Hoeffding's independence test
Encyclopedia
In statistics
, Hoeffding's test of independence, named after Wassily Hoeffding
, 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
functions.
Hoeffding derived an unbiased estimator of
that can be used to test for independence, and is consistent
for any continuous alternative. The test should only be applied to data drawn from a continuous distribution, since
has a defect for discontinuous 
, namely that it is not necessarily zero when 
.
A recent paper 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.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, Hoeffding's test of independence, named after Wassily Hoeffding
Wassily Hoeffding
Wassily Hoeffding was an American statistician and probabilist...
, 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 distributionMarginal 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. The term marginal variable is used to refer to those variables in the subset of variables being retained...
functions.
Hoeffding derived an unbiased estimator of
that can be used to test for independence, and is consistentConsistent estimator
In statistics, a sequence of estimators for parameter θ0 is said to be consistent if this sequence converges in probability to θ0...
for any continuous alternative. The test should only be applied to data drawn from a continuous distribution, since
has a defect for discontinuous , namely that it is not necessarily zero when .A recent paper 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
- CorrelationCorrelationIn statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
- Kendall's tau
- Spearman's rank correlation coefficientSpearman's rank correlation coefficientIn statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho or as r_s, is a non-parametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can...
- Distance correlationDistance correlationIn statistics and in probability theory, distance correlation is a measure of statistical dependence between two random variables or two random vectors of arbitrary, not necessarily equal dimension. Its important property is that this measure of dependence is zero if and only if the random...
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.