By Patrice Bertail, Paul Doukhan, Philippe Soulier
This e-book offers a close account of a few contemporary advancements within the box of likelihood and data for established info. The ebook covers a variety of issues from Markov chain concept and vulnerable dependence with an emphasis on a few fresh advancements on dynamical structures, to powerful dependence in instances sequence and random fields. a unique part is dedicated to statistical estimation difficulties and particular functions. The ebook is written as a succession of papers by means of a few experts of the sphere, alternating normal surveys, typically at a degree available to graduate scholars in chance and information, and extra basic study papers almost always compatible to researchers within the field.
The first a part of the publication considers a few contemporary advancements on vulnerable based time sequence, together with a few new effects for Markov chains in addition to a few advancements on new notions of vulnerable dependence. This half additionally intends to fill a niche among the likelihood and statistical literature and the dynamical procedure literature. the second one half offers a few new effects on robust dependence with a distinct emphasis on non-linear techniques and random fields at the moment encountered in purposes. ultimately, within the final half, a few basic estimation difficulties are investigated, starting from price of convergence of extreme chance estimators to effective estimation in parametric or non-parametric time sequence types, with an emphasis on purposes with non-stationary data.
Patrice Bertail is researcher in records at CREST-ENSAE, Malakoff and Professor of information on the University-Paris X. Paul Doukhan is researcher in information at CREST-ENSAE, Malakoff and Professor of records on the collage of Cergy-Pontoise. Philippe Soulier is Professor of records on the University-Paris X.
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Extra info for Dependence in Probability and Statistics
3 in Bertail & Cl´emen¸con (2004c)), Theorem 3. 1 . Remark 5. The condition αn = o(n−1/2 ) as n → ∞ may be ensured by smoothness conditions satisﬁed by the transition kernel p(x, y): under H¨ older constraints of order s such rates are achieved as soon as s > 1, that is a rather weak assumption. , studentized ) sample mean by ςn = n1/2 σ −1 (f )(µn (f ) − µ(f )) , tn = n1/2 σn (f )−1 (µn (f ) − µ(f )) . A M The following theorem straightforwardly results from Theorem 3. Regeneration-based statistics for Harris recurrent Markov chains 25 Theorem 4.
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This can be done by adapting standard SLLN arguments for U -statistics. 7 Robust functional parameter estimation Extending the notion of inﬂuence function and/or robustness to the framework of general time series is a diﬃcult task (see K¨ unsch (1984) or Martin & Yohai (1986)). Such concepts are important not only to detect ”outliers” among the data or inﬂuential observations but also to generalize the important notion of eﬃcient estimation in semiparametric frameworks (see the recent discussion in Bickel & Kwon (2001) for instance).