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 Xi with frequency fi from the aggregate.
The frequency will be fi-1 by subtracting 1 from fi.
 We interpret in this situation that a negative unit of Xi is "added"
to the aggregate. 
 In ordinary case, the frequency fi is positive or possibly zero.
According to the new concept, fi 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=��(fiXi)/��fi
   where Xi:value of the unit i,
        fi:frequency of the unit i,
        ��:summation sign.
   In this situation, m can be smaller than the smallest Xi or greater
   than the greatest Xi. If ��fi=0, then the average has no value.
   If fi is positive for all i, m is identical to well known average.

(2)Variance of the aggregate
   ��2(X)=��fi(Xi-m)2/��fi
   The variance can be negative and the standard error be imaginary value, if the 
   aggregate has negative fi.

 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)