10 Jahre Wiederaufbau

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Monograph

Identifikator:: 1768152721

URN:: urn:nbn:de:zbw-retromon-148079

Document type:: Monograph

Title:: 10 Jahre Wiederaufbau

Place of publication:: Wien

Publisher:: Wirtschaftszeitungs-Verlags-Ges. M.B.H.

Year of publication:: 1928

Scope:: 664 S.; Ill.

Digitisation:: 2021

Collection:: Economics Books

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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Die Entwicklung des Post- und Telegraphenwesens / von Ministerialrat Dr. Ferdinand Weiß

Collection:: Economics Books

XIL—PARTIAL CORRELATION. 205 the coefficients 6,4 . . . . , etc, and so on: they are sometimes termed the normal equations. If the student will follow the pro- cess by which (5) was obtained, he will see that when the con- dition is expressed that 4,,,, _ _, shall possess the “least-square ” value, z, enters into the product-sum with #4; ... .,; when the same condition is expressed for d,5,, .. ., ; enters into the product-sum, and so on. Taking each regression in turn, in fact, every x the suffix of which is included in the secondary suffixes of #05... , enters into the product-sum. The normal equations of the form (5) are therefore equivalent to the theorem— The product-sum of any deviation of order zero with any deviation of higher order is zero, provided the subscript of the former occur among the secondary subscripts of the latter. 8. But it follows from this that (2134... 234...n) =ZZ1s4...n(@p—boss...n.Tg— 0. —Donss... n=1. Tn) =3(21.34 ...n. Tg) Similarly, (21.34... n. T2354... n) = Z(x, «X34... 0) Similarly again, Z(@134...n- T2354... (0-1) = (L134... mn. Ty), and so on. Therefore, quite generally, 2(Tray nea n=3(x , ... n=1)+®234 ....m) = Tyg... nm) = BE «0 ETSY 3 irs nerd) =2%15... n-&) Comparing all the equal product-sums that may be obtained in this way, we see that the product-sum of any two deviations is unaltered by omitting any or all of the secondary subscripts of either which are common to the two, and, conversely, the product-sum of any deviation of order | with a deviation of order p+q, the p subscripts being the same in each case, is unaltered by adding to the secondary subscripts of the former any or all of the q additional subscripts of the latter. It follows therefore from (5) that any product-sum is zero if ali the subscripts of the one deviation occur among the secondary sub- scripts of the other. As the simplest case, we may note that a, is uncorrelated with z,;, and z, uncorrelated with z,,. The theorems of this and of the preceding paragraph are of fundamental importance, and should be carefully remembered. Rf

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