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Regularity of arbitrary order for a filtered statistical model. (Régularité d’ordre quelconque pour un modèle statistique filtré.) (French) Zbl 0737.62087
Séminaire de probabilités, Lect. Notes Math. 1485, 140-161 (1991).
[For the entire collection see Zbl 0733.00018.]
The author generalizes his results obtained in Probab. Theory Relat. Fields 86, No. 3, 305-335 (1990; Zbl 0677.62001), on the relation between “likelihood processes” $$Z_ \theta$$ and “partial likelihood processes” $$\overline Z_ \theta$$. The notion of partial likelihood process was introduced by him in Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 2, 299-329 (1990; Zbl 0727.60037). In the paper cited above, he proved that the condition that the map $$\theta\to(Z_ \theta)^{1/2}$$ is differentiable at $$\theta=0$$ in $$L^ 2(P_ 0)$$ and “locally uniformly” in time, implies a similar condition for the map $$\theta\to(\overline Z_ \theta)^{1/2}$$.
Here he proves an analogous result. It is shown that the condition that the map $$\theta\to(Z_ \theta)^{1/r}$$ is differentiable at $$\theta=0$$ in $$L^ k(P_ 0)$$ and “locally uniformly” in time, implies a similar condition for the map $$\theta\to(\overline Z_ \theta)^{1/r}$$ for $$1\leq r\leq k$$ where $$k$$ and $$r$$ are real.
##### MSC:
 62M99 Inference from stochastic processes 60G99 Stochastic processes
##### Keywords:
likelihood processes; partial likelihood processes
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