7
conditionally distribution-free nonparametric tests for the
null hypothesis of bivariate symmetry versus alternatives
that the marginal distributions differed only in location,
or that the marginal distribution differed only in scale, or
that the marginal distributions differed in both location
and scale. The basic idea behind his tests is the
I I
following. Under Hq, the pairs (xii>X2i^ i=l2,...,n are a
random sample from an exchangeable continuous
distribution. He pools all the elements into one sample (of
size N=2n), ignoring the fact the original observations were
bivariate pairs and then ranks this combined sample. From
this, Sen obtains what he refers to as the rank matrix,
n / R11
R 1 2
R1 n
RN "
\ R 2 1
R 2 2 *
R2n
where R^^ is the rank of X^^ in the pooled sample j=l,2
i=l,2,...,n. Let S(R^) be the set of all rank matrices that
can be obtained from R^ by permuting within the same column
of R^ for one or more columns. Under HQ, each of the 2n
elements of S(R^) is equally likely and thus, if Tn is a
statistic with a probability distribution (given S(RN) and
H ) which depends only on the 2n equally likely permutations
of Rn, Tn is conditionally distri bution-free (conditional on
the given R^ and thus S(R^) observed). Sen's statistic Tn
can be defined as
T
n
n
n
l
i = l
,R
1 i