The Rao-Wu Rescaling Bootstrap: From theory to practice

At Statistics Canada, variance estimation for complex surveys is mainly carried out using replication methods. The two replication methods which have been used in the last decade are the delete-one Primary Sampling Unit (PSU) jackknife and, more recently, the bootstrap. As Valliant (2007) rightly points out there are several variants of the bootstrap introduced by Efron (1979) in the i.i.d. case which are being used in survey sampling and we have to be clear which one is being referred to; at Statistics Canada we use solely the Rao-Wu rescaling bootstrap for production. Even though it was introduced in 1988 (see Rao and Wu (1988)), first implementations occurred only in the late 1990s. The Rao-Wu bootstrap is performed on both with and without replacement designs for which it yields sensible variance estimates for a variety of estimators including percentiles, a claim not matched by the delete-one PSU jackknife. Because of this, surveys which initially relied on the jackknife often make the switch to the bootstrap when the occasion arises, like during a survey re-design. The reader can get a good overview of what has been tried in survey sampling with regard to the bootstrap from Rust and Rao (1996), Shao (2003) and Lahiri (2003). The RaoWu variant of the original bootstrap procedure is appealing to survey users because it is simple to implement, it yields adequate variance estimates when the sample sizes are small (which is common with stratified designs) and it comes in the form of bootstrap weights.