Prepivoting to Reduce Level Error of Confidence Sets

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function's sampling distribution. We call this function the root of the confidence set. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. When the root is not a pivot, in the sense of classical statistics, the actual level of the approximate confidence set may differ substantially from the intended level. Prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function. Prepivoting can be iterated. Bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root. The first prepivoting is nearly equivalent to studentizing, when that operation is appropriate. Further iterations of prepivoting make higher order corrections automatically.

Bootstrap Hypothesis Testing

“…This is the basic idea of the iterated bootstrap, of which a special case is the double bootstrap, proposed in Beran (1987Beran ( , 1988.…”

Bootstrapping the mode

“…In contrast, the bootstrap approach is automatic, and the asymptotic validity of this approach is the subject of this paper. As well as being applicable quite generally, the bootstrap is known to outperform conventional asymptotic approximations in some situations (see Beran (1987) for example). An interesting application (Silverman (1981)) of the bootstrap is testing whether a density is unimodal.…”

Analytical approximations for iterated bootstrap confidence intervals

“…This nomenclature is adopted merely for convenience, since d0 and Io may be made arbitrarily close to one another by choice of a sufficiently large B. Although the percentile method is simple to implement and generally yields intervals with fairly stable lengths and endpoints, it typically produces intervals with very poor coverage accuracy; see, for example, Beran (1987), Schenker (1987), Hall (1988), Hall el al. (1989), and Martin (1990a,b).…”

“…In complex problems, an accurate variance estimate may be very difficult to construct, especially in small samples. Hall et al (1989) suggest that iterated bootstrap techniques introduced by Hall (1986) and Beran (1987), and discussed by Beran (1988), Hall and Martin (1988), Martin (1990a,b) and Hinkley and Shi (1990), can be used in complex situations to construct confidence intervals that have both high coverage accuracy and stable lengths and endpoints. Described briefly, the idea behind the construction of iterated bootstrap confidence intervals is that the bootstrap can be used to estimate and correct for coverage error in bootstrap confidence intervals.…”

“…Although iterated bootstrap techniques offer significant advantages in terms of improved accuracy in a variety of statistical problems, they typically impose an exacting toll in terms of computer resources, being an order of magnitude more 'greedy' than their standard bootstrap counterparts. The iterated bootstrap was introduced in the confidence interval context by Hall (1986) and Beran (1987) and has been discussed by several authors, including Hall and Martin (1988), Beran (1988), Hinkley and Shi (1990), and Martin (1990a,b). Efron (1983) introduced 0960-3174/92 9 1992 Chapman & Hall an iterated bootstrap construction in a different context to that considered here.…”