### Abstract

Irrespective of the statistical model under study, the derivation of lim- its, in the Le Cam sense, of sequences of local experiments (see [7]-[10]) often follows along very similar lines, essentially involving differentiability in quadratic mean of square roots of (conditional) densities. This chapter establishes two ab- stract and very general results providing sufficient and nearly necessary conditions for (i) the existence of a quadratic expansion, and (ii) the asymptotic linearity of local log-likelihood ratios (asymptotic linearity is needed, for instance, when un- specified model parameters are to be replaced, in some statistic of interest, with some preliminary estimator). Such results have been established, for locally asymp- totically normal (LAN) models involving independent and identically distributed observations, by, e.g., [1], [11] and [12]. Similar results are provided here for mod- els exhibiting serial dependencies which, so far, have been treated on a case-by-case basis (see [4] and [5] for typical examples) and, in general, under stronger regularity assumptions. Unlike their i.i.d. counterparts, our results extend beyond the context of LAN experiments, so that non-stationary unit-root time series and cointegration models, for instance, also can be handled (see [6]).

Original language | English |
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Title of host publication | Mathematical Statistics and Limit Theorems |

Subtitle of host publication | Festschrift in Honour of Paul Deheuvels |

Editors | M. Hallin, D.M. Mason, D. Pfeifer, J. Steinebach |

Place of Publication | Cham |

Publisher | Springer |

Pages | 147-165 |

Number of pages | 19 |

ISBN (Electronic) | 9783319124421 |

ISBN (Print) | 9783319124414 |

DOIs | |

Publication status | Published - 8 Apr 2015 |

### Publication series

Name | Springer Proceedings in Mathematics & Statistics |
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## Cite this

Hallin, M., van den Akker, R., & Werker, B. J. M. (2015). On quadratic expansions of log-likelihoods and a general asymptotic linearity result. In M. Hallin, D. M. Mason, D. Pfeifer, & J. Steinebach (Eds.),

*Mathematical Statistics and Limit Theorems : Festschrift in Honour of Paul Deheuvels*(pp. 147-165). (Springer Proceedings in Mathematics & Statistics). Springer. https://doi.org/10.1007/978-3-319-12442-1_9