Bootstrap Methods: Another Look at the Jackknife

We discuss the following problem: given a random sample X = (X₁, X₂, , Xₙ) from an unknown probability distribution F, estimate the sampling distribution of some prespecified random variable R (X, F), on the basis of the observed data x. (Standard jackknife theory gives an approximate mean and variance in the case R (X, F) = (F) - (F), some parameter of interest. ) A general method, called the "bootstrap, " is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.