The greater part of contemporary nonparametric inference employs
methods that are linear in the data.  The exceptions to this rule
generally involve estimators with empirically chosen tuning
parameters; examples of those parameters include the bandwidth in
kernel-type estimation, and the threshold in wavelet methods.
Nevertheless, the estimator is still ``intrinsically'' linear, not
least because its first-order theoretical properties are equivalent
to those of a linear estimator.

    This linearity is largely a reflection of the fact that no
constraints are imposed.  Indeed, in the context of unconstrained
inference there exist several results (for example, those of
Jianqing Fan in the case of local linear regression) which show
that often, no substantial improvement in mathematical performance
can be gained by using other than linear methods.

    However, if qualitative order-type constraints are to be imposed
on a curve estimator (for example, constraints of monotonicity or
unimodality of a regression estimator or a density estimator) then
it will often be necessary for estimators to depart significantly
from linearity, at least in places where a linear estimator would
violate the constraint.  In such instances, first-order asymptotic
properties of the estimator may be altered.  We shall discuss ways
of achieving qualitative constraints.

    Other, quantitative constraints are sometimes of interest.  They
include constraints that reduce the size of bias or variance.  In
the former case, the mean squared error of nonparametric curve
estimators can be reduced by an order of magnitude, and in the
latter, relatively robust estimators in a variety of different
settings may be obtained.  Quantitative constraints are often also
important when using nonparametric empirical methods (for example,
the bootstrap) to test a hypothesis.  In particular, when the
bootstrap is employed to test the hypothesis that a sampled
distribution has zero skewness, against the complementary
alternative, we usually wish to resample in such a way that the
empirical distribution reflects the null hypothesis.  This
requires a quantitative constraint to be imposed on the empirical
distribution.

    The talk will address a variety of problems of this type.

    Time series describing the intensity of radiation from stars can be
used to classify the stars into types, particularly if the radiation
is periodic or can be expressed as the convolution of a small number
of periodic functions.  Signals of the latter type are conveniently
referred to as ``multiperiodic functions.''  Classification can
involve accessing the individual periodic components, which generally
correspond to different sources of radiation and have intrinsic
physical meaning.  Therefore they need to be ``deconvolved'' from the
mixture.  We shall discuss a combination of kernel and orthogonal
series methods for performing the deconvolution, and show how to
estimate both the sequence of periods and the periodic functions
themselves.  Particular attention will be paid to the issue of
identifiability, in a nonparametric sense, of the components.  This
aspect of the problem exhibits unusual features, and has connections
to number theory, as too do convergence rates of estimators.

 
    Techniques for digital signal analysis involve sampling data
repeatedly in time.  Recent research suggests that the sampling
rate should vary temporally, and should depend on empirical
assessment of signal complexity.  A signal of given duration,
sampled the same total number of times, can potentially be
recovered at higher fidelity if the sampling rate is varied so
that relatively more data are sampled when the signal is more
complex.  In some, perhaps most, applications this requires
real-time analysis of signal complexity, as well as algorithms
for rapidly switching from one sampling rate to another.  It has
been suggested that procedures of this type be used for both
data transmission and storage.  Properties of these procedures
will be discussed, in the contexts of wavelet-based as well as
more conventional denoising techniques.