Download PDF by Dauxois J.-Y., Druihlet P., Pommeret D.: A Bayesian Choice Between Poisson, Binomial and Negative

By Dauxois J.-Y., Druihlet P., Pommeret D.

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Additional info for A Bayesian Choice Between Poisson, Binomial and Negative Binomial Models

Example text

26) to xà ¼ ej , j ! 1 where ðej Þ is an orthonormal basis of H, and by summing the obtained equalities. In this context one may use the simpler but less precise preference relation: p1 0 1 p 2 , Eu k p 1 À g k 2 Eu k p2 À g k2 ; u 2 Q: Clearly p1 0 p2 ) p1 0 1 p2 : Now, the results concerning sufficient statistics remain valid in the multidimensional case. 2 to xà ðpÞ and xà ðgÞ shows that, if SðXÞ is P-sufficient, one has ES ðpÞ < p. A similar method allows us to extend the results concerning BUP.

In the current section we glance at various preference relations and some empirical predictors. 1 29 Loss function A loss function L : R  R ! R is a positive measurable function such that Lða; aÞ ¼ 0, a 2 R. It generates a risk function defined as Ru ðg; pÞ ¼ Eu ½LðgðX; Y; uÞ; pðXÞފ; u 2 Q which measures the accuracy of p when predicting g. The resulting preference relation is p1 0 p2 , Ru ðg; p1 Þ Ru ðg; p2 Þ; u 2 Q: The following extension of the Rao–Blackwell theorem holds. 11 Let 0 be a preference relation defined by a loss function Lðx; yÞ which is convex with respect to y.

Now b ucðTÞ ! s. yields (see Billingsley 1968) D ucðTÞ ÞÀ! N1 Á N2 ; DT ðb then if PðjN1 N2 j va=2 Þ ¼ 1 À a 2 ð0 < a < 1Þ and    ! hva=2 hva=2 b b IT;h ¼ eÀucðTÞ h XT À pffiffiffiffi ; eÀucðTÞ h XT þ pffiffiffiffi ; T T we have a à 2 IT;h Þ ! N ð0; 1Þ: Now define 0 aðTÞ a 11=2   À2b ucðTÞ h hva=2 1 À e Àb ucðTÞ h @ A ¼ Àna=2 þe XT À pffiffiffiffi T 2b ucðTÞ 52 ASYMPTOTIC PREDICTION and 0 bðTÞ a 11=2   À2b ucðTÞ h hva=2 1 À e Àb ucðTÞ h @ A ¼ Àna=2 þe XT þ pffiffiffiffi T 2b ucðTÞ where na=2 Þ ¼ 1 À PðjNj a 2 d ðN ¼ N ð0; 1ÞÞ; then ðTÞ lim Pu ðXTþh 2 ½aðTÞ a ; ba ŠÞ !

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A Bayesian Choice Between Poisson, Binomial and Negative Binomial Models by Dauxois J.-Y., Druihlet P., Pommeret D.

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