Q2 Z2,(Q2) Z2(Q2) - Institute for Water Resources - U.S. Army

These results are disappointing, but should have been expected.
On the one hand, one might think that each X ij is normally distributed
over the totai-sample:of the orie.ns óf the three modes. There are no
a priori reasons to expect another distribution. Therefore, one might,
speculate that they are normally distributed as "pure" random variables
are. But, if the total sample is normally .distributed, each separate
,
sub-Sample :cannot be so distributed'. This can be seen if one considers
that—the X-'s j which make up the ith mode sub-sampl correspond largely
to the X. 's in one of the tails of the total sample. distribution. Ii
the extreme case where the's would be insignificant, the distribution
of theX ij 's would correspond to a chunk of the former distribution and
could not be normal. On the other hand, these observations are also
conditionalby the actual geographic circumstances of the experiment,
and some of the possible value ofX ij 's within their observed range are
not at all represented in the sample. These distributions are essentially
reflecting the geographic pattern of corn production.
This situation raises a double question. First, if it is not
possible to justify the hypothesis of normality - on empirical or theoretical
-
grounds - is it still possible to use it as a purely ad hoc device? This
is the question of the robustness of the proposed statistical procedure.
Second, is there not another method, a distribution-free method, which -
could be used in place of the procedure with normality assuMption? ,
As an answer to the first question, M.G. Kendall'nd A. Stuart
propose to apply the discriminators, or classification rules, to each member
of the samples on which they are based and to observe the errors in the
samp1es. (1) This procedure has been applied to the non-weighed January sample,
and the results reported in Table VIII.
(1) M.G. Kendall and A. Stuart 'The Advanced Theorx_of Statistics(, Vol. 3,
1966, pp. 324-.325.