Abstract
Stochastic expansions of likelihood quantities are usually derived through oridinary Taylor expansions, rearranging terms according to their asymptotic order. The most convenient form for such expansions involves the score function, the expected information, higher order log-likelihood derivatives and their expectations. Expansions of this form are called expected/observed. If the quantity expanded is invariant or, more generally, a tensor under reparameterisations, the entire contribution of a given asymptotic order to the expected/observed expansion will follow the same transformation law. When there are no nuisance parameters, explicit representations through appropriate tensors are available. In this paper, we analyse the geometric structure of expected/observed likelihood expansions when nuisance parameters are present. We outline the derivation of likelihood quantities which behave as tensors under interest-respectign reparameterisations. This allows us to write the usual stochastic expansions of profile likelihood quantities in an explicitly tensorial form.
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References
Barndorff-Nielsen, O. E. (1986). Strings, tensorial components and Bartlett adjustments,Proceedings of the Royal Society of London Series A,406, 127–137.
Barndorff-Nielsen, O. E. (1987). Differential geometry and statistics: Some mathematical aspects,Indian Journal of Mathematics,29, 335–350.
Barndorff-Nielsen, O. E. (1994). Adjusted versions of profile likelihood and directed likelihood, and extended likelihood,Journal of the Royal Statistical Society Series B,56, 125–140.
Barndorff-Nielsen, O. E. and Blæsild, P. (1987). Strings: Mathematical theory and statistical examples,Proceedings of the Royal Society of London Series A,411, 155–176.
Barndorff-Nielsen, O. E. and Cox, D. R. (1994).Inference and Asymptotics, Chapman and Hall, London.
Barndorff-Nielsen, O. E. and Jupp, P. E. (1988). Differential geometry, profile likelihood L-sufficiency and composite transformation models,Annals of Statistics,16, 1009–1043.
Cox, D. R. and Hinkley, D. V. (1974).Theoretical Statistics, Chapman and Hall, London.
DiCiccio, T. J. and Stern, S. E. (1994). Frequentist and Bayesian Bartlett correction of test statistics based on adjusted profile likelihoods,Journal of the Royal Statistical Society Series B,56, 397–408.
Li, B. (2001). Sensitivity of Rao’s score test, the Wald test and the likelihood ratio test to nuisance parameters,Journal of Statistical Planning and Inference,97, 57–66.
McCullagh, P. (1987).Tensor Methods in Statistics, Chapman and Hall, London.
McCullagh, P. and Cox, D. R. (1986). Invariants and likelihood ratio statistics,Annals of Statistics,14, 1419–1430.
McCullagh, P. and Tibshirani, R. (1990). A simple method for the adjustment of profile likelihoods,Journal of the Royal Statistical Society Series B,52, 325–344.
Pace, L. and Salvan, A. (1994). The geometric structure of the expected/observed, likelihood expansions,Annals of the Institute of Statistical Mathematics,46, 649–666.
Pace, L. and Salvan, A. (1997).Principles of Statistical Inference from a Neo-Fisherian Perspective, Advanced Series on Statistical Science and Applied Probability, Vol. 4, World Scientific, Singapore.
Sartori, N., Salvan, A. and Pace, L. (2003). A note on directed adjusted profile likelihoods,Journal of Statistical Planning and Inference,110, 1–9.
Severini, T. (2000).Likelihood Methods in Statistics, Oxford University Press, Oxford.
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Pace, L., Salvan, A. Tensors and likelihood expansions in the presence of nuisance parameters. Ann Inst Stat Math 56, 511–528 (2004). https://doi.org/10.1007/BF02530539
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DOI: https://doi.org/10.1007/BF02530539