Marriott, Paul, 1961- and Salmon, Mark H. (Mark Howard), 1949- (2000) An introduction to differential geometry in econometrics. Working Paper. University of Warwick: Warwick Business School Financial Econometrics Research Centre. (Working Papers Series).

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Abstract

In this introductory chapter we seek to cover sufficient differential geometry in order to understand its application to Econometrics. It is not intended to be a comprehensive review of either differential geometric theory, nor of all the applications which geometry has found in statistics. Rather it is aimed as a rapid tutorial covering the material needed in the rest of this volume and the general literature. The full abstract power of a modern geometric treatment is not always necessary and such a development can often hide in its abstract constructions as much as it illuminates. In Section 2 we show how econometric models can take the form of geometrical objects known as manifolds, in particular concentrating on classes of models which are full or curved exponential families. This development of the underlying mathematical structure leads into Section 3 where the tangent space is introduced. It is very helpful, to be able view the tangent space in a number of different, but mathematically equivalent ways and we exploit this throughout the chapter. Section 4 introduces the idea of a metric and more general tensors illustrated with statistically based examples. Section 5 considers the most important tool that a differential geometric approach offers, the affine connection. We look at applications of this idea to asymptotic analysis, the relationship between geometry and information theory and the problem of the choice of parameterisation. The last two sections look at direct applications of this geometric framework. In particular at the problem of inference in curved families and at the issue of information loss and recovery. Note that while this chapter aims to give a reasonably precise mathematical development of the required theory an alternative and perhaps more intuitive approach can be found in the chapter by Critchley, Marriott and Salmon later in this volume. For a more exhaustive and detailed review of current geometrical statistical theory see Kass and Vos (1997) or from a more purely mathematical background, see Murray and Rice (1993).

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Social Sciences > Warwick Business School > Financial Econometrics Research Centre Faculty of Social Sciences > Warwick Business School
Library of Congress Subject Headings (LCSH):	Geometry, Differential, Geometrical models in statistics
Series Name:	Working Papers Series
Publisher:	Warwick Business School Financial Econometrics Research Centre
Place of Publication:	University of Warwick
Date:	22 January 2000
Volume:	Vol.1999
Number:	No.10
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
References:	Amari S-I, (1987). Differential geometric theory of statistics. In Differential Geometry in Statistical Inference (eds. S.I. Amari, O.E. Barndorff-Nielsen, R.E. Kass, S.L. Lauritzen and C.R. Rao), 217-240. Institute of Mathematical Statistics, Hayward, CA. Amari S-I, (1990), Differential-Geometric Methods in Statistics. Springer, Lecture Notes in Statistics, 28, Berlin. Barndorff-Nielsen, O.E., (1978) Information and Exponential Families in Statistical Theory. Wiley, London. Barndorff-Nielsen, O.E., (1987), Differential geometry and statistics: some mathematical aspects. Indian J. Math. 29, Ramanujan Centenary Volume. Barndorff-Nielsen, O.E., (1988) Parametric Statistical Families and Likelihood. Springer, New York. Bandorff-Nielsen and Cox (1994),Inference and Asymptotics Monographs on Statistics and Applied Probability, 52, Chapman and Hall, London Barndorff-Nielsen, O.E. and Jupp, P., (1989), Approximating exponential models, Ann. Statist. Math. 41, 247-267. Barnard, G.A. and Sprott, D.A., (1971), A note on Basu’s examples of anomalous ancillary statistics (with discussion). In Foundations of Statistical Inference eds V.P. Godambe abd D.A. Sprott). Holt, Rinehart and Winston, Toronto, pp. 163-76 Basu, D., (1964), Recovery of ancillary information, Sankya A, 26, 3-16. Bates, M. and Watts, D.G., (1988) Nonlinear Regression Analysis & its Applications Wiley, London. Blæsild, P., (1987), Yokes: Elemental properties with statistical applications. In Geometrization of Statistical Theory, Proceedings for the GST Workshop, University of Lancaster (ed C.T.J. Dodson) 193-198. ULDM Publications, Univ. Lancaster. Blæsild, P., (1991). Yokes and tensors derived from yokes. Ann. Inst. Statist. Maths. 43, 95-113 Bröcker, T.H. and Jänich, K., (1982), Introduction to Differential Geometry, CUP, Cambridge. Brown, L.D., (1986), Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Institute of Mathematical Statistics, Haywood, CA. Cox, D.R., (1971), The choice between alternative ancillary statistics. J. R. Statist. Soc. C, 33, 251-5. Cox, D.R., and Hinkley D.V., (1974) Theoretical Statistics, Chapman and Hall, London. Critchley, F., Marriott P.K., and Salmon, M., (1993), Preferred point geometry and statistical manifolds. Ann. Statist. 21, 1197-1224. Critchley, F., Marriott P.K., and Salmon, M. (1994) On the local differential geometry of the Kullback-Liebler divergence,Annals Statist 22 p1587-1602. Critchley, F., Marriott P.K., and Salmon, M., (1996), On the differential geometry of the Wald test with nonlinear restrictions. Econometrica 64, 1213-1222. Critchley, F., Marriott P.K., and Salmon, M., (1998), An Elementary account of Amari’s Expected geometry, Applications of Differential Geometry in Econometrics Ed. P.K. Marriott and M. Salmon, CUP, Cambridge Dodson, C.T. and Poston, T., (1991), Tensor Geometry: The Geometric Viewpoint and Its Uses, 2nd ed. Springer-Verlag, New York. Efron, B., (1975), Defining the curvature of a statistical problem (with applications to second order efficiency). Ann. Statist. 3, 1189-1217. Efron, B., (1978), The geometry of exponential families. Ann. Statist. 6, 362-376. Efron, B., (1982), Maximum likelihood and decision theory. Ann. Statist. 10, 340-356. Efron, B. and Hinkley, D.V., (1978), Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information, Biometrika 65, 457-481. Fisher, R.A., (1925), Theory of statistical estimation, Proceedings of the Cambridge Philosophical Society, 22, 700-725. Hougaard, P., (1982), Parametrisations of nonlinear models. J. Roy Statist. Soc. Series B Methodological 44, 244-252. Kass R.E. and Vos P.W., (1997), Geometrical Foundations of Asymptotic Inference, Wiley series in Probability and Statistics, J. Wiley and Sons, New York. Lauritzen, S.L. (1987), Statistical manifolds. In Differential Geometry in Statistical Inference (eds. S.I. Amari, O.E. Barndorff-Nielsen, R.E. Kass, S.L. Lauritzen and C.R. Rao), 217- 240. Institute of Mathematical Statistics, Hayward, CA. Marriott, P.K., (1989), Applications of differential geometry to statistics Ph.D. dissertation. University of Warwick. McCullagh, P., (1987), Tensor Methods in Statistics, Monographs on Statistics and Applied Probability, 29, Chapman and Hall, London. Murray, M.K. and Rice, J.W., (1993), Differential Geometry and Statistics, Monographs on Statistics and Applied Probability, 48, Chapman and Hall, London. Ravishanker, N (1994), Relative Curvature Measures of Nonlinearity for Time-Series Models, Communications in Statistics—Simulation and Computation, 23, No.2, pp.415-430 Rao, C.R., (1945), Asymptotic effciency and limiting information. In Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1, 531-545. Rao, C.R., (1987), Differential metrics in probability spaces. In Differential Geometry in Statistical Inference (eds. S.I. Amari, O.E. Barndorff-Nielsen, R.E. Kass, S.L. Lauritzen and C.R. Rao), 217-240. Institute of Mathematical Statistics, Hayward, CA. Rudin, W., (1976), Principles of Mathematical Analysis, 3rd edition, McGraw Hill, New York. Spivak, M. (1979), A Comprehensive Introduction to Differential Geometry, 2nd Edition, Publish and Perish, Boston. Sweeting, T., (1996), Bayesian Stats Approximate Bayesian computation based on signed roots of log-density ratios (with discussion). Bayesian Statistics 5 (J. M. Bernardo, J. O. Berger, A. P. Dawid, A. F. M. Smith, eds.), Oxford: University Press, 427-444. Willmore, T.J., (1959), An Introduction to Differential Geometry, Clarendon, Oxford.
URI:	http://wrap.warwick.ac.uk/id/eprint/1839

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