An introduction to statistical finance by Bouchaud J.-P.

By Bouchaud J.-P.

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Continuous) is frequently chosen. The previous theorem shows that the family of 9minimum sufficient subfields is the family of subfields which are ^-equal to the smallest subfield ^ * making the chosen determination of densities measurable. Thus there exists a minimum sufficient subfield if and only if the intersection of all subfields ^-equal to J'* is still sufficient. This will be true only if Ω is countable. Now, without loss of generality, suppose that a determination of densities has been chosen.

1 will show that in most cases, one can consider estimators only. On the contrary, in testing hypotheses, one cannot restrict oneself to deterministic strategies. Moreover, we shall see in Chapters 5 and 6 that we cannot estimate a parameter by testing each possible value and, conversely, that we cannot test two hypotheses by estimating their indicator functions. Remark 4 We shall not consider set estimation as a statistical decision problem since we cannot choose a subfield of Ή for which d becomes a measurable mapping.

Dominated by the special distribution P*. If an event A exists such that P*(A\m = x, (ae]0,l[), then P(A) = OL, Proof (1) VPe0>. The subfield M is sufficient and we have P{A\») = P*(A\£) = a, VPe^. Integration leads to (1). | We note that Theorem 2 is useful only if & is strictly included in se. Otherwise the event A does not exist. 3 shows that a free event having probability a exists if and only if an event A exists such that P*(A\@) = a. In addition, this theorem reduces the problem of free events to the study of P*.

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