% -*- texinfo -*-
% @deftypefn {Function File} {} kendall (@var{x}, @var{y})
% Compute Kendall's @var{tau} for each of the variables specified by
% the input arguments.
%
% For matrices, each row is an observation and each column a variable;
% vectors are always observations and may be row or column vectors.
%
% @code{kendall (@var{x})} is equivalent to @code{kendall (@var{x},
% @var{x})}.
%
% For two data vectors @var{x}, @var{y} of common length @var{n},
% Kendall's @var{tau} is the correlation of the signs of all rank
% differences of @var{x} and @var{y}; i.e., if both @var{x} and
% @var{y} have distinct entries, then
%
% @tex
% $$ \tau = {1 \over n(n-1)} \sum_{i,j} {\rm sign}(q_i-q_j) {\rm sign}(r_i-r_j) $$
% @end tex
% @ifnottex
% @example
% @group
% 1
% tau = ------- SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
% n (n-1) i,j
% @end group
% @end example
% @end ifnottex
%
% @noindent
% in which the
% @tex
% $q_i$ and $r_i$
% @end tex
% @ifnottex
% @var{q}(@var{i}) and @var{r}(@var{i})
% @end ifnottex
% are the ranks of
% @var{x} and @var{y}, respectively.
%
% If @var{x} and @var{y} are drawn from independent distributions,
% Kendall's @var{tau} is asymptotically normal with mean 0 and variance
% @tex
% ${2 (2n+5) \over 9n(n-1)}$.
% @end tex
% @ifnottex
% @code{(2 * (2@var{n}+5)) / (9 * @var{n} * (@var{n}-1))}.
% @end ifnottex
% @end deftypefn