Stochastic calculus, statistical asymptotics, Taylor strings and phyla
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 3 (1994) no. 1, pp. 5-62. 
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     author = {Barndorff-Nielsen, Ole E. and Jupp, Peter E. and Kendall, Wilfrid S.},
     title = {Stochastic calculus, statistical asymptotics, {Taylor} strings and phyla},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {5--62},
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     number = {1},
     year = {1994},
     mrnumber = {1270515},
     zbl = {0815.62002},
     language = {en},
     url = {http://archive.numdam.org/item/AFST_1994_6_3_1_5_0/}
}
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Barndorff-Nielsen, Ole E.; Jupp, Peter E.; Kendall, Wilfrid S. Stochastic calculus, statistical asymptotics, Taylor strings and phyla. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 3 (1994) no. 1, pp. 5-62. http://archive.numdam.org/item/AFST_1994_6_3_1_5_0/

[1] Amari (S.-I.) .- Differential Geometric Methods in Statistics, Lecture Notes in Statistics 28, Springer-Verlag, New York - Heidelberg - Berlin (1985). | MR | Zbl

[2] Amari (S.-I.), Barndorff-Nielsen (O.E.), Lauritzen (S.L.) and Rao (C.R.) . Differential Geometry in Statistical Inference, Lecture Notes-Monograph Series 10, Institute of Mathematical Statistics, Hayward, Califormia - (1987). | MR | Zbl

[3] Ambrose (W.), Palais (R.S.) and Singer (I.M.) .- Sprays, Anais Acad. Bras. de Ciências 32 (1960), pp. 163-178. | MR | Zbl

[4] Andrews (D.F.) and Stafford (J.E.) .- Tools for the symbolic computation of asymptotic expansions, J. Roy. Statist. Soc. B 55 (1993), pp. 613-627. | MR | Zbl

[5] Antonelli (P.L.), Chapin (J.) and Voorhees (B.H.) .- The geometry of random genetic drift VI: a random selection diffusion model, Adv. in Appl. Probab. 12 (1980), pp. 50-58. | MR | Zbl

[6] Barndorff-Nielsen (O.E.) .- On a formula for the distribution of the maximum likelihood estimator, Biometrika 70 (1983), pp. 343-365. | MR | Zbl

[7] Barndorff-Nielsen (O.E.) .- Strings, tensorial combinants, and Bartlett adjustments, Proc. Roy. Soc. London A 406 (1986), pp. 127-137. | MR | Zbl

[8] Barndorff-Nielsen (O.E.) . - Differential geometry and statistics: some mathematical aspects, Indian J. Math. 29 (1987), pp. 335-350. | MR | Zbl

[9] Barndorff-Nielsen (O.E.) .- Invited contribution to the discussion of R. E. Kass: The Geometry of Asymptotic Inference, Statist. Science 4 (1989), pp. 222-227; Amendments in Statist. Science 5 (1990), p. 370. | MR

[10] Barndorff-Nielsen (O.E.) and Blæsild (P.) .- Strings: Mathematical theory and statistical examples, Proc. Roy. Soc. London A 411 (1987), pp. 155-176. | MR | Zbl

[11] Barndorff-Nielsen (O.E.) and Blæsild (P.) .- Derivative strings: contravariant aspect, Proc. Roy. Soc. London A 411 (1987), pp. 421-444. | MR | Zbl

[12] Barndorff-Nielsen (O.E.) and Blæsild (P.) .- Coordinate-free definition of structurally symmetric derivative strings, Adv. in Appl. Math. 9 (1988), pp. 1-6. | MR | Zbl

[13] Barndorff-Nielsen (O.E.), Blæsild (P.), Carey (A.L.), Jupp (P.E.), Mora (M.) and Murray (M.K.) .- Finite-dimensional algebraic representations of the infinite phylon group, Acta Appl. Math. 28 (1992), pp. 219-252. | MR | Zbl

[14] Barndorff-Nielsen (O.E.), Blæsild (P.) and Mora (M.) .- Derivative strings and higher order differentiation, Dept. of Theoretical Statistics, Institute of Mathematics, Univ. of Aarhus, Memoir no 11 (1988). | MR

[15] Barndorff-Nielsen (O.E.), Blæsild (P.) and Mora (M.) .- Generalized higher-order differentiation, Acta Appl. Math. 16 (1989), pp. 243-259. | MR | Zbl

[16] Barndorff-Nielsen (O.E.), Cox (D.R.) and Reid (N.) .- The role of differential geometry in statistical theory, Internat. Statist. Rev. 54 (1986), pp. 83-96. | MR | Zbl

[17] Blæsild (P.) .- Yokes: Orthogonal and extended normal coordinates, Dept. of Theoretical Statistics, Institute of Mathematics, Univ. of Aarhus, Research Report n° 205 (1990).

[18] Blæsild (P.) . - Yokes and tensors derived from yokes, Ann. Inst. Statist. Math. 43 (1991), pp. 95-113. | MR | Zbl

[19] Blæsild (P.) and Mora (M.) .- Differential strings: basic theory, Exposition. Math. 10 (1991), pp. 249-273. | MR | Zbl

[20] Bowman (R.H.) .- On differentiable extensions, Tensor (N.S.) 21 (1970), pp. 139-150. | MR | Zbl

[21] Carey (A.L.) and Murray (M.K.) .- Higher-order tensors, strings and new tensors, Proc. Roy. Soc. London A 430 (1990), pp. 423-432. | MR | Zbl

[22] Craig (H.V.) .- Vector and Tensor Analysis, McGraw-Hill, New York (1943). | MR | Zbl

[23] Critchley (F.), Marriott (P.K.) and Salmon (M.H.) .- Preferred point geometry and statistical manifolds, Ann. Statist. (to appear). | MR | Zbl

[24] Dekrét (A.) .- On quasi-jets, Gas. Pest. Mat. 111 (1986), pp. 345-352. | Zbl

[25] Dellacherie (C.) and Meyer (P.-A.) .- Probabilities and Potential A, North-Holland, Amsterdam (1978). | MR | Zbl

[26] Dellacherie (C.) and Meyer (P.-A.) .- Probabilities and Potential B, North-Holland, Amsterdam (1982). | MR | Zbl

[27] Dellacherie (C.) and Meyer (P.-A.) .- Probabilities and Potential C, North-Holland, Amsterdam (1988). | MR | Zbl

[28] Ehresmann (C.) .- Les prolongements d'une variété différentiable. I. Calcul des jets, prolongement principal, C.R. Acad. Sci. Paris 239 (1951), pp. 598-600. | MR | Zbl

[29] Ehresmann (C.) .- Extension du calcul des jets aux jets non holonomes, C.R. Acad. Sci. Paris 239 (1954), pp. 1762-1764. | MR | Zbl

[30] Emery (M.) . - Stochastic Calculus in Manifolds (with an appendix by P.-A. Meyer), Springer-Verlag, New York - Heidelberg - Berlin (1989). | MR | Zbl

[31] Epstein (D.B.A.) and Thurston (W.P.) .- Transformation groups and natural bundles, Proc. London Math. Soc. 38 (1979), pp. 219-236. | MR | Zbl

[32] Foster (B.L.) .- Differentiation on manifolds without a connection, Michigan Math. J. 5 (1958), pp. 183-190. | MR | Zbl

[33] Foster (B.L.) . - Some remarks on tensor differentiation, Ann. Mat. Pura Appl. 54 (1961), pp. 143-146. | MR | Zbl

[34] Foster (B.L.) .- Would Leibniz lie to you? (Three aspects of the affine connection), Math. Intelligencer 8 (1986), pp. 34-40, 57. | MR | Zbl

[35] Foster (B.L.) . - Higher derivatives in geometry and physics, Proc. Roy. Soc. London A 423 (1989), pp. 443-455. | MR | Zbl

[36] Ikeda (N.) and Watanabe (S.) . - Stochastic Differential Equations and Diffusion Processes, North-Holland and Kodansha, Amsterdam and Tokyo (1981). | MR | Zbl

[37] Itô (K.) .- Stochastic differentials, Appl. Math. Optim. 1 (1965), pp. 374-381. | MR | Zbl

[38] Jupp (P.E.) .- Derivative strings, differential strings, and semi-holonomic jets, Proc. Roy. Soc. London A 436 (1992), pp. 89-98. | MR | Zbl

[39] Kass (R.E.) . - The geometry of asymptotic inference (with discussion), Statist. Science 4 (1989), pp. 188-234. | MR | Zbl

[40] Kendall (W.S.) .- Computer algebra, Brownian motion, and the statistics of shape, Geometrization of Statistical Theory, Proceedings, GST Workshop, Univ. of Lancaster (28-31 Oct. 1987); C.T.J. Dodson ed., Univ. of Lancaster, ULDM Publications, pp. 171-192. | MR

[41] Kendall (W.S.) . - Stochastic differential geometry: an introduction, Acta Appl. Math. 9 (1987), pp. 29-60. | MR | Zbl

[42] Kendall (W.S.) .- Symbolic computation and the diffusion of shapes of triads, Adv. in Appl. Probab. 20 (1988), pp. 775-797. | MR | Zbl

[43] Kendall (W.S.) .- Martingales on manifolds and harmonic maps, The Geometry of Random Motion, R. Durrett & M. Pinsky eds., American Mathematical Society, Providence, RI (1988), pp. 121-157 (but see Huang and Kendall, 1991). | Zbl

[44] Kendall (W.S.) . - Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence, Proc. London Math. Soc. 61 (1990), pp. 371-406. | MR | Zbl

[45] Kendall (W.S.) .- Computer algebra and stochastic calculus, Notices Amer. Math. Soc. 37 (1990), pp. 1254-1256.

[46] Kendall (W.S.) .- Diffusion of Euclidean shape, Disorder in Physical Systems, G. Grimmett & D. Welsh eds., Oxford Univ. Press, Oxford (1990), pp. 203-217. | Zbl

[47] Kendall (W.S.) .- Symbolic Itô calculus: an introduction, Probabilités Numériques, N. Bouleau & D. Talay eds., INRIA, Rocquencourt, France (1991), pp. 186-192.

[48] Kendall (W.S.) .- Statistical asymptotics using the computer package "REDUCE", Warwick Univ. Dept. of Statistics, Research Report in preparation.

[49] Kendal (W.S.) .- A remark on the proof of Itô's formula for C2 functions of continuous semimartingales, J. Appl. Probab. 29 (1992), pp. 216-221. | MR | Zbl

[50] Kobayashi (S.) .- Canonical forms on vector bundles of higher order contact, Proc. Symp. Pure Math. 3 (1961), pp. 186-193. | MR | Zbl

[51] Mccullagh (P.) .- Tensor Methods in Statistics, Chapman and Hall, London (1987). | MR | Zbl

[52] Mccullagh (P.) and Cox (D.R.) .- Invariants and likelihood ratio statistics, Ann. Statist. 14 (1986), pp. 1419-1430. | MR | Zbl

[53] Meyer (P.-A.) .- A differential geometric formalism for the Itô integral, Stochastic Integrals (Proc. Durham 1980), D. Williams ed., Lect. Notes in Math. 851, Springer-Verlag, New York - Heidelberg - Berlin (1981), pp. 256-270. | MR | Zbl

[54] Meyer (P.-A.) .- Géométrie stochastique sans larmes, Sém. Probab. XV, Lect. Notes in Math 850, Springer-Verlag, New York - Heidelberg - Berlin (1981), pp. 44-102. | Numdam | MR | Zbl

[55] Meyer (P.-A.) .- Qu'est-ce qu'une différentielle d'ordre n ? Exposition. Math. 7 (1989), pp. 249-264. | MR | Zbl

[56] Mora (M.) .- Geometrical expansions for the distributions of the score vector and the maximum likelihood estimator, Ann. Inst. Statist. Math. 44 (1992), pp. 63-83. | MR | Zbl

[57] Murray (M.K.) . - Coordinate systems and Taylor series in statistics, Proc. Roy. Soc. London A 415 (1988), pp. 445-452. | MR | Zbl

[58] Palais (R.S.) and Terng (C.L.) .- Natural vector bundles have finite order, Topology 16 (1977), pp. 271-277. | MR | Zbl

[59] Protter (P.) .- Stochastic Integration and Differential Equations, a new approach, Springer-Verlag, New York - Heidelberg - Berlin (1990). | MR | Zbl

[60] Revuz (D.) and Yor (M.) . - Continuous Martingales and Brownian Motion, Springer-Verlag, New York - Heidelberg - Berlin (1991). | MR | Zbl

[61] Rogers (L.C.G.) and Williams (D.) .- Diffusions, Markov Processes, and Martingales Vol. II: Itô calculus, Wiley, Chichester (1987). | MR | Zbl

[62] Saunders (D.J.) . - The Geometry of Jet Bundles, London Mathematical Society Lecture Notes 142, Cambridge University Press, Cambridge (1989). | MR | Zbl

[63] Schwartz (L.) .- Géométrie différentielle du deuxième ordre, semi-martingales et équations différentielles stochastiques sur une variété différentielle, Sém. Probab. XVI (supplément : géométrie différentielle stochastique), Lect. Notes in Math. 921, Springer-Verlag, New York - Heidelberg - Berlin, (1982), pp. 1-150. | Numdam | MR | Zbl

[64] Schwartz (L.) .- Semimartingales and their Stochastic Calculus on Manifolds, Les Presses de l'Université de Montréal, Montréal (1984). | MR | Zbl

[65] Synge (J.-L.) and Schild (A.) .- Tensor Calculus, Mathematical Expositions 5, University of Toronto Press, Toronto (1949). | MR | Zbl

[66] Terng (C.-L.) . - Natural vector bundles and natural diffential operators, Amer. J. Math. 100 (1978), pp. 775-828. | MR | Zbl

[67] White (J.E.) .- The Method of Iterated Tangents with Applications in Local Riemannian Geometry, Monographs and Studies in Mathematics 13, Pitman, Boston, (1982). | MR | Zbl

[68] Yuen (P.C.) .- Higher order frames and linear connections, Cahiers Topologie Géom. Différentielle 12 (1971), pp. 333-374. | Numdam | MR | Zbl