to that of a Wilcoxon signed rank statistic. Thus, any
member in the class can be used to provide an exact test
which is distribution-free for the null hypothesis. The
statistic based on Kendall's tau is not distribution-free
for small sample sizes and thus, a permutation test based on
the statistic is recommended in these cases. For large
samples, a modified version of the Kendall's tau statistic
is shown to be asymptotically distri bution-free.
For the second and more general alternative, a small
sample permutation test is proposed based on the quadratic
form Wn = T^ j; ^ T where T' is a 2-vector of statistics
composed of a statistic designed to detect location
differences and a statistic designed to detect scale
differences and | is the variance-covariance matrix for
T For large samples, a distribution-free approximation
for T' 1 T is recommended.
~ n T ~ n
Monte Carlo results are presented which compare the two
types of statistics for detecting alternative (1), for
sample sizes of 25 and 40. Quadratic form statistics Wn
using different scale statistic components are also compared
in a simulation study for samples of size 35. For the
alternative involving scale differences only, the statistic
based on Kendall's tau performed best overall but requires a
computer to do the calculations for moderate sample sizes.
For the more general alternative of location and/or scale
differences, the quadratic form using the scale statistic
based on Kendall's tau performed the best overall.
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