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Extra resources for Die Grundlagen des natuerlichen Systems, der vergleichenden Anatomie und der Phylogenetik. Leipzig

Example text

1). In general, it is not easy to find a conditional distribution under the condition that a given statistic has a given value. 4) = f(O, g(w»)k(w) for some functions f: 8 xii -+ R+, k: U -+ R+. 1) for any a, b, a < b, the density can be represented as P",b(Xl' ... 5) where y and z are values of Xw and X Cnh respectively. 2). bJ(Z) «b-a)II)-t and kixl, ... +oo'(Y). Now we apply the factorization criterion to any two-element family {PI' P 2} of distributions with positive densities Pt , P2 with respect to a given measure.

Clearly, a measurable function g: D ~ ii which is a sufficient statistic for a certain family of distributions defined on D is not necessarily a sufficient statistic for the enlarged family. For example, consider n independent random variables Xl' ... l' = {U(a, b): a, bE R, a < b}. 1) Let X(j) denote the i-th order statistic, i = 1, ... , n, in the sequence Xl, ... , X n . J}) since for arbitrary a, bER, the value at the point (xt. ,xn ) the density of the conditional distribution of (Xl' ...

In the literature there are no monographs on measurement theory in statistics. Basic information about measurement theory can be found in the monographs of Pfanzagl (1968), Roberts (1979) and the paper of Bromek et at. (1984). 1. FORMALIZATION OF THE GOAL OF RESEARCH In statistical investigations we aim at obtaining some information about the phenomenon which interests us by exploiting that information about it which is available. In Chapter I we have presented a formalism permitting a description of the available information; now we shall concentrate on a formal description of the required information.

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