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somewhat the number of 'errors' as defined above and for the fact that a few peaks, not significant for practical purposes, will be forecast, it seems possible to develop a method along these lines that will forecast turns 'correctly', i.e., within plus or minus 4 months, about 3 times out of 4 with a lead of initial forecast long enough to be useful.

Appendix ESTIMATION METHODS

If a is a random variable and X1, X2 - - - Xn are the observations of a sample of n observations arranged in order of size, Xi is said to be the i-th order statistic of the sample.28 The date of the first specific turn observed in the sample frequency distribution of the text may be regarded as the first order statistic, the date of the second turn as the second order statistic, and so on. A problem of grouping arises when two or more turns fall in the same month but this can be handled with a small loss of accuracy by distributing the peaks throughout the month in which they occur.

It is assumed that the sample is from N(α, σ) a normal population with mean α and standard deviation σ. Let mi/n (mi) be the expected value of the i-th order statistic in a sample of size n from this population (Fig. 3).

Figure 3 Estimation of Standard Deviation

Let ci/n (or ci) be the expected value of the i-th order statistic in a sample of size n drawn from N (O, 1), a normal population with zero mean and unit standard deviation29 (Fig. 3). Then:
mi = α + ciσ
mj = α + cjσ; and
mi - mj = (ci - cj

____________________
28
Cf. note 25. See also S. S. Wilks, Mathematical Statistics ( Princeton University Press, 1943), p. 89.
29
Exact values of ci/n for n ≤ 10 and an asymptotic formula for ci/n, n > 10 are given by Hastings, Mosteller, Tukey, and Winson, op. cit.

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