In
statistics, Hoeffding's test of independence, named after
Wassily Hoeffding
Wassily Hoeffding (June 12, 1914 – February 28, 1991) was a Finnish statistician and probabilist. Hoeffding was one of the founders of nonparametric statistics, in which Hoeffding contributed the idea and basic results on U-statistics.
In pr ...
, is a test based on the population measure of deviation from independence
:
where
is the
joint distribution function of two random variables, and
and
are their
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
functions.
Hoeffding derived an
unbiased estimator
In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In st ...
of
that can be used to test for
independence
Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the s ...
, and is
consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consisten ...
for any continuous
alternative
Alternative or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...
. The test should only be applied to data drawn from a
continuous distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, since
has a defect for discontinuous
, namely that it is not necessarily zero when
. This drawback can be overcome by taking an
integration
Integration may refer to:
Biology
* Multisensory integration
* Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technolo ...
with respect to
. This modified measure is known as Blum–Kiefer–Rosenblatt coefficient.
A paper published in 2008
[Wilding, G.E., Mudholkar, G.S. (2008) "Empirical approximations for Hoeffding's test of bivariate independence using two Weibull extensions", ''Statistical Methodology'', 5 (2), 160-–170 ] describes both the calculation of a sample based version of this measure for use as a test statistic, and calculation of the null distribution of this test statistic.
See also
*
Correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statisti ...
*
Kendall's tau
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter τ, tau), is a statistic used to measure the ordinal association between two measured quantities. A τ test is a ...
*
Spearman's rank correlation coefficient
In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation ( statistical dependence betw ...
*
Distance correlation In statistics and in probability theory, distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal, dimension. The population distance correlation coefficient is ze ...
References
Primary sources
* Wassily Hoeffding, A non-parametric test of independence, ''Annals of Mathematical Statistics'' 19: 293–325, 1948.
JSTOR
* Hollander and Wolfe, Non-parametric statistical methods (Section 8.7), 1999. Wiley.
Covariance and correlation
Nonparametric statistics
Statistical tests
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