20
applied. Figure 4 shows what happens to the contour given
in Figure 2 (i.e., under H ) when this transformation is
applied. Note, as can be seen in Figure 3, under this
I
transformation and H Y, and Y01 are not correlated
o 11 11
t
although Xjj and X2 ^ possibly were. Similarly, as can be
seen in Figure 4, under this transformation and and
f
Y21 are correlated (negatively in this case). Thus, the
original problem of testing for unequal marginal scales has
been transformed into the problem of testing for correlation
I
between Y^ and Y2^ Kepner (1979) suggested the use of
T
Kendall's tau to test for correlation between Y^ and Y9y .
Kendall's tau was chosen, due to the fact it is a
U-statistic and, thus, the many established results for
U-statistics could be applied.
The test statistic which will be presented in this
section is very similar to the above mentioned statistic.
However, when censoring is present, the true observed value
If
of X ^ ^ 0^ X 21
(or
both) is not known,
a nd
thus
Y11 Y21
(or both) are
also
affected. To take
this
into
account a
modified Kendall's tau will be used which was presented by
Oakes (1982) to test for independence in the presence of
censoring. First though, some additional notation must be
int roduced.
I I
Recall, (Xjj,X2i) denotes bivariate random variables
t 1
which are distributed as (xii>^21^* Let 0^,02, Cn denote
the censoring random variables which are independent and
identically distributed (i.i.d.) with continuous