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.