A survey of high dimension low sample size asymptotics
Makoto Aoshima1, Dan Shen2, Haipeng Shen3
1Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan.
Australian & New Zealand Journal of Statistics
|September 11, 2018
Summary
This survey explores Peter Hall's lesser-known work on high-dimensional, low-sample size asymptotics. It highlights surprising yet mathematically direct concepts in this area of mathematical statistics.
Area of Science:
- Mathematical Statistics
- Statistical Theory
Background:
- Peter Hall's significant contributions to statistical thought, including the bootstrap and smoothing.
- Exploration of lesser-known areas within mathematical statistics.
Purpose of the Study:
- To survey a specific area of Peter Hall's work: high-dimensional, low-sample size (HDLSS) asymptotics.
- To examine the conceptual depth and mathematical underpinnings of HDLSS asymptotics.
Main Methods:
- Review of seminal papers in HDLSS asymptotics, starting from 2005.
- Analysis of the conceptual and mathematical characteristics of the research.
Main Results:
- Identification of deep, insightful, and often counter-intuitive concepts in HDLSS asymptotics.
- Observation that the mathematical proofs for these concepts are generally direct and accessible.
Conclusions:
- Peter Hall's work on HDLSS asymptotics offers profound insights.
- The field is characterized by surprising results with straightforward mathematical foundations.
Related Concept Videos
02:16Surveys
16.9K
Often, psychologists develop surveys as a means of gathering data. Surveys are lists of questions to be answered by research participants, and can be delivered as paper-and-pencil questionnaires, administered electronically, or conducted verbally. Generally, the survey itself can be completed in a short time, and the ease of administering a survey makes it easy to collect data from a large number of people.
16.9K
01:19Sample Size Calculation
6.7K
Knowledge of the sample size is the first requirement to conduct random sampling or an experiment. The sample size is the total number of units, observations, or groups (in some cases) used to get the data to estimate a population parameter. As the name suggests, the sample size is that of the sample drawn from the population and differs from the population size.
The sample size for the given experiment or sampling effort is fundamental to any study design. Sample size decides the number of...
The sample size for the given experiment or sampling effort is fundamental to any study design. Sample size decides the number of...
6.7K
01:27Slant Asymptotes
137
A function's behavior is often guided by asymptotic constraints, where one term dominates another, defining a limiting trend. In the given scenario, the mathematical pattern follows a rational function: a cubic term in the numerator is divided by a squared term in the denominator. This results in a function with distinct characteristics, including an oblique asymptote, critical points, and undefined regions.The function's validity is determined by the denominator, which must be nonzero. This...
137
01:15One-Way ANOVA: Unequal Sample Sizes
6.7K
One-way ANOVA can be performed on three or more samples of unequal sizes. However, calculations get complicated when sample sizes are not always the same. So, while performing ANOVA with unequal samples size, the following equation is used:
6.7K
01:15One-Way ANOVA: Equal Sample Sizes
4.1K
One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the...
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the...
4.1K
01:30Asymptotes in Rational Functions
255
A rational function is defined as the quotient of two polynomials: where Q(x)≠0, These functions often exhibit asymptotes, which are the lines that the graph approaches but never touches. These asymptotes are classified based on how the function behaves near specific values of the input.Vertical asymptotes occur where the denominator is zero, and the numerator is not, causing the function to be undefined. These are found by solving Q(x)=0. For example: has a vertical...
255