Van Dantzig Seminar

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Van Dantzig Seminar: April 11, 2014

Programme: (click names or scroll down for titles and abstracts)

14:00 - 14:05 Opening
14:05 - 15:05 Johan Segers (Université catholique de Louvain)
15:05 - 15:25 Break
15:25 - 15:55 Richard Gill (Mathematical Institute, Leiden)
15:55 - 16:25 Tina Nane (Centre for Science and Technology Studies, Leiden)
16:30 - 17:30 Reception
Location: Mathematical Institute, Leiden University, Snellius Building, Room 312 (Directions)

Titles and abstracts

  • 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.

    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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Supported by

BTK, Amsterdam 2014