable classical pivot; the correlation has no corresponding distribution-free elementary pivot. An infinite-variance distribution would require other theory rather than this ordinary standard-error argument.