A survey on multivariate chi-square distributions and their applications in testing multiple hypotheses

@article{Dickhaus2014ASO,
  title={A survey on multivariate chi-square distributions and their applications in testing multiple hypotheses},
  author={Thorsten Dickhaus and Thomas Royen},
  journal={Statistics},
  year={2014},
  volume={49},
  pages={427 - 454}
}
We are concerned with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and consider the computation of multiplicity-adjusted p-values under the respective global hypothesis. By means of numerical examples, we demonstrate how much gain in level exhaustion or, equivalently, power can be achieved with… 
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References

SHOWING 1-10 OF 49 REFERENCES
Multiple Point Hypothesis Test Problems and Effective Numbers of Tests for Control of the Family-Wise Error Rate
Abtsrcat We consider a specific class of multiple test problems, consisting of M simultaneous point hypothesis tests in local statistical experiments. Under certain structural assumptions the global
Multiple Comparisons of Polynomial Distributions
Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial
Use of Ranks in One-Criterion Variance Analysis
Abstract Given C samples, with n i observations in the ith sample, a test of the hypothesis that the samples are from the same population may be made by ranking the observations from from 1 to Σn i
Generalized maximum range tests for pairwise comparisons of several populations
A multiple comparison procedure (MCP) is proposed for the comparison of all pairs of several independent samples. This MCP is essentially the closed procedure with union-intersection tests based on
Simultaneous Statistical Inference
TLDR
A variety of classical and modern type I and type II error rates in multiple hypotheses testing are defined, some relationships between them are analyzed, and different ways to cope with structured systems of hypotheses are considered.
The Simes Method for Multiple Hypothesis Testing with Positively Dependent Test Statistics
Abstract The Simes method for testing intersection of more than two hypotheses is known to control the probability of type I error only when the underlying test statistics are independent. Although
Simultaneous Statistical Inference in Dynamic Factor Models
Based on the theory of multiple statistical hypothesis testing, we elaborate simultaneous statistical inference methods in dynamic factor models. In particular, we employ structural properties of
Computing and approximating multivariate chi-square probabilities
TLDR
Techniques from matrix factorization and probability theory are applied and a system of MATLAB programs implementing the proposed algorithms for evaluating and approximating multivariate chi-square probabilities are provided.
SIMULTANEOUS TEST PROCEDURES-SOME THEORY OF MULTIPLE COMPARISONS'
1. Introduction and summary. When a hypothesis is tested by a significanice test and is not rejected, it is generally agreed that all hypotheses implied by that hypothesis (its "components") must
...
1
2
3
4
5
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