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Kendal’s tau

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Kendal’s tau is a measure of rank correlation and measures the similarity of orderings of data when ranked by each of the quantities.

 

If are a set of joint observations from two random variables  and  then pairs  and   are said to be ‘concordant’ if both  and  (or if both  and ). They are said to be ‘discordant’ if  and  or if  and  (and neither concordant or discordant if if  or  , which will not happen if all the  and all the  are unique).

 

Kendall’s  coefficient is then:

 

 

As there are  pairs in total, the coefficient is in the range .

 

[N.B. There are various different ways of handling ties]

 

It is a non-parametric statistic and focuses just on ordering, i.e. on behaviour of the copula, and not the individual marginal distributions. This accords with how copulas are specified. For certain copula families, the parameter that selects between different members of the family has a one-to-one relationship with Kendal’s tau (e.g. the Clayton copula). A natural way of empirically selecting between members of such a family is thus to calculate the empirical Kendal tau (i.e. the one derived from the observations) and then to identify the choice of parameter that reproduces this value.

 

See MnKendalTauCoefficient or MnKendalTauCoefficients for Nematrian web functions that can be used to calculate Kendal’s tau for a single pair of series or for multiple pairs simultaneously.

 


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