Hanna K. Jankowski

Expectations of Random Sets and Their Boundaries Using Oriented Distance Functions

Shape estimation and object reconstruction are common problems in image analysis. Mathematically, viewing objects in the image plane as random sets reduces the problem of shape estimation to inference about sets. Currently existing definitions of the expected set rely on different criteria to construct the expectation. This paper introduces new definitions of the expected set and the expected boundary, based on oriented distance functions. The proposed expectations have a number of attractive properties, including inclusion relations, convexity preservation and equivariance with respect to rigid motions. The paper introduces a special class of decomposable oriented distance functions for parametric sets and gives the definition and properties of decomposable random closed sets. Further, the definitions of the empirical mean set and the empirical mean boundary are proposed and empirical evidence of the consistency of the boundary estimator is presented. In addition, the paper discusses loss functions for set inference in frequentist framework and shows how some of the existing expectations arise naturally as optimal estimators. The proposed definitions are illustrated on theoretical examples and real data.

Convergence of linear functionals of the Grenander estimator under misspecification

Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate

Nonparametric estimation of a convex bathtub-shaped hazard function

n^{1/3}

Computation of nonparametric convex hazard estimators via profile methods

if the true density is curved [Sankhy\={a} Ser. A 31 (1969) 23-36] and at rate

Identifying Skeleton Curves in Noisy Data

n^{1/2}

Multiple Comparisons Using Composite Likelihood in Clustered Data

if the density is flat [Ann. Probab. 11 (1983) 328-345; Canad. J. Statist. 27 (1999) 557-566]. In the case that the true density is misspecified, the results of Patilea [Ann. Statist. 29 (2001) 94-123] tell us that the global convergence rate is of order

Computing confidence intervals for log-concave densities

n^{1/3}

Estimating a discrete log-concave distribution in higher dimensions

in Hellinger distance. Here, we show that the local convergence rate is

A random set approach to confidence regions with applications to the effective dose with combinations of agents

n^{1/2}

A note on statistical and biological communication: a case study of the 2009 H1N1 pandemic.

at a point where the density is misspecified. This is not in contradiction with the results of Patilea [Ann. Statist. 29 (2001) 94-123]: the global convergence rate simply comes from locally curved well-specified regions. Furthermore, we study global convergence under misspecification by considering linear functionals. The rate of convergence is