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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
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