on
not distribution free, even when H is true. Thus, if f is
o rn
the variance-covariance of W the quadratric form W'
~ n n ~nrn~n
will not be distribution-free. A consistent estimator of
A
j] n j-n t>e introduced in Section 4.5 and a test based
the asymptotic distribution-free statistic U' i W will be
recommended for large sample sizes. For small sample sizes
a permutation test will be recommended. First though, we
introduce the TE statistic by Popovich (1983) with a slight
change in notation to accommodate this thesis.
Let
Di
= Xli
- x2i
a nd
R( |
Di|)
be
the
absolute
Di
for i
= 1,2
> >
n, that
is ,
R(
1 Di 1
is
the
rank of |
among (|
Dll>
1 ^2 1
|Dn
| ).
Define
1
if
Z .
^ 0
T .
1
= ¥(D1)
= <
1
0
if
Z .
1
O
V
Let
TE
nl
a nd
TEn
c
be defined
to
be
t he
following:
nl
TE
nl
i
II 0^1
H-*
4'1 R (
lDi
)
and
TEn = N N2 .
c
Notice that TE
is the Wilcoxon signed rank statistic
applied to the n^ totally uncensored pairs. Popovich (1983)
showed under Hq, N^ is distributed as a Binomial random
variable with parameters nc and p = V2 ?2(0) = V2 P(type 2 or
3 pair). With a slight modification from Popovich, the
n 1 n<
statistic TE
is