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Bartlett identities and large deviations in likelihood theory. (English) Zbl 0951.62014

Summary: The connection between large and small deviation results for the signed square root statistic \(R\) is studied, both for likelihoods and for likelihood-like criterion functions. We show that if \(p-1\) Barlett identities are satisfied to first order, but the \(p\)th identity is violated to this order, then \(\text{cum}_q (R)=O (n^{-q/2})\) for \(3\leq q<p\), whereas \(\text{cum}_p (R)=\kappa_p n^{-(p-2)/2} +O(n^{-p/2})\). We also show that the large deviation behavior of \(R\) is determined by the values of \(p\) and \(\kappa_p\). The latter result is also valid for more general statistics. Affine (additive and/or multiplicative) correction to \(R\) and \(R^2\) are special cases corresponding to \(p=3\) and 4.
The cumulant behavior of \(R\) gives a way of characterizing the extent to which \(R\)-statistics derived from criterion functions other than log likelihoods can be expected to behave like ones derived from true log likelihoods, by looking at the number of Bartlett identities that are satisfied. Empirical and nonparametric survival analysis type likelihoods are analyzed from this perspective via the device of “dual criterion functions”.

MSC:

62E20 Asymptotic distribution theory in statistics
62F10 Point estimation
60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
62J99 Linear inference, regression
62M99 Inference from stochastic processes
62M09 Non-Markovian processes: estimation
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