10
D
n
/[ fFu-y)
- Fn(y,x)}2dFn(x,y)
where
is the bivariate empirical c.d.f. He notes that nDn is not
distribution-free nor asymptotically distribution-free when
Hq is true, and thus proposed a conditional test in which
the conditioning process is based on the 2n data points
(J i )
{((xHx2i)
V 0 or 1
for k = 1,2,...,n}
>
which are equally likely under HQ Here we let
(s,t)^) = (s,t) and (s,t)^^ = (t,s). This statistic
performs well even for extremely small sample sizes (n=5)
with one major drawback as mentioned by Hollander which is
the computer time which it takes to evaluate nDn. It
becomes very prohibitive for even moderate n. Koziol (1979)
developed the critical values for nDn for large sample
sizes, which work much better than the large sample critical
value approximations originally suggested by Hollander.
Kepner (1979) proposed tests based on the transformed
observations (Y^,Y2) of Bell and Haller for the null
hypothesis of bivariate symmetry versus the alternatives
that the marginal distributions differ in scale or that the
marginal distributions differ in location and/or scale. For
the alternative of differences in scale, he proposed a test