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June 1999 Bartlett identities and large deviations in likelihood theory
Per Aslak Mykland
Ann. Statist. 27(3): 1105-1117 (June 1999). DOI: 10.1214/aos/1018031270


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 $\operator{cum}_q(R) = O(n^{-q/2})$ for $3 \leq q < p$, whereas $\operator{cum}_p(R) = \kappa_p n^{-(p-2)/2} + O(n^{-p/2})$. 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 th 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.”


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Per Aslak Mykland. "Bartlett identities and large deviations in likelihood theory." Ann. Statist. 27 (3) 1105 - 1117, June 1999.


Published: June 1999
First available in Project Euclid: 5 April 2002

zbMATH: 0951.62014
MathSciNet: MR1724043
Digital Object Identifier: 10.1214/aos/1018031270

Primary: 62E20 , 62M09 , 62M10
Secondary: 60G42 , 60G44 , 62J99 , 62M99

Keywords: Approximate likelihoods , Bartlett correction , convergence of cumulants , dual criterion functions , empirical likelihood , Survival analysis

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 3 • June 1999
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