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By Prof. Dr. Hans Schneeweiss, Prof. Dr. Heinrich Strecker (auth.), Prof. Dr. Hans Schneeweiss, Prof. Dr. Heinrich Strecker (eds.)

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SchlieBlich wird gezeigt, daB die erhaltenen Bedingungen fur die Modelle der einfachen und zweifachen Klassifikation erfullt sind. O} O. E. i=1 1. 1. 1. E = (E1' ... 'E~)' is a vector composed of independent random variables. Let U = (U 1 , ... I 1. ni )E, where In. is a unit-matrix of appropriate order. If E(E) 0 and E(EE') Ir' 1. m m then Ey = X0, Cov y = UVU', where V L 0 2. V. x diag i=1 1. 1. i=1 1. (0, ... I n. , ... 0). In Drygas [4]itmsbeen discussed when a best inva- riant qu~dratic unbiased estimator in the quasi-normal case is also best invariant quadratic unbiased estimator in the non-normal case.

The papers of pfouts (1972) and Hild/Hacker (1978) did not result in explicit index formulas, whereas section 4 of the present paper provides an explicit formula which is quadratic in the current prices. Though there exists a great variety of price index numbers it is well known that no other index is so frequently and worldwide used as the Laspeyres index. Undoubtedly this index has a decisive comparative advantage stemming from the availability of the data required. Presumably an index formula is acceptable for practitioners and will be calculated periodically only if the data requirements do not exceed the data requirements of Laspeyres' index.

T) p. (t) m~n /(1) ~ I[p(l),p(t),q(1)] ~ max ~ ~ 1 i Pi 34 Proof: The validity of the first four tests or axioms can be in- ferred directly from (19). The right hand side of the mean value test can be verified by application of Schwarz' inequality on the last term under the square root in (19), (20) This leads to n Pitt) L: q::' p. (l)p. (t) p. (1) i=l l l l l n L: q2 p. (l)p. (t) i=l l l l Pi (t) :S max - - (1-) i Pi The left hand side of the mean value test is verified most conveniently by using the alternative representation I b t =-- A- e obtainable from (17); the symbols duced in b,A,e have the meaning intro- and (16).

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