Non-linear curve fitting and the true method of least squares

Journal of the Royal Statistical Society Series D: The Statistician

Volumn 47, Issue 1, 1998, Pages 137-147

  Farebrother, Richard William ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Computational procedures; Gauss Newton procedures; Gompertz function; Logistic function; Method of averages; Method of least squares; Method of moments; Method of selected points; Modified exponential function; Pearl and Reed's procedure; Polynomial curves

Indexed keywords

EID: 0039477240     PISSN: 00390526     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9884.00119     Document Type: Article

References (49)

1. Broadbent, J., Clements, K. W. and Johnson, L. W. (1981) Growth in holdings of a new financial asset: a logistic analysis. Aust. J. Mangmnt, 6, 1-6.
2. Campbell, N. R. (1920) The adjustment of observations 1. Phil. Mag. Ser. 6, 39, 177-194.
3. Croxton, F. E. and Cowden, D. J. (1939) Applied General Statistics. Englewood Cliffs: Prentice Hall.
4. _ (1955) Applied General Statistics, 2nd edn. London: Pitman.
5. Dahlquist, G., Sjöberg, B. and Svenssen, S. (1968) Comparison of the method of averages with the method of least squares. Math. Computn, 22, 833-846.
6. Davis, H. T. (1941a) The Theory of Econometrics. Bloomington: Principia.
7. _ (1941b) The Analysis of Economic Time Series. Bloomington: Principia.
8. Eisenhart, C. (1963) The background and evolution of the method of least squares. International Statistical Institute Conf., Ottawa.
9. Farebrother, R. W. (1995) The statistical estimation of the standard linear model 1755-1853. In Proc. 1st Int. Tampere Sem. Linear Statistical Models and Their Applications, University of Tampere, Tampere (eds T. Pukkila and S. Puntanen), pp. 77-99.
10. _ (1994) Notes on the early history of elemental set methods. Tampere Wrkshp Matrix Methods for Statistics, Tampere, May.
11. Gauss, C. F. (1809) Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium. Hamburg: Perthes and Besser. (Engl. transl. by C. H. Davis: Little, Brown and Company, Boston, 1857).
12. _ (1811) Disquisito de elementis ellipticis palladis ex oppositionibus annorum 1803, 1804, 1805, 1807, 1808, 1809. Comment. Soc. Reg. Sci. Gotting. Rec., 1.
13. _ (1823) Anwendung der Wahrscheinlichkeitsrechnung auf Eine Aufgabe der praktische Geometrie. Astron. Nach., 1, 81-86.
14. Guest, P. G. (1951) The fitting of polynomials by the method of weighted grouping. Ann. Math. Statist., 22, 537-548.
15. _ (1954) Group methods in the fitting of polynomials to equally spaced observations. Biometrika, 41, 62-76.
16. Hald, A. (1990) A History of Probability and Statistics and Their Applications before 1750. New York: Wiley.
17. Hansen, L. P. (1982) Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029-1054.
18. Hawkins, D. M. (1993) The accuracy of elemental set approximations for regression. J. Am. Statist, Ass., 88, 580-589.
19. Hotelling, H. (1927) Differential equations subject to error and population estimates. J. Am. Statist. Ass., 22, 283-314.
20. Huntington, E. V. (1924) Curve fitting by least squares and the method of moments. In Handbook of Mathematical Statistics (ed. H. L. Rietz), pp. 62-70. New York: Houghton Mifflin.
21. Kuznets, S. S. (1930) Secular Movements in Production and Prices. Boston: Houghton Mifflin.
22. Leach, D. (1981) Re-evaluation of the logistic curve for human populations. J. R. Statist. Soc. A, 144, 94-103.
23. Mills, F. C. (1938) Statistical Methods Applied to Economics and Business, revised edn. New York: Holt.
24. _ (1965) Statistical Methods Applied to Economics and Business, 3rd edn. London: Pitman.
25. Nair, K. R. (1954) The fitting of growth curves. In Statistics and Mathematics in Biology (eds O. Kempthorne, T. A. Bancroft, J. W. Cowen and J. L. Lush). Ames: Iowa State College Press.
26. Oliver, F. R. (1964) Methods of estimating the logistic growth function. Appl. Statist., 13, 57-66.
27. _ (1966) Aspects of maximum likelihood estimation of the logistic growth function. J. Am. Statist., Ass., 61, 697-705.
28. _ (1969) Another generalisation of logistic growth functions. Econometrika, 37, 144-147.
29. _ (1981) Tractors in Spain: a further logistic analysis. J. Oper. Res. Soc., 32, 499-502.
30. _ (1982) Notes on the logistic curve for human populations. J. R. Statist. Soc. A, 145, 359-363.
31. _ (1987) Personal communication.
32. Pearl, R. (1924) Studies in Human Biology. Baltimore: Williams and Wilkins.
33. Pearl, R. and Reed, L. J. (1920) On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proc. Natn. Acad. Sci. USA, 6, 275-288.
34. Pearson, K. (1895) Contributions to the mathematical theory of evolution: I, Skew variation in homogeneous material. Phil. Trans. R. Soc. Lond. A, 186, 343-371.
35. _ (1902) On the systematic fitting of curves to observations and measurements. Biometrika, 1, 265-303.
36. Peters, C. C. and van Voorhis, W. R. (1940) Statistical Procedures and Their Mathematical Bases. New York: McGraw-Hill.
37. Plackett, R. L. (1988) Data analysis before 1750. Int. Statist. Rev., 56, 181-195.
38. Ratkowsky, D. A. (1983) Nonlinear Regression Modelling: a Unified Practical Approach. New York: Dekker.
39. Rhodes, E. C. (1940) Population mathematics - III. J. R. Statist. Soc., 103, 362-387.
40. Sandland, R. L. (1983) Mathematics and the growth of organism, some historical incidents. Math. Scient., 19, 11-30.
41. Schultz, H. (1930) The standard error of forecast from a curve. J. Am. Statist. Ass., 25, 139-185.
42. Sheynin, O. B. (1993a) The treatment of observations in early astronomy. Arch. Hist. Exact Sci., 46, 153-192.
43. _ (1993b) On the hostory of the principle of least squares. Arch. Hist. Exact Sci., 46, 39-54.
44. Stigler, S. M. (1981) Gauss and the method of least squares. Ann. Statist., 9, 465-474.
45. _ (1994) Jevons on the King-Davenant law of demand: a simple solution of a historical puzzle. Hist. Polit. Econ., 26, 185-191.
46. Stone, J. R. N. (1980) Sigmoids. Bull. Appl. Statist., 7, 59-119.
47. Tintner, G. (1952) Econometrics. New York: Wiley.
48. Verhulst, P. R. (1845) Recherches mathematiques sur la loi d'accroissement de la population. Nouv. Mem. Acad. R. Sci. Belles Lett. Brux., 18, 1-38.
49. Yule, G. U. (1925) The growth of population and the factors which control it. J. R. Statist. Soc., 88, 1-62.