Dou-Pollard-Zhou.pdf =
Functional regression for general exponential families
(with Wei Dou and Harry Zhou)
The paper derives a minimax lower bound for rates of convergence for
an infinite-dimensional parameter in an exponential family model. An
estimator
that achieves the optimal rate is constructed by maximum likelihood
on finite-dimensional approximations with parameter dimension that
grows
with sample size.
Concentration.14jan07.pdf = A note on Talagrand's convex hull
concentration inequality
The paper reexamines an argument by Talagrand
that leads to a remarkable exponential tail bound for the concentration
of probability near a set. The main novelty is the
replacement of a mysterious calculus inequality by an application
of Jensen's inequality.
Tusnady3.pdf = Tusnady's inequality revisited (with
Andrew Carter)
Tusnady's inequality is the key ingredient in the KMT/Hungarian
coupling of the empirical distribution function with a Brownian
Bridge. We
present an elementary proof of a result that sharpens the Tusnady
inequality,
modulo constants. Our method uses the beta integral representation of
Binomial
tails, simple Taylor expansion, and some novel bounds for the ratios
of
normal tail probabilities.
thoughts.pdf = Some thoughts on Le Cam's statistical decision theory
The paper contains some musings about the abstractions
introduced by Lucien
Le Cam into the asymptotic theory of statistical inference and
decision theory. A short, selfcontained
proof of a key result (existence of randomizations via convergence in
distribution
of likelihood ratios), and an outline of a proof of a local asymptotic
minimax theorem, are
presented as an illustration of how Le Cam's approach leads to conceptual simplifications of
asymptotic theory.
pollardBDI.1july02.pdf = Maximal inequalities via bracketing with
adaptive truncation
Abstract. The paper provides a recursive interpretation for the
technique
known as bracketing with adaptive truncation. By way of illustration,
a simple
bound is derived for the expected value of the supremum of an
empirical
process, thereby leading to a simpler derivation of a functional
central limit
limit due to Ossiander. The recursive method is also abstracted into a
framework
that consists of only a small number of assumptions about processes
and functionals indexed by sets of functions. In particular, the
details of the
underlying probability model are condensed into a single inequality
involving
finite sets of functions. A functional central limit theorem of
Doukhan, Massart
and Rio, for empirical processes defined by absolutely regular
sequences,
motivates the generalization.
convex.pdf = Asymptotics for minimisers of convex
processes (with Nils Lid Hjort)
By means of two simple convexity arguments we are able to
develop a general
method for proving consistency and asymptotic normality of estimators
that are
defined by minimisation of convex criterion functions. This method is
then applied to a
fair range of different statistical estimation problems, including Cox
regression, logistic
and Poisson regression, least absolute deviation regression outside
model conditions, and
pseudo-likelihood estimation for Markov chains.
Our paper has two aims. The first is to exposit the method itself,
which in many
cases, under reasonable regularity conditions, leads to new proofs
that are simpler than
the traditional proofs. Our second aim is to exploit the method to its
limits for logistic
regression and Cox regression, where we seek asymptotic results under
as weak regularity
conditions as possible. For Cox regression in particular we are able
to weaken previously
published regularity conditions substantially.
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