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**1 - 4**of**4**### Statistical Inference for Cluster Trees

"... Abstract A cluster tree provides a highly-interpretable summary of a density function by representing the hierarchy of its high-density clusters. It is estimated using the empirical tree, which is the cluster tree constructed from a density estimator. This paper addresses the basic question of quan ..."

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Abstract A cluster tree provides a highly-interpretable summary of a density function by representing the hierarchy of its high-density clusters. It is estimated using the empirical tree, which is the cluster tree constructed from a density estimator. This paper addresses the basic question of quantifying our uncertainty by assessing the statistical significance of topological features of an empirical cluster tree. We first study a variety of metrics that can be used to compare different trees, analyze their properties and assess their suitability for inference. We then propose methods to construct and summarize confidence sets for the unknown true cluster tree. We introduce a partial ordering on cluster trees which we use to prune some of the statistically insignificant features of the empirical tree, yielding interpretable and parsimonious cluster trees. Finally, we illustrate the proposed methods on a variety of synthetic examples and furthermore demonstrate their utility in the analysis of a Graft-versus-Host Disease (GvHD) data set.

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"... My core statistical research focuses on semiparametric/nonparametric methodology and large sample theory — efficient estimation in semiparametric models, nonparametric function es-timation (with emphasis on shape constrained estimation), and bootstrap based inference in non-standard problem. I am al ..."

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My core statistical research focuses on semiparametric/nonparametric methodology and large sample theory — efficient estimation in semiparametric models, nonparametric function es-timation (with emphasis on shape constrained estimation), and bootstrap based inference in non-standard problem. I am also actively involved in interdisciplinary research in astronomy. My research has applications in broad areas such as genomics (multiple testing problems), economics (utility and production function estimation and binary response models), causal inference (conditional independence and dimension reduction) and astronomy (understanding the accretion history of galaxies), among other fields. In the following, first I give a brief overview of my research. The second part of the document contains a detailed description of my research as well as the related future research directions. Summary of Research Mixture Models: Two-component mixture models arise in multiple testing problems and more generally in contamination problems. We studied these model without any para-metric or nonparametric assumptions. We developed the first distribution-free and

### Uncertainty Measures and Limiting Distributions for Filament Estimation

"... A filament is a high density, connected region in a point cloud. There are several methods for estimating filaments but these methods do not provide any measure of uncer-tainty. We give a definition for the uncertainty of estimated filaments and we study statistical properties of the estimated filam ..."

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A filament is a high density, connected region in a point cloud. There are several methods for estimating filaments but these methods do not provide any measure of uncer-tainty. We give a definition for the uncertainty of estimated filaments and we study statistical properties of the estimated filaments. We show how to estimate the uncertainty mea-sures and we construct confidence sets based on a bootstrap-ping technique. We apply our methods to astronomy data and earthquake data. Categories and Subject Descriptors G.3 [PROBABILITY AND STATISTICS]: Multivari-ate statistics, Nonparametric statistics