{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Independent Data Bootstrap

Independent Data Bootstrap - Bootstrap Methods in...

This preview shows pages 1–2. Sign up to view the full content.

As Hall (1995) notes, the idea behind the bootstrap has a long history. If we wish to estimate a functional of a distribution function F , such as a population mean = Z xdF ( x ) ; we may use the same functional of the empirical distribution function 1 F n X n = Z xdF n ( x ) : The estimator so constructed is a bootstrap estimator. Yet not all functionals can be estimated in this way. For example, a probability density cannot be so estimated, as the empirical distribution function is pointwise discontinuous. In 1979, Efron expanded the power of the bootstrap by showing that for many estimation problems in which the bootstrap estimator is di¢ cult to com- pute, the estimator could be approximated by monte carlo resampling. In essence, resamples of size n are drawn repeatedly from the original sample, the estimate is calculated for each resample, and the bootstrap estimator is ap- proximated by taking an average of the appropriate function of the resample estimates. The accuracy of the resampling approximation increases with the number of resamples. Such a procedure provides an accurate approximation for bootstrap estimators, such as X n , which are expressed as an expectation condi- tional on the sample (or, equilivalently, as an integral with respect to F n ( x ) ). The power of resampling to provide a rich distributional approximation holds even for small sample sizes. Consider the distribution of the sample mean with a sample size of n = 7 . Because the number of possible distinct resamples is n n and the number of possible distinct resample estimators of the sample mean is 2 n 1 n 1 ± , there are 1700 di/erent possible values for the sample mean constructed from a resample if n = 7 . In fact, for any smooth statistic that does not depend on order, the bootstrap distribution of the statistic is roughly continuous even for small sample sizes. However, for quantiles (such as the median) things are not quite so rosy. If n is odd, there are only n distinct values for the sample median constructed from a resample. Because the term bootstrap estimator has become synonomous with the ap- proximate bootstrap estimator constructed from resampling, in what follows we refer to the resampling approximation as the bootstrap estimator. We focus on applying the bootstrap to statistics whose asymptotic distributions are indepen- dent of unknown population parameters. A statistic whose asymptotic distribu- tion is independent of unknown population parameters is called asymptotically pivotal. Simple bootstrap procedures provide improved approximations to the distributions of asymptotically pivotal statistics but not to the distributions 1 The empirical distribution function is the probability measure that assigns to a set a measure equal to the proportion of sample values that lie in the set. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 10

Independent Data Bootstrap - Bootstrap Methods in...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online