Quadratic Approximation for Log-Likelihood Ratio Processes
We consider a sequence of general filtered statistical models with a finite-dimensional parameter. It is tacitly assumed that a proper rescaling of the parameter space is already done (so we deal with a local parameter) and also time rescaling is done if necessary. Our first and main purpose is to give sufficient conditions for the existence of certain uniform in time linear–quadratic approximations of log-likelihood ratio processes. Second, we prove general theorems establishing LAN, LAMN and LAQ properties for these models based on these linear–quadratic approximations. Our third purpose is to prove three theorems related to the necessity of the conditions in our main result. These theorems assert that these conditions are necessarily satisfied if (1) an approximation of a much more general form exists and a (necessary) condition of asymptotic negligibility of jumps of likelihood ratio processes holds, or (2) we have LAN property at every moment of time and the limiting models are continuous in time, or (3) we have LAN property, Hellinger processes are asymptotically degenerate at the terminal times, and the condition of asymptotic negligibility of jumps holds.
KeywordsContiguity Filtered statistical experiment Hellinger process Likelihood ratio process Limit theorems Local asymptotic mixed normality Local asymptotic normality Local asymptotic quadraticity
This research was supported by Suomalainen Tiedeakatemia (A.A. Gushchin), by Academy of Finland grants 210465 and 212875 (E. Valkeila), and by the program Tête-à-tête in Russia (Euler International Mathematical Institute; both authors). For the first author, this work has been funded by the Russian Academic Excellence Project ‘5-100’. The first author is deeply grateful to the referees for a number of comments aimed at improving the exposition.
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