We first discuss the construction of bootstrap confidence bands for the distribution function F of a population for simple random sampling but do not assume that F is continuous. The known alternative approach is to use the quantiles from the tabled Kolmogorov distribution. This approach is known to be conservative. It has already been shown in Bickel and Freedman (1981) that the bootstrap confidence band has the correct coverage probability asymptotically. We show by simulation that the bootstrap works well for small samples and outperforms the conservative approach, particularly for distributions that have small carriers. We also investigate the analogous problem of finding bootstrap confidence bands for the distribution function F of a population for a more complicated situation, stratified random sampling. The conservative approach in the previous situation is extended to this case when sampling is with replacement (we expect that it holds for sampling without replacement) and the supports of...

Confidence Bands for a Distribution Function Using the Bootstrap

Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

In a certain sense sample heterogeneity can only decrease the variance of a linear function of order statistics compared with that for a homogeneous sample, and thus the variance of those statistics will tend to be overestimated. The magnitude of this effect for robust estimators of location and scale is described, with particular reference to contamination models. It is shown that with asymmetric contamination, extreme observations may have much less influence on these estimators than is generally believed, if trimming is employed.

The Effect of Sample Heterogeneity on Linear Functions of Order Statistics, with Applications to Robust Estimation

Residual median for non-identically distributed random variables

/pdf/weak-convergence-of-the-empiric-process-for-independent-15924u687x.pdf

R. Rechtschaffen

Papers

Weak Convergence of the Empiric Process for Independent Random Variables