Simplified quasi-likelihood analysis for a locally asymptotically quadratic random field

Abstract

The asymptotic decision theory by Le Cam and Hajek has been given a lucid perspective by the Ibragimov-Hasminskii theory on convergence of the likelihood random field. Their scheme has been applied to stochastic processes by Kutoyants, and today this plot is called the IHK program. This scheme ensures that asymptotic properties of an estimator follow directly from the convergence of the random field if a large deviation estimate exists. The quasi-likelihood analysis (QLA) proved a polynomial type large deviation (PLD) inequality to go through a bottleneck of the program. A conclusion of the QLA is that if the quasi-likelihood random field is asymptotically quadratic and if a key index reflecting identifiability the random field has is non-degenerate, then the PLD inequality is always valid, and as a result, the IHK program can run. Many studies already took advantage of the QLA theory. However, not a few of them are using it in an inefficient way yet. The aim of this paper is to provide a reformed and simplified version of the QLA and to improve accessibility to the theory. As an example of the effects of the theory based on the PLD, the user can obtain asymptotic properties of the quasi-Bayesian estimator by only verifying non-degeneracy of a key index.