arXiv subjects

Piet Groeneboom

Publications and source records attributed to Piet Groeneboom.

40 records · Page 3Linked to original sources

arXiv2008

Current status data with competing risks: Limiting distribution of the MLE

We study nonparametric estimation for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider a simpler ``naive estimator.'' Groeneboom, Maathuis and Wellner [Ann. Statist. (2008) 36 1031--1063] proved that both types of estimators converge globally and locally at rate $n^{1/3}$. We use these results to derive the local limiting distributions of the estimators. The limiting distribution of the naive estimator is given by the slopes of the convex minorants of correlated Brownian motion processes with parabolic drifts. The limiting distribution of the MLE involves a new self-induced limiting process. Finally, we present a simulation study showing that the MLE is superior to the naive estimator in terms of mean squared error, both for small sample sizes and asymptotically.

math.ST↗

arXiv2006

Second class particles and cube root asymptotics for Hammersley's process

We show that, for a stationary version of Hammersley's process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North--East path $L(t,t)$ from $(0,0)$ to $(t,t)$ is equal to $2\mathbb {E}(t-X(t))_+$, where $X(t)$ is the location of a second class particle at time $t$. This implies that both $\mathbb {E}(t-X(t))_+$ and the variance of $L(t,t)$ are of order $t^{2/3}$. Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of Cator and Groeneboom [Ann. Probab. 33 (2005) 879--903].

math.PR↗

arXiv2005

Hammersley's process with sources and sinks

We show that, for a stationary version of Hammersley's process, with Poisson ``sources'' on the positive x-axis, and Poisson ``sinks'' on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley's process. This allows us to show that Hammersley's process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199-213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley's process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t,t)/t converges to 2, as t\to\infty, where L(t,t) is the length of a longest North-East path from (0,0) to (t,t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke's theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.

math.PR↗

arXiv2004

The suppport reduction algorithm for computing nonparametric function estimates in mixture models

Vertex direction algorithms have been around for a few decades in the experimental design and mixture models literature. We briefly review this type of algorithm and describe a new member of the family: the support reduction algorithm. The support reduction algorithm is applied to the problem of computing nonparametric estimates in two inverse problems: convex density estimation and the Gaussian deconvolution problem. Usually, VD algorithms solve a finite dimensional (version of the) optimization problem of interest. We introduce a method to solve the true infinite dimensional optimization problem.

math.ST↗