Some early statistical contributions to the theory and practice of linear algebra

Linear Algebra and Its Applications

Volumn 237-238, Issue , 1996, Pages 205-224

  Farebrother, Richard William ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

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[No Author keywords available]

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EID: 21344434726     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/0024-3795(95)00600-1     Document Type: Article

References (100)

1. A. C. Aitken, On least squares and linear combination of observations, Proc. Roy. Soc. Edinburgh Part A 55:42-47 (1935).
2. A. C. Aitken, Determinants and Matrices, Oliver and Boyd, Edinburgh, 1939; Greenwood Press, London, 1983.
3. A. C. Aitken, On the estimation of many statistical parameters, Proc. Roy. Soc. Edinburgh Part A 62:33-67 (1948).
4. E. Benoît, Note sur une méthode de résolution des équations normales (procédé du Commandant Cholesky), Bull. Géodésique Toulouse, 1924, pp. 67-77.
5. A. Ben-Tal and M. Teboule, A geometric property of the least-squares solution of linear equations, Linear Algebra Appl. 139:165-170 (1990).
6. I. J. Bienaymé, Sur la probabilité des erreurs d'après la methode des moindres carrés, J. Math. Pures Appl. Ser. 1 17:33-78 (1852).
7. I. J. Bienaymé, Remarques sur les différences qui distinguent l'interpolation de M. Cauchy de la méthode des moindres carrés, et qui assurent la supériorité de cette méthode, C. R. Acad. Sci. Paris, 37:5-13 (1853).
8. J. P. M. Binet, Sur quelques formules d'algèbre, sur leur application à des expressions qui ont rapport avec axes conjugués des corps, Nouveau Bull. Sci. Soc. Philomatique 2:389-392 (1811).
9. A. Bravais, Analyse mathématique sur les probabilités des erreurs de situation d'un point, Mém. Présentées Acad. Roy. Sci. Inst. France 9:255-332 (1846).
10. A. L. Cauchy, Cours d'Analyse de l'Ecole Royale Polytechnique I, Paris, 1821; reprinted in his Oeuvres (Ser. 2), Vol. 1, Gauthier-Villars, Paris, 1906.
11. A. L. Cauchy, Cours d'Analyse de l'Ecole Royale Polytechnique I, Paris, 1821; reprinted in his Oeuvres (Ser. 2), Vol. 1, Gauthier-Villars, Paris, 1906.
12. A. L. Cauchy, Sur les centres, les plans principaux et les axes principaux des surfaces du second degré, Éxercises Math. 3 (1828); reprinted in Oeuvers (Ser. 2), Vol. 8, Gauthier-Villars, Paris, 1890, pp. 9-35.
13. A. L. Cauchy, Sur les centres, les plans principaux et les axes principaux des surfaces du second degré, Éxercises Math. 3 (1828); reprinted in Oeuvers (Ser. 2), Vol. 8, Gauthier-Villars, Paris, 1890, pp. 9-35.
14. A. L. Cauchy, Sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des mouvements des planètes, Exercises Math. 4:140-160 (1829); reprinted in Oeuvres (Ser. 2), Vol. 9, Gauthier-Villars, Paris, 1891, pp. 174-195.
15. A. L. Cauchy, Sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des mouvements des planètes, Exercises Math. 4:140-160 (1829); reprinted in Oeuvres (Ser. 2), Vol. 9, Gauthier-Villars, Paris, 1891, pp. 174-195.
16. A. L. Cauchy, On a new formula for solving the problem of interpolation in a manner applicable to physical investigations, Philos. Mag. (Ser. 3) 49:459-468 (1836).
17. A. Cayley, Memoir on the theory of matrices, Philos. Trans. Roy. Soc. London 148:17-37 (1858).
18. P. L. Chebyshev, Sur les functions continués (1855), French transi, by I. J. Bienaymé, J. Math. Pures Appl. 3:289-323 (1858); reprinted in Oeuvres, Vol. 1, St. Petersburg, 1899, and Chelsea, New York, 1902, pp. 203-230.
19. P. L. Chebyshev, Sur les functions continués (1855), French transi, by I. J. Bienaymé, J. Math. Pures Appl. 3:289-323 (1858); reprinted in Oeuvres, Vol. 1, St. Petersburg, 1899, and Chelsea, New York, 1902, pp. 203-230.
20. P. L. Chebyshev, Sur les functions continués (1855), French transi, by I. J. Bienaymé, J. Math. Pures Appl. 3:289-323 (1858); reprinted in Oeuvres, Vol. 1, St. Petersburg, 1899, and Chelsea, New York, 1902, pp. 203-230.
21. C. J. de la Vallée Poussin, Sur la Méthode de l'approximation minimum, Ann. Soc. Sci. Bruxelles 35:1-16 (1911).
22. W. F. Donkin, An essay on the theory of combinations of observations, Trans. Ashmolean Soc., 1844.
23. W. F. Donkin, La théorie de la combinaison des observations, J. Math. Pures Appl. Ser. 1 15:297-322 (1850).
24. M. H. Doolittle, Method employed in this office in the solution of normal equations and in the adjustment of a triangulation, in Report of the Superintendent, US Coast and Geodetic Survey (for 1878), 1881, Appendix 8, paper 3, pp. 115-120.
25. C. Eckart and G. Young, A principal axis transformation for non-Hermitian matrices, Bull. Amer. Math. Soc. 45:118-121 (1939).
26. R. W. Farebrother, An historical note on recursive residuals, J. Roy. Statist. Soc. Ser. B 40:373-375 (1978).
27. R. W. Farebrother, Linear unbiased approximators of the disturbances of the standard linear model, Linear Algebra Appl. 67:259-274 (1985).
28. ∞ estimation, Comm. Statist. Theory Methods 14:1941-1962 (1985).
29. R. W. Farebrother, Linear Least Squares Computations, Marcel Dekker, New York, 1988.
30. 1-Statistical Analysis and Related Methods (Y. Dodge, Ed.), North-Holland, Amsterdam, 1992, pp. 337-349.
31. R. W. Farebrother, Boscovich's method for correcting discordant observations, In R. J. Boscovich: His Life and Scientific Work (P. Bursill-Hall, Ed.), Ist. Enciclopedia Italiana, Rome, 1993, pp. 255-261.
32. J. Farkas, Theorie der einfachen Ungleichungen, J. Reine Angew. Math. 124:1-27 (1902).
33. G. Frobenius, Über das Pfaffsche Problem, J. Reine Angew. Math. 82:230-315 (1877).
34. C. F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium, F. Perthes and I. H. Besser, Hamburg, 1809; reprinted in Werke, Vol. 7, F. Perthes, Gotha, 1871; English transl. by C. H. Davis, Little, Brown, Boston, 1857; French transl. by E. Dubois, Paris, 1864; German transl. by C. Haase, Hanover, 1865.
35. C. F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium, F. Perthes and I. H. Besser, Hamburg, 1809; reprinted in Werke, Vol. 7, F. Perthes, Gotha, 1871; English transl. by C. H. Davis, Little, Brown, Boston, 1857; French transl. by E. Dubois, Paris, 1864; German transl. by C. Haase, Hanover, 1865.
36. C. F. Gauss, Disquisitio de elementis ellipticis Palladis ex oppositionibus annorum 1803, 1804, 1805, 1807, 1808, 1809, Comment. Soc. Regiae Sci. Gottingensis Recentiores, 26 pp. 1811; reprinted in Werke, Vol. 6, Göttingen, 1874, pp. 1-24.
37. C. F. Gauss, Disquisitio de elementis ellipticis Palladis ex oppositionibus annorum 1803, 1804, 1805, 1807, 1808, 1809, Comment. Soc. Regiae Sci. Gottingensis Recentiores, 26 pp. 1811; reprinted in Werke, Vol. 6, Göttingen, 1874, pp. 1-24.
38. C. F. Gauss, Theoria combinationis erroribus minimis obnoxiae: Pars prior, Comment. Soc. Regiae Sci. Gottingensis Recentiores 5:33-62 (1823); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 3-26; English transl. by G. W. Stewart, Classics Appl. Statist. II, SIAM Publ. Philadelphia, 1995.
39. C. F. Gauss, Theoria combinationis erroribus minimis obnoxiae: Pars prior, Comment. Soc. Regiae Sci. Gottingensis Recentiores 5:33-62 (1823); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 3-26; English transl. by G. W. Stewart, Classics Appl. Statist. II, SIAM Publ. Philadelphia, 1995.
40. C. F. Gauss, Theoria combinationis erroribus minimis obnoxiae: Pars prior, Comment. Soc. Regiae Sci. Gottingensis Recentiores 5:33-62 (1823); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 3-26; English transl. by G. W. Stewart, Classics Appl. Statist. II, SIAM Publ. Philadelphia, 1995.
41. C. F. Gauss, Theoria combinationis erroribus minimis obnoxiae: Pars posterior, Comment. Soc. Regiae Sci. Gottingensis Recentiores 5:63-90 (1823); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 29-53; English transl. by G. W. Stewart, Classics Appl. Statist. II, SIAM Publ., Philadelphia, 1995.
42. C. F. Gauss, Theoria combinationis erroribus minimis obnoxiae: Pars posterior, Comment. Soc. Regiae Sci. Gottingensis Recentiores 5:63-90 (1823); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 29-53; English transl. by G. W. Stewart, Classics Appl. Statist. II, SIAM Publ., Philadelphia, 1995.
43. C. F. Gauss, Theoria combinationis erroribus minimis obnoxiae: Pars posterior, Comment. Soc. Regiae Sci. Gottingensis Recentiores 5:63-90 (1823); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 29-53; English transl. by G. W. Stewart, Classics Appl. Statist. II, SIAM Publ., Philadelphia, 1995.
44. C. F. Gauss, Supplementum theoriae combinationis erroribus minimis obnoxiae, Comment. Soc. Sci. Regiae Gottingensis Recentiores 6:57-93 (1828); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 57-93.
45. C. F. Gauss, Supplementum theoriae combinationis erroribus minimis obnoxiae, Comment. Soc. Sci. Regiae Gottingensis Recentiores 6:57-93 (1828); reprinted in Werke, Vol. 4, Göttingen, 1880, pp. 57-93.
46. J. W. L. Glaisher, On the method of least squares, Mem. Roy. Astron. Soc. 40:600-614 (1879).
47. I. Grattan-Guinness, "A new type of question." On the prehistory of linear and non-linear programming, in The History of Modern Mathematics (E. Knoblock and D. Rowe, Eds.), Vol. 3, Academic, New York, 1994, pp. 43-89.
48. I. Grattan-Guinness and W. Ledermann, Matrix theory, in Companion Encyclo-pedia to the History and Philosophy of the Mathematical Sciences (I. Grattan-Guinness, Ed.), Routledge, London, 1994, pp. 775-786.
49. A. Haar, Über lineare Ungleichungen, Acta Litt. Sci. Regiae Univ. Hungar. Francisco-Josephinae Sect. Sci. Math. 2:1-14 (1924-1926).
50. A. Hald, T. N. Thiele's contribution to statistics, Internat. Statist. Rev. 49:1-20 (1981).
51. D. M. Hawkins, The accuracy of elemental set approximations for regression, J. Amer. Statist. Assoc. 88:580-589 (1993).
52. T. W. Hawkins, Cauchy and the theory of matrices, Historia Math. 2:1-29 (1975).
53. T. W. Hawkins, The theory of matrices in the 19th century, in Proceedings of the International Congress of Mathematics, Vancouver, Canad. Math. Congress, 1975, Vol. 2, pp. 561-570.
54. F. R. Helmert, Über die Berechnung des wahrscheinlichen Fahlers aus einer endlichen Anzahl Beobachtungsfehler, Math. u. Phys. 20:300 (1875).
55. F. R. Helmert, Über die Wahrscheinlichkeit der Potenzsummen der Beobach-tungsfehler und über einige damit im Zusammenhange stehende Fragen, Z. Math. u. Phys. 21:192 (1876).
56. C. G. J. Jacobi, De formatione et proprietatibus determinatium, J. Reine Angew. Math. 22:285-318 (1841); reprinted in Werke, Vol. 3, Georg Reimer, Berlin, 1884, pp. 355-392.
57. C. G. J. Jacobi, De formatione et proprietatibus determinatium, J. Reine Angew. Math. 22:285-318 (1841); reprinted in Werke, Vol. 3, Georg Reimer, Berlin, 1884, pp. 355-392.
58. J. L. Lagrange, Recherches sur la méthode de maximis et minimus, Miscellanea Turinensis 7 (1759); reprinted in Oeuvres, Vol. 1, Gauthier-Villars, Paris, 1858, pp. 3-21.
59. J. L. Lagrange, Recherches sur la méthode de maximis et minimus, Miscellanea Turinensis 7 (1759); reprinted in Oeuvres, Vol. 1, Gauthier-Villars, Paris, 1858, pp. 3-21.
60. P. S. Laplace, Mémoire sur la figure de la Terre, Mém. Acad. Roy. Sci. Paris; reprinted in Oeuvres Complètes, 1786, Vol. 11, pp. 17-46; Gauthier-Villars, Paris, 1895, pp. 3-32.
61. P. S. Laplace, Mémoire sur la figure de la Terre, Mém. Acad. Roy. Sci. Paris; reprinted in Oeuvres Complètes, 1786, Vol. 11, pp. 17-46; Gauthier-Villars, Paris, 1895, pp. 3-32.
62. P. S. Laplace, Théorie Analytique des Probabilitiés, Mme. Couricer, Paris, 1812; 3rd ed. with an introduction and three supplements, Mme. Courcier, Paris, 1820; reprinted in Oeuvres Complètes, Vol. 17, Imprimerie Royale, Paris, 1847, and Gauthier-Villars, Paris, 1886.
63. P. S. Laplace, Théorie Analytique des Probabilitiés, Mme. Couricer, Paris, 1812; 3rd ed. with an introduction and three supplements, Mme. Courcier, Paris, 1820; reprinted in Oeuvres Complètes, Vol. 17, Imprimerie Royale, Paris, 1847, and Gauthier-Villars, Paris, 1886.
64. P. S. Laplace, Théorie Analytique des Probabilitiés, Mme. Couricer, Paris, 1812; 3rd ed. with an introduction and three supplements, Mme. Courcier, Paris, 1820; reprinted in Oeuvres Complètes, Vol. 17, Imprimerie Royale, Paris, 1847, and Gauthier-Villars, Paris, 1886.
65. P. S. Laplace, Premier Supplément to Laplace [47], 1816.
66. P. S. Laplace, Deuxième Supplément to Laplace [47], 1818.
67. A. M. Legendre, Nouvelles Méthodes pour la Détermination des Orbites des Comètes, Courcier, Paris, 1805.
68. Y. Li and S. Du, Chinese Mathematics: A Concise History, English transl. by J. N. Crossley and A. W. C. Lun, Oxford U.P., Oxford, 1987.
69. A. A. Markov, The Calculus of Probability (in Russian), St. Petersburg, 1900; German transl. 2nd ed. (1908), Leipzig, 1912.
70. J. T. Mayer, Abhandlung über die Umwalzung des Monds um seine Axe, Kosmographische Nachr. u. Sammlungen 1:52-183 (1750); English transl. pp. 146-159 by G. Trenkler, Univ. of Dortmund, 1986.
71. Y. Mikami, The Development of Mathematics in China and Japan, B. G. Taubner, Leipzig, 1913; Chelsea, New York, 1974.
72. T. Muir, The Theory of Determinants in the Historical Order of Development, St Martin's Press, 1906; Dover, New York, 1960.
73. P. Pizzetti, I Fondamenti Matematici per la Critica dei Risultati Sperimentali, Reale Ist. per Sordomuti, Genoa, 1891; reprinted in Atti Univ. Genova, 1892.
74. P. Pizzetti, I Fondamenti Matematici per la Critica dei Risultati Sperimentali, Reale Ist. per Sordomuti, Genoa, 1891; reprinted in Atti Univ. Genova, 1892.
75. R. L. Plackett, A historical note on the method of least squares, Biometrika 36:458-460 (1949).
76. R. L. Plackett, Karl Pearson and the chi-square-test, Internat. Statist. Rev. 51:59-72 (1983).
77. C. R. Rao, Unified theory of least squares, Comm. Statist. 1:1-18 (1972).
78. C. R. Rao, A lemma on optimization of a matrix function and a review of the unified theory of linear estimation, in Statistical Data Analysis and Inference (Y. Dodge, Ed.), Elsevier Science, Amsterdam 1989, pp. 397-418.
79. D. W. Robinson, Gauss and generalized inverses, Historia Math. 7:118-125 (1980).
80. P. J. Rousseeuw, Least median of squares regression, J. Amer. Statist. Assoc. 79:871-880 (1984).
81. O. B. Sheynin, R. J. Boscovich's work on probability, Arch. Hist. Exact Sci. 9:306-324 (1973), 28:173 (1983).
82. O. B. Sheynin, R. J. Boscovich's work on probability, Arch. Hist. Exact Sci. 9:306-324 (1973), 28:173 (1983).
83. O. B. Sheynin, A. A. Markov's work on probability, Arch. Hist. Exact Sci. 39:337-377, 40:387 (1989).
84. O. B. Sheynin, A. A. Markov's work on probability, Arch. Hist. Exact Sci. 39:337-377, 40:387 (1989).
85. O. B. Sheynin, On the history of the principle of least squares, Arch. Hist. Exact Sci. 46:39-54 (1993).
86. D. E. Smith, A Source Book of Mathematics, McGraw-Hill, New York, 1929; Chelsea, New York, 1959.
87. B. Stay (annotated by R. J. Boscovich), Philosophiae Naturalis ... Versibus Traditae, Tomus II, N. and M. Palearini, Rome, 1760.
88. E. L. Stiefel, Note on Jordan elimination, linear programming and Tchebycheff approximation, Numer. Math. 2:1-17 (1960).
89. S. M. Stigler, The History of Statistics: The Quantification of Uncertainty before 1900, Harvard U.P., Cambridge, Mass., 1986.
90. 1-Statistical Analysis and Related Methods (Y. Dodge, Ed.), North-Holland, Amsterdam, 1992, pp. 103-112.
91. G. P. H. Styan, On the efficiency of a linear unbiased estimator and on matrix versions of the Cauchy-Schwarz inequality, presented at International Workshop on Matrix Methods for Statistics, Auckland, Dec. 1992.
92. M. Subrahmanyam, A property of simple least squares estimates, Sankhyā Ser. B 34:355-356 (1972).
93. T. N. Thiele, The Theory of Observations, Layton, London, 1903; reprinted in Ann. Math. Statist. 2:165-308 (1931).
94. T. N. Thiele, The Theory of Observations, Layton, London, 1903; reprinted in Ann. Math. Statist. 2:165-308 (1931).
95. B. L. van der Waerden, Geometry of Algebra in Ancient Civilizations, Springer-Verlag, Berlin, 1983.
96. W. C. Waterhouse, Gauss's first argument for least squares, Arch. Hist. Exact Sci. 41:41-52 (1990).
97. K. Weierstrass, Zur Theorie der bilinearen und quadratischen Formen, Monatsber. Berlin. Akad. Wiss., 1868; reprinted in Mathematische Werke, Vol. 2, Mayer & Muller, Berlin, 1895, pp. 19-44.
98. K. Weierstrass, Zur Theorie der bilinearen und quadratischen Formen, Monatsber. Berlin. Akad. Wiss., 1868; reprinted in Mathematische Werke, Vol. 2, Mayer & Muller, Berlin, 1895, pp. 19-44.
99. E. T. Whittaker and G. Robinson, The Calculus of Observations, Blackie and Son, London, 1924; 4th ed. (1944) reprinted by Dover, New York, 1967.
100. C. F. J. Wu, Jackknife, bootstrap and other resampling methods in regression analysis, Ann. Statist. 14:1261-1295 (1986).