Abstract
Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.
Similar content being viewed by others
References
Barndorff-Nielsen, O. E. (1986). Strings, tensorial combinants and Bartlett adjustments,Proc. Roy. Soc. London Ser. A,406, 127–137.
Barndorff-Nielsen, O. E. (1987). Differential geometry and statistics: some mathematical aspects,Indian J. Math.,29, 335–350.
Barndorff-Nielsen, O. E. (1989). Comment to Kass, R. E.: the geometry of asymptotic inference,Statist. Sci.,4, 188–234.
Barndorff-Nielsen, O. E. and Blæsild, P. (1987). Strings: mathematical theory and statistical examples,Proc. Roy. Soc. London Ser. A,411, 155–176.
Barndorff-Nielsen, O. E., Blæsild, P., Pace, L. and Salvan, A. (1991a). Formulas for asymptotic statistical calculations, Research Rep., No. 207, Department of Theoretical Statistics, Aarhus University, Denmark.
Barndorff-Nielsen, O. E., Jupp, P. E. and Kendall, W. S. (1991b). Stochastic calculus, statistical asymptotics, Taylor strings and phyla,Annales de Toulouse (to appear).
Blæsild, P. (1990). Yokes: orthogonal and extended normal coordinates, Research Rep., No. 205, Department of Theoretical Statistics, Aarhus University, Denmark.
Blæsild, P. (1991). Yokes and tensors derived from yokes,Ann. Inst. Statist. Math.,43, 95–113.
Kendall, W. S. (1992). Computer algebra and yoke geometry I: when is an expression a tensor?, Research Rep., No. 237, Department of Statistics, University of Warwick, U.K.
Lawley, D. M. (1956). A general method for approximating to the distribution of the likelihood-ratio criteria,Biometrika,43, 295–303.
McCullagh, P. (1987).Tensor Methods in Statistics, Chapman and Hall, London.
McCullagh, P. and Cox, D. R. (1986). Invariants and likelihood ratio statistics,Ann. Statist.,14, 1419–1430.
Murray, M. K. (1988). Coordinate systems and Taylor series in statistics,Proc. Roy. Soc. London Ser. A.,415, 445–452.
Murray, M. K. and Rice, J. W. (1993).Differential Geometry and Statistics, Chapman and Hall, London.
Author information
Authors and Affiliations
Additional information
This research was partially supported by the Italian National Research Council grant n.93.00824.CT10.
About this article
Cite this article
Pace, L., Salvan, A. The geometric structure of the expected/observed likelihood expansions. Ann Inst Stat Math 46, 649–666 (1994). https://doi.org/10.1007/BF00773474
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773474