Parameter estimation in nonlinear regression modelsFootnote¹

¹Paper given at the 2nd International Summer School on Problems of Model Choice and Parameter Estimation in Regression Analysis, Reinhardsbrunn (G.D.R.), November 1975.

References

• Bunke H. Bunke O. The foundation of statistical randomization 1977 To appear
   Google Scholar
• Bunke , O. 1975 . Improved inference in linear models with additional information . Math. Operationsforsch. Statist , 6 : 817 – 829 .
   Google Scholar
• Humak , K.M.S. 1976 . “ Statistische Methoden zur Modellbildung ” . In Statistische Inferenz für lineare Parameter , Vol. 1 , Berlin : Akademie-Verlag .
   Google Scholar
• Nelder , J.A. 1961 . The fitting of a generalization of the logistic curve . Biometrics , 17 : 89 – 110 .
   Web of Science ®Google Scholar
• Nelder , J.A. 1962 . An alternative form of a generalized logistic function . Biometrics , 18 : 614 – 616 .
   Web of Science ®Google Scholar
• Rasch , D. 1967 . Schätzprobleme bei eigentlich nichtlinearen Regressionsfunktionen , 121 – 128 . Abh. Dt. Akad. Wiss .
   Google Scholar
• Agha , M. 1971 . A direct method for fitting linear combinations of exponentials . Biometrics , 27 : 399 – 413 .
   Web of Science ®Google Scholar
• Goldfeld , S.M. and Quandt , R.E. 1972 . Nonlinear Methods in Econometrics , Amsterdam : North-Holland Publishing Company . London
   Google Scholar
• Goldberger , A.S. 1968 . The interpretation and estimation of Cobb-Douglas functions . Econometrica , 36 : 464 – 472 .
   Web of Science ®Google Scholar
• Box , M.J. 1971 . Bias in nonlinear estimation . J. Roy. Statist. Soc. B , 33 : 171 – 201 .
   Google Scholar
• Barham , R.H. and Drane , W. 1972 . An algorithm ior least squares estimation of nonlinear parameters when some of the parameters are linear . Technometrics , 14 : 757 – 766 .
   Web of Science ®Google Scholar
• Lawton , H. and Sylvestre , E.A. 1971 . Elimination of linear parameters in nonlinear regression . Technometrics , 13 : 461 – 467 .
   Web of Science ®Google Scholar
• Bunke , H. 1973 . Approximation of regression functions . Math. Operationsforsch. Statist , 4 : 315 – 325 .
   Google Scholar
• Bunke , H. and Bunke , O. 1974 . Identiflability and estimability . Math, Operationsforsch. Statist , 5 : 223 – 233 .
   Google Scholar
• Hoffmann, K. (1977). Approgression with random regressors. To appear.
   Google Scholar
• Petersen , J. 1973 . Linear minimax estimation for inadequate models in L 2-metric . Math. Operations forsch. Statist , 4 : 463 – 471 .
   Google Scholar
• Ermakov , S.M. 1970 . On optimal unbiased designs for regression experiments . Trudy Mat. Inst. Steklov , 111 : 303 – 309 .
   Google Scholar
• Beale , E.M.L. 1960 . Confidence regions in non-linear estimation . J. Roy. Statist. Soc. B , 22 : 41 – 76 .
   Google Scholar
• Malintvaud , E. 1970 . The consistency of nonlinear regressions . Ann, Math, Statist , 41 : 956 – 969 .
   Google Scholar
• Jennrich , R.I. 1969 . Asymptotic properties of non-linear least squares estimators . Ann, Math, Statist , 40 : 633 – 643 .
   Google Scholar
• Himmelblau , D.M. 1970 . Process Analysis by Statistical Method New York
   Google Scholar
• Chambers , J.M. 1973 . Fitting nonlinear models: numerical techniques . Biomeirika , 60 : 1 – 13 .
   Web of Science ®Google Scholar
• Hartley , H.O. and Booker , A. 1965 . Nonlinear least squares estimation . Ann. Math. Statist , 36 : 638 – 650 .
   Google Scholar
• Hooke , R. and Jeeves , T.V. 1961 . Direct search solution of numerical and statistical problems . J. Association of Computing Machinery , 8 : 212 – 221 .
   Google Scholar
• Spendley , N. , Hext , G.R. and Himsworth , F.R. 1962 . Technometrics , 4 : 441
   Web of Science ®Google Scholar
• Nelder , J.A. and Mead , R. 1965 . Computer-Journal , 7 : 308
   Web of Science ®Google Scholar
• Hartley , H.O. 1961 . The modified Gauss-Newton method for the fitting of nonlinear regression functions by leastsquares . Technometrics , 3 : 269 – 280 .
   Web of Science ®Google Scholar
• Box , M.J. 1966 . A comparison of several current optimization methods and the use of transformations in constrained problems . Computer-Journal , 9 : 67 – 77 .
   Web of Science ®Google Scholar
• Fletcher , R. 1965 . Function minimization without evaluating derivatives – a review . Computer-Journal , 8 : 33 – 41 .
   Web of Science ®Google Scholar
• Fletcher , R. 1971 . A general quadratic programming algorithm . J. Inst. Math. Appl , 7 : 76 – 91 .
   Google Scholar
• Powell , M.J.D. 1970 . A survey of numerical methods for unconstrained optimization . S.I.A.M. Rev , 12 : 79 – 97 .
   Web of Science ®Google Scholar
• Marquardt , D.W. 1963 . An algorithm for least squares estimation of nonlinear parameters . S.I.A.M. J. Appl. Math , 11 : 431 – 441 .
   Web of Science ®Google Scholar
• Marquardt , D.W. 1970 . Generalized inverses, ridge regression, biased linear estimation and nonlinear estimation . Technometrics , 12 : 591 – 612 .
   Web of Science ®Google Scholar
• Henschke, K. (1977). Confidence regions for linear models with additional information, to appear
   Google Scholar
• Williams , E.J. 1962 . Exact fiducial limits in non-linear estimation . J. Roy. Statist. Soc, B , 24 : 125 – 139 .
   Google Scholar
• Halperin , M. 1963 . Confidence interval estimation in nonlinear regression . J. Roy Statist. Soc. B , 25 : 330 – 333 .
   Google Scholar
• Box , G.E.P. and Coutie , G.A. 1963 . Application of digital computers in the exploration of functional relationship . Proc. I. E. E. Part B , 25 ( 1 ) : 100 – 107 .
   Google Scholar
• Box , G.E.P. and Cox , D.R. 1964 . An analysis of transformations . J. Roy. Statist. Soc. B , 26 ( 1 ) : 211 – 252 .
   Google Scholar
• Cox , D.R. and Snell , E.J. 1968 . A general definition of residuals . J. Roy. Statist. Soc, B , 80 ( 1 ) : 248 – 275 . with discussion
   Google Scholar
• Cox , D.R. 1971 . On test statistics calculated from residuals . Biometrika , 58 ( 1 ) : 589 – 594 .
   Web of Science ®Google Scholar
• Lawton , W.H. , Sylvestre , E.A. and Maggio , M.S. 1972 . Self modeling nonlinear regression . Technometrics , 14 ( 1 ) : 513 – 532 .
   Web of Science ®Google Scholar
• Draper , N.R. and Smith , H. 1966 . Applied Regression Analysis , New York : Wiley .
   Google Scholar
• Ross , G.J.S. 1971 . The efficient use of function minimization in nonlinear maximum likelihood estimation . Appl.Statist , 20 : 205 – 221 .
   Google Scholar
• Box , G.E.P. and Tidwell , P.W. 1962 . Transformation of the independent variables . Technometrics , 4 : 531 – 550 .
   Web of Science ®Google Scholar
• Box , G.E.P. and Hunter , W.G. 1962 . A useful method for model building . Technometrics , 4 : 301 – 318 .
   Web of Science ®Google Scholar
• Bunke, H. (1977). Simple consistent estimation in nonlinear regression by data transformation and design of experiments, to appear.
   Google Scholar
• Jones , A. 1970 . Spiral-A new algorithm for nonlinear parameter estimation using least squares . Computer-Journal , 13 : 301 – 307 .
   Web of Science ®Google Scholar
• Nagel , G. and Wolff , W. 1974 . Ein Yerfahren zur Minimierung einer Quadrat-summe nichtlinearer Funktionen . Biomeir. Z , 16 : 431 – 439 .
   Google Scholar