# 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…

## 15 Citations

Multivariate Multiple Test Procedures

- MathematicsHandbook of Multiple Comparisons
- 2021

Dependencies among data points are present in virtually all modern statistical applications. This holds especially true for studies with multiple endpoints which are all measured for the same…

Simultaneous statistical inference in dynamic factor models: Chi-square approximation and model-based bootstrap

- MathematicsComput. Stat. Data Anal.
- 2019

Non-Central Multivariate Chi-Square and Gamma Distributions

- Mathematics
- 2016

A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank.…

Uncertainty quantification for the family-wise error rate in multivariate copula models

- Mathematics
- 2015

We derive confidence regions for the realized family-wise error rate (FWER) of certain multiple tests which are empirically calibrated at a given (global) level of significance. To this end, we…

Computing and approximating multivariate chi-square probabilities

- Computer Science, Mathematics
- 2014

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.

Some probability inequalities for multivariate gamma and normal distributions

- Mathematics
- 2015

The Gaussian correlation inequality for multivariate zero-mean normal probabilities of symmetrical n-rectangles can be considered as an inequality for multivariate gamma distributions (in the sense…

Empirical Likelihood for the Analysis of Experimental Designs

- Mathematics
- 2021

Empirical likelihood enables a nonparametric, likelihood-driven style of inference without restrictive assumptions routinely made in parametric models. We develop a framework for applying empirical…

Asymptotic properties of statistical estimators using multivariate Chi-squared measurements

- MathematicsDigit. Signal Process.
- 2020

An Exploration of Three New Methods to Facilitate the Calculation of the Significance Levels of Periodogram Peaks by Simulation

- MathematicsThe Astronomical Journal
- 2021

At a given frequency ν, calculation of the periodogram entails evaluating two linear combinations and of the observations. It is pointed out that C and S are normally distributed even for relatively…

Interim monitoring in sequential multiple assignment randomized trials.

- PsychologyBiometrics
- 2021

This work introduces group sequential methods to SMARTs to facilitate interim monitoring based on the multivariate chi-square distribution and demonstrates that the proposed interim monitoring in SMART (IM-SMART) maintains the desired type I error and power with reduced expected sample size compared to the classical SMART.

## 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

- Mathematics
- 2013

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

- Mathematics
- 1984

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

- Mathematics
- 1952

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

- Mathematics
- 1989

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

- MathematicsSpringer Berlin Heidelberg
- 2014

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

- Mathematics
- 1997

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

- Mathematics, Computer Science
- 2012

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

- Computer Science, Mathematics
- 2014

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'

- Mathematics
- 1969

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…