A survey of high dimension low sample size asymptotics

Abstract

Peter Hall's work illuminated many aspects of statistical thought, some of which are very well known including the bootstrap and smoothing. However, he also explored many other lesser known aspects of mathematical statistics. This is a survey of one of those areas, initiated by a seminal paper in 2005, on high dimension low sample size asymptotics. An interesting characteristic of that first paper, and of many of the following papers, is that they contain deep and insightful concepts which are frequently surprising and counter-intuitive, yet have mathematical underpinnings which tend to be direct and not difficult to prove.

Classification of Chemical Compounds Based on the Correlation Between in Vitro Gene Expression Profiles

“…It is difficult to obtain in vitro gene expression data for a large number of chemicals, due to the time and cost involved in conducting the necessary experiments; and scenarios in which the number of chemical compounds is small, but each compound has a host of characteristic parameters, are currently very common. Thus, several statistical methods have been proposed, in the recent years, to address such scenarios (Aoshima et al, 2018). Investigation of chemical compound classification methods based on such new statistical methods presents a further challenge.…”

Second Order Expansions for High-Dimension Low-Sample-Size Data Statistics in Random Setting

“…Scientific areas where these settings have proven to be very useful include genetics and other types of cancer research, neuroscience, and also image and shape analysis. See a recent survey on HDLSS asymptotics and its applications in Aoshima et al [3].…”

“…These interesting results imply that the data converge to the vertices of a deterministic regular simplex. These properties were extended for non-normal sample under some assumptions (see Hall et al [1] and Aoshima et al [3]).…”

“…In Kawaguchi et al [4], the relations (1)- (3) were refined by constructing second order asymptotic expansions for distributions of all three basic statistics. The refinements of (1) and (2) were achieved by using the idea of Ulyanov et al [5] who obtained the computable error bounds of order O(m −1 ) for the chi-squared approximation of transformed chi-squared random variables with m degrees of freedom.…”

Geometric consistency of principal component scores for high‐dimensional mixture models and its application

“…Moreover, they created a new principal component analysis (PCA) called the cross-data-matrix methodology that is applicable to construct an unbiased estimator in HDLSS nonparametric settings. Based on the cross-data-matrix methodology, Aoshima and Yata (2011) developed a variety of inference for HDLSS data such as given-bandwidth confidence region, two-sample test, classification, variable selection, regression, pathway analysis, and so on (see Aoshima et al, 2018 for the review).…”