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21 (Sequence of Poisson processes) Let ðNt; j ; t ! 0Þ, j ! 0 be a sequence of independent P homogeneous Poisson processes with respective intensity lj, j ! 1 such that j lj < 1. , therefore Mt ¼ ðNt;j ; j ! 0Þ defines an ‘2 -valued random variable, where ‘2 is the Hilbert P 1=2 P 2 : space fðxj Þ 2 RN ; j x2j < 1g with norm k ðxj Þ k¼ j xj One observes MðTÞ ¼ ðMt ; 0 t TÞ and wants to predict MTþh ðh > 0Þ. It is easy to see that MT is a P-sufficient statistic, then one only considers predictors of the form pðMT Þ.
Lðg; ES ðpÞÞ þ L0 ðg; ES ðpÞÞðp À ES ðpÞÞ where L0 is the right derivative of Lðg; ÁÞ. 22). 3). We let the reader verify that various optimality results given above remain valid if the quadratic error is replaced by a convex loss function. 2 Location parameters Suppose that L is associated with a location parameter m defined by ELðZ; mÞ ¼ min ELðZ; aÞ a2R with PZ 2 P L , a family of probability measures on R. 23) is minimum for p0 ðXÞ ¼ mu ðXÞ, where mu ðXÞ is the location parameter associated with PXu;g .
And various estimators (maximum likelihood, Bayesian estimators, . ). 4. bTþh . The first statement provides rate We now study asymptotic behaviour of X of convergence. 5), it suffices to study the asymptotic behaviour of D1;T . 1(i) entails D1;T Eu ½ðak ðTÞZ 2k Þðbk ðTÞ k b ucðTÞ À u k2k Þ T and from the ? Davydov inequality? 6). d. sequences, we have, for these sequences, Ã XTþh ¼ Eu ðXTþh Þ :¼ rðuÞ: Consequently, if bðTÞÀ1 is the minimax quadratic rate for estimating u and rðuÞ, it is also the minimax quadratic rate for predicting XTþh .