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Johan Segers
When uniform weak convergence fails: empirical processes for dependence functions and residuals via epi- and hypographs
In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing
the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of
bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those
spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical
processes in linear regression models in case the underlying distributions lack a certain degree of smoothness.
To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak
convergence theory is developed. Convergence with respect to the new metric is related to epi- and
hypoconvergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent
to locally uniform convergence, whereas under mild side conditions, it implies Lp convergence. For the
examples mentioned above, weak convergence with respect to the new metric is established in situations where
it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties
of resampling procedures and goodness-of-fit tests.
Joint work with Axel Bücher and Stanislav Volgushev (Ruhr-Universität Bochum)
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Richard Gill
Estimating a probability mass function with unknown labels
In the context of a species sampling problem we discuss a non-parametric maximum likelihood estimator
for the underlying probability mass function. The estimator is known in the computer science literature
as the high profile estimator. We prove strong consistency and obtain rates of convergence.
We also study a sieved estimator for which similar consistency results are derived. Numerical
computation of the sieved estimator is of interest for practical problems, such as forensic
DNA analysis, and we present a computational algorithm based on combining stochastic
approximation, Metropolis-Hastings, and EM.
http://arxiv.org/abs/1312.1200
Joint work with Dragi Anevski (Lund) and Stefan Zohren (Rio de Janeiro)
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Tina Nane
Shape constrained nonparametric estimation in the Cox model
Within survival analysis, Cox proportional hazards model is one of the most
acknowledged approaches to model right-censored time to event data in
the presence of covariates. Different functionals of the lifetime distribution are
commonly investigated. The hazard function is of particular interest, as it
represents an important feature of the time course of a process under study,
e.g., death or the onset or relapse of a certain disease. Even though the baseline
hazard can be left completely unspecified, in practice, it is often reasonable
to assume a qualitative shape. This can be done by assuming the baseline
hazard to be monotone, for example, as suggested by Cox himself. Various
studies have indicated that a monotonicity constraint should be imposed
on the baseline hazard, which complies with the medical expertise.
The main objective is therefore to derive nonparametric baseline hazard
and baseline density estimators under monotonicity constraints and
investigate their asymptotic behavior. We consider the nonparametric
maximum likelihood estimator of a nondecreasing baseline hazard and
we propose a Grenander-type estimator, defined as the left-hand slope
of the greatest convex minorant of the Breslow estimator. The two
estimators are then shown to be strongly consistent and asymptotically
equivalent. Moreover, we derive their common limit distribution at a fixed
point. The two equivalent estimators of a nonincreasing baseline hazard
and their asymptotic properties are acquired similarly. Furthermore, we
introduce a Grenander-type estimator of a nonincreasing baseline density,
defined as the left-hand slope of the least concave majorant of an estimator
of the baseline cumulative distribution function derived from the Breslow
estimator. This estimator is proven to be strongly consistent and its
asymptotic distribution at a fixed point is derived.
Based on join work with Rik Lopuhaä
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