Non-asymptotic inference for high and infinite dimensional data

点击翻译按钮获取中文摘要

When analyzing high and infinite dimensional data, novel statistical techniques that go beyond classical asymptotic theory are required. One class of such alternatives are the exact and permutation tests that aim to characterize the finite sample distribution of the data [Pigoli et al., 2014, Cabassi et al., 2017]. However, such approaches are often mired in the computational complexity of combinatorial enumeration and thus cannot be fully realized. This research proposal aims to attack the same problem of characterizing the finite sample distribution of the data by the careful construction and implementation of concentration inequalities, the so-called non-asymptotic theory of independence [Boucheron et al., 2013]. There are three facets to this research.The first is investigations into non-asymptotic methods for high dimensional data. The main focus is on covariance and precision matrix estimation for high dimensional data using concentration inequalities and other non-asymptotic tools. The secondary focus is on the applicability of such estimators to actual inferential problems. The second facet is research into log-concave stochastic processes for functional data. The main goal is to carefully construct nonparametric classes of stochastic processes that have enough nice properties--e.g. strong concentration behaviour--to be used for inference on functional data problems including speech analysis and neuroimaging. The final topic is inference on transformation invariant distributions. One problem of interest is to determine a testing methodology for the detection of invariance of data to some transformation. For example, the multivariate Gaussian distribution has ellipsoidal symmetry. A second problem is to investigate concentration and contraction properties of random variables when such transformations are applied. This is an extension of the Rademacher averages, which yield fascinating and highly useful properties. The impact of this research proposal can be quite far reaching. Primarily, it aims to develop methodology, which will be widely applicable to some of the most complex and inscrutable data available to scientific researchers. This includes high dimensional data such as gene expressions and epidemiological studies. It also includes infinite dimensional or functional data such as neuroimaging and other medical imaging. It furthermore can be applied to various types of spatial temporal data such as collections of climate measurements. Beyond such applications, this research will have close connections with topics in probability theory and functional analysis leading to many well cited papers in top journals for both statistical methodology and statistical theory.