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Piet Groeneboom

  1. Estimators for the interval censoring problem.

    Thu, 09/29/2011 - 15:01 — SomNambul
    Authors: Piet Groeneboom, Tom Ketelaars
    Subjects: Statistics
    Abstract

    We study three estimators for the interval censoring case 2 problem, a
    histogram-type estimator, proposed in Birg\'e (1999), the maximum likelihood
    estimator (MLE) and the smoothed MLE, using a smoothing kernel. Our focus is on
    the asymptotic distribution of the estimators at a fixed point. The estimators
    are compared in a simulation study.

  2. A maximum smoothed likelihood estimator in the current status continuous mark model.

    Wed, 09/07/2011 - 15:01 — SomNambul
    Authors: Piet Groeneboom, Geurt Jongbloed, Birgit Witte
    Subjects: Statistics
    Abstract

    We consider the problem of estimating the joint distribution function of the
    event time and a continuous mark variable based on censored data. More
    specifically, the event time is subject to current status censoring and the
    continuous mark is only observed in case inspection takes place after the event
    time. The nonparametric maximum likelihood estimator (MLE) in this model is
    known to be inconsistent. We propose and study an alternative likelihood based
    estimator, maximizing a smoothed log-likelihood, hence called a maximum
    smoothed likelihood estimator (MSLE).

  3. Likelihood ratio type two-sample tests for current status data.

    Tue, 02/22/2011 - 16:01 — SomNambul
    Authors: Piet Groeneboom
    Subjects: Statistics
    Abstract

    We introduce fully nonparametric two sample tests for testing the null
    hypothesis that the two samples come from the same distribution if the values
    are only indirectly given via current status censoring. The tests are based on
    the likelihood ratio principle and allow the observation distributions to be
    different for the two samples, in contrast with earlier proposals for this
    situation. A bootstrap method is given for determining critical values and
    asymptotic theory is developed for one of these tests.

  4. Smooth and non-smooth estimates of a monotone hazard.

    Tue, 02/22/2011 - 16:01 — SomNambul
    Authors: Piet Groeneboom, Geurt Jongbloed
    Subjects: Statistics
    Abstract

    We discuss a number of estimates of the hazard under the assumption that the
    hazard is monotone on an interval [0,a]. The usual isotonic least squares
    estimators of the hazard are inconsistent at the boundary points 0 and a. We
    use penalization to obtain uniformly consistent estimators. Moreover, we
    determine the optimal penalization constants, extending related work in this
    direction by Woodroofe and Sun (1993) and Woodroofe and Sun (1999). Two methods
    of obtaining smooth monotone estimates based on a non-smooth monotone estimator
    are discussed.

  5. Testing monotonicity of a hazard: asymptotic distribution theory.

    Wed, 01/19/2011 - 16:01 — SomNambul
    Authors: Piet Groeneboom, Geurt Jongbloed
    Subjects: Statistics
    Abstract

    Two new test statistics are introduced to test the null hypotheses that the
    sampling distribution has an increasing hazard rate on a specified interval
    [0,a]. These statistics are empirical L_1-type distances between the isotonic
    estimates, which use the monotonicity constraint, and either the empirical
    distribution function or the empirical cumulative hazard. They measure the
    excursions of the empirical estimates with respect to the isotonic estimates,
    due to local non-monotonicity.

  6. Isotonic L_2-projection test for local monotonicity of a hazard.

    Tue, 01/18/2011 - 16:01 — SomNambul
    Authors: Piet Groeneboom, Geurt Jongbloed
    Subjects: Statistics
    Abstract

    We introduce a new test statistic for testing the null hypothesis that the
    sampling distribution has an increasing hazard rate on a specified interval
    [0,a]. It is based on a comparison of the empirical distribution function with
    an isotonic estimate, using the restriction that the hazard is increasing, and
    measures the excursions of the empirical distribution above the isotonic
    estimate, due to local non-monotonicity.

  7. Vertices of the least concave majorant of Brownian motion with parabolic drift.

    Tue, 11/02/2010 - 16:01 — SomNambul
    Authors: Piet Groeneboom
    Subjects: Probability
    Abstract

    It was shown in Groeneboom (1983) that the least concave majorant of
    one-sided Brownian motion without drift can be characterized by a jump process
    with independent increments, which is the inverse of the process of slopes of
    the least concave majorant. This result can be used to prove the result of
    Sparre Andersen (1954) that the number of vertices of the smallest concave
    majorant of the empirical distribution function of a sample of size n from the
    uniform distribution on [0,1] is asymptotically normal, with an asymptotic
    expectation and variance which are both of order log n.

  8. The maximum of Brownian motion minus a parabola.

    Tue, 11/02/2010 - 16:01 — SomNambul
    Authors: Piet Groeneboom
    Subjects: Probability
    Abstract

    We derive a simple integral representation for the distribution of the
    maximum of Brownian motion minus a parabola, which can be used for computing
    the density and moments of the distribution, both for one-sided and two-sided
    Brownian motion.

  9. Elementary Statistics on Trial (the case of Lucia de Berk).

    Tue, 09/07/2010 - 15:01 — SomNambul
    Authors: Piet Groeneboom, Richard D. Gill, Peter de Jong
    Subjects: Applications
    Abstract

    In the conviction of Lucia de Berk an important role was played by a simple
    hypergeometric model, used by the expert consulted by the court, which produced
    very small probabilities of occurrences of certain numbers of incidents. We
    want to draw attention to the fact that, if we take into account the variation
    among nurses in incidents they experience during their shifts, these
    probabilities can become considerably larger. This points to the danger of
    using an oversimplified discrete probability model in these circumstances.

  10. Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model.

    Wed, 01/13/2010 - 16:01 — SomNambul
    Authors: Piet Groeneboom, Geurt Jongbloed, Birgit I. Witte
    Subjects: Statistics
    Abstract

    We consider the problem of estimating the distribution function, the density
    and the hazard rate of the (unobservable) event time in the current status
    model. A well studied and natural nonparametric estimator for the distribution
    function in this model is the nonparametric maximum likelihood estimator (MLE).
    We study two alternative methods for the estimation of the distribution
    function, assuming some smoothness of the event time distribution. The first
    estimator is based on a maximum smoothed likelihood approach.

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