A NEW THEORY ON AGGREGATE IN STATISTICS(SUMMARY)
This paper presents a new concept "Negative existence of statistical unit".
Let us suppose that we remove a unit X_{i} with frequency f_{i} from the aggregate.
The frequency will be f_{i}-1 by subtracting 1 from f_{i}.
We interpret in this situation that a negative unit of X_{i} is "added"
to the aggregate.
In ordinary case, the frequency f_{i} is positive or possibly zero.
According to the new concept, f_{i} can be negative. Hence the new aggregate
has higher degree of freedom in its nature. Let us show some examples.
(1)Average of the aggregate
m=(f_{i}X_{i})/f_{i}
where X_{i}:value of the unit i,
f_{i}:frequency of the unit i,
:summation sign.
In this situation, m can be smaller than the smallest X_{i} or greater
than the greatest X_{i}. If f_{i}=0, then the average has no value.
If f_{i} is positive for all i, m is identical to well known average.
(2)Variance of the aggregate
^{2}(X)=f_{i}(X_{i}-m)^{2}/f_{i}
The variance can be negative and the standard error be imaginary value, if the
aggregate has negative f_{i}.
The theory is interesting in its applications to, for instance, ratio estimation, etc.
Original paper was issued in Japanese, March 2001, THE BULLETIN OF ECONOMIC STUDIES,
Volume 32, Number 1.2, MEISEI UNIVERSITY, TOKYO; and(continued) December 2001, same
publication, Volume 33, Number 1.(PDF below)
(PDF)ORIGINAL PAPER(1) IN JAPANESE March 2001
(PDF)ORIGINAL PAPER(2-1) IN JAPANESE December 2001
(PDF)ORIGINAL PAPER(2-2) IN JAPANESE December 2001(continued)

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