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The bootstrap is a popular and convenient method for quantifying theauthority of an empirical ordering of attributes, for example of a ranking ofthe performance of institutions or of the influence of genes on a responsevariable. In the first of these examples, the number, $p$, of quantities beingordered is sometimes only moderate in size; in the second it can be very large,often much greater than sample size. However, we show that in both types ofproblem the conventional bootstrap can produce inconsistency. Moreover, thestandard $n$-out-of-$n$ bootstrap estimator of the distribution of an empiricalrank may not converge in the usual sense; the estimator may converge indistribution, but not in probability. Nevertheless, in many cases the bootstrapcorrectly identifies the support of the asymptotic distribution of ranks. Insome contemporary problems, bootstrap prediction intervals for ranks areparticularly long, and in this context, we also quantify the accuracy ofbootstrap methods, showing that the standard bootstrap gets the order ofmagnitude of the interval right, but not the constant multiplier of intervallength. The $m$-out-of-$n$ bootstrap can improve performance and producestatistical consistency, but it requires empirical choice of $m$; we suggest atuning solution to this problem. We show that in genomic examples, where itmight be expected that the standard, ``synchronous'' bootstrap willsuccessfully accommodate nonindependence of vector components, that approachcan produce misleading results. An ``independent component'' bootstrap canovercome these difficulties, even in cases where components are not strictlyindependent.

Using the bootstrap to quantify the authority of an empirical ranking