Particularly in genomics, but also in other fields, it has become commonplace
to undertake highly multiple Student's $t$-tests based on relatively small
sample sizes. The literature on this topic is continually expanding, but the
main approaches used to control the family-wise error rate and false discovery
rate are still based on the assumption that the tests are independent.

We propose a method for incorporating variable selection into local
polynomial regression. This can improve the accuracy of the regression by
extending the bandwidth in directions corresponding to those variables judged
to be are unimportant. It also increases our understanding of the dataset by
highlighting areas where these variables are redundant. The approach has the
potential to effect complete variable removal as well as perform partial
removal when a variable redundancy applies only to particular regions of the
data.

The notion of probability density for a random function is not as
straightforward as in finite-dimensional cases. While a probability density
function generally does not exist for functional data, we show that it is
possible to develop the notion of density when functional data are considered
in the space determined by the eigenfunctions of principal component analysis.
This leads to a transparent and meaningful surrogate for density defined in
terms of the average value of the logarithms of the densities of the
distributions of principal components for a given dimension.

Student's $t$ statistic is finding applications today that were never
envisaged when it was introduced more than a century ago. Many of these
applications rely on properties, for example robustness against heavy tailed
sampling distributions, that were not explicitly considered until relatively
recently. In this paper we explore these features of the $t$ statistic in the
context of its application to very high dimensional problems, including feature
selection and ranking, highly multiple hypothesis testing, and sparse, high
dimensional signal detection.

The bootstrap is a popular and convenient method for quantifying the
authority of an empirical ordering of attributes, for example of a ranking of
the performance of institutions or of the influence of genes on a response
variable. In the first of these examples, the number, $p$, of quantities being
ordered is sometimes only moderate in size; in the second it can be very large,
often much greater than sample size. However, we show that in both types of
problem the conventional bootstrap can produce inconsistency.

Increasing practical interest has been shown in regression problems where the
errors, or disturbances, are centred in a way that reflects particular
characteristics of the mechanism that generated the data. In economics this
occurs in problems involving data on markets, productivity and auctions, where
it can be natural to centre at an end-point of the error distribution rather
than at the distribution's mean.