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Related Concept Videos

Introduction to Test of Independence01:21

Introduction to Test of Independence

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In statistics, the term independence means that one can directly obtain the probability of any event involving both variables by multiplying their individual probabilities. Tests of independence are chi-square tests involving the use of a contingency table of observed (data) values.
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The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to conclude whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence. The hypotheses for the test for homogeneity can...
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The test of independence is a chi-square-based test used to determine whether two variables or factors are independent or dependent. This hypothesis test is used to examine the independence of the variables. One can construct two qualitative survey questions or experiments based on the variables in a contingency table. The goal is to see if the two variables are unrelated (independent) or related (dependent). The null and alternative hypotheses for this test are:
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The Wald-Wolfowitz runs test, commonly referred to as the runs test, is a nonparametric test used to assess the randomness of ordered data. The test evaluates the number of runs, which are consecutive sequences of similar elements within the data. If the number of runs is significantly higher or lower than expected, the data is considered non-random, indicating a detectable pattern or structure.
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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.
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ASYMPTOTICALLY INDEPENDENT U-STATISTICS IN HIGH-DIMENSIONAL TESTING.

Yinqiu He1, Gongjun Xu1, Chong Wu2

  • 1Department of Statistics, University of Michigan.

Annals of Statistics
|December 3, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces U-statistics for high-dimensional hypothesis testing, offering unbiased estimators for feature norms. The adaptive testing procedure combines p-values for robust power against various alternatives.

Keywords:
62F0362F05High-dimensional hypothesis testU-statisticsadaptive testing

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Area of Science:

  • Statistics
  • High-Dimensional Data Analysis
  • Hypothesis Testing

Background:

  • High-dimensional data analysis often involves testing marginal or low-dimensional features of complex distributions.
  • Existing methods for testing mean vectors, covariance matrices, and regression coefficients face challenges with high dimensionality.

Purpose of the Study:

  • To construct a novel family of U-statistics for unbiased estimation of ℓ-norms of high-dimensional features.
  • To develop an adaptive hypothesis testing procedure that leverages these U-statistics for improved power and robustness.

Main Methods:

  • Construction of a family of U-statistics as unbiased estimators for ℓ-norms.
  • Asymptotic analysis of U-statistics under the null hypothesis, demonstrating independence and normal distribution.
  • Investigation of asymptotic independence between U-statistics and maximum-type test statistics with extreme value distributions.

Main Results:

  • U-statistics of different finite orders are asymptotically independent and normally distributed under the null hypothesis.
  • U-statistics exhibit asymptotic independence with maximum-type test statistics.
  • The proposed adaptive testing procedure effectively combines p-values from U-statistics of varying orders.

Conclusions:

  • The developed U-statistics provide reliable estimators for high-dimensional feature norms.
  • The adaptive testing procedure demonstrates high power across a range of alternative hypotheses.
  • This approach offers a powerful and flexible framework for high-dimensional hypothesis testing.