Strengths and Limitations of Some Miscellaneous Statistical Procedures - Common Errors in Statistics - page 79

One of the first proposed uses of the bootstrap, illustrated in Chapter 4,
was in providing an interval estimate for the sample median. Because the
median or 50th percentile is in the center of the sample, virtually every
element of the sample contributes to its determination. As we move out
into the tails of a distribution, to determine the 20th percentile or the
90th, fewer and fewer elements of the sample are of assistance in making
the estimate.
For a given size sample, bootstrap estimates of percentiles in the tails
will always be less accurate than estimates of more centrally located per-
centiles. Similarly, bootstrap interval estimates for the variance of a distrib-
ution will always be less accurate than estimates of central location such as
the mean or median because the variance depends strongly upon extreme
values in the population.
One proposed remedy is the tilted bootstrap 3 in which instead of
sampling each element of the original sample with equal probability, we
weight the probabilities of selection so as to favor or discourage the
selection of extreme values.
If we know something about the population distribution in
advance—for example, if we know that the distribution is symmetric,
or that it is chi-square with six degrees of freedom—then we may be
able to take advantage of a parametric or semiparametric bootstrap as
described in Chapter 4. Recognize that in doing so, you run the risk
of introducing error through an inappropriate choice of parametric
framework.
Problems due to the discreteness of the bootstrap statistic are usually
evident from plots of bootstrap output. They can be addressed using a
smooth bootstrap as described in Davison and Hinkley [1997, Section
3.4].
BAYESIAN METHODOLOGY
Since being communicated to the Royal Society in 1763, 4 Bayes' Theorem
has exerted a near fatal attraction on those exposed to it. 5 Much as a bell
placed on the cat would magically resolve so many of the problems of the
average house mouse, Bayes' straightforward, easily grasped mathematical
formula would appear to provide the long-awaited basis for a robotic
judge free of human prejudice.
On the plus side, Bayes' Theorem offers three main advantages:
3 See, for example, Hinkley and Shi [1989] and Phipps [1997].
4 Philos. Tran . 1763; 53:376-398. Reproduced in Biometrika 1958; 45: 293-315.
5 The interested reader is directed to Keynes [1921] and Redmayne [1998] for some
accounts.