Conditional bootstrap methods in the mean-shift model

SUMMARY Bootstrap methods are not inherently conditional, but they can be made so by appropri- ate stratification of the simulated samples which bootstrap produces. We show how stratification can work in a bootstrap analysis of mean-shift in Nile river flow data. The results are compared with both parametric and semiparametric likelihood analyses. The paper ends with some general remarks on conditional bootstraps. Bootstrap methods (Efron, 1982) are simulation methods for assessing frequency properties of statistical analyses, the simulation model being that which is fitted to the actual data. Often the simulation involves Monte Carlo, rather than theoretical, evaluation. Unfortunately the bootstrap as usually described is inherently unconditional; Rubin's (1981) discussion bears on this point. However, if the bootstrap simulation results can be properly stratified, then it is possible to estimate relevant conditional probabilities. Stratification can sometimes be effected in the model itself, as when fixing the values of sample sizes or explanatory variables. Otherwise one must run the unconditional bootstrap and stratify its output. In the present paper we illustrate the latter situation in the analysis of a mean-shift model. Section 2 describes the application and its analysis by parametric methods. Sections 3 and 4 produce alternative bootstrap analyses, the first of which uses the stratification of an unconditional Monte Carlo simulation; the second bootstrap analysis involves estimation of a likelihood function. The final section contains some brief general remarks about conditional bootstraps.