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**Polynomial tapered two-stage least squares method in nonlinear regression.**
*(English)*
Zbl 1290.62048

Summary: Nonlinear models play an important role in various scientific disciplines and engineering. The parameter estimation of these models should be efficient to make better decisions. Ordinary least squares (OLS) method is used for estimating the parameters of nonlinear regression models when all regression assumptions are satisfied. If there is a problem with these assumptions, OLS fails to give efficient results. This paper examines the efficiency of parameter estimation under the problem of autocorrelated errors. Some methods have been proposed in order to overcome the problem and obtain efficient parameter estimates especially for autoregressive (AR) processes. One of the most commonly used method is two-stage least squares (2SLS). This method is based on generalized least squares. In this paper, a novel approach is proposed for 2SLS method by evaluating a polynomial tapering procedure on autocorrelated errors. This new method is called tapered two-stage least squares (T2SLS). The finite sample properties and improvements of T2SLS are explored by means of some real life examples and a Monte Carlo simulation study. Both numerical and experimental results reveal that T2SLS can give more efficient parameter estimates especially in small samples under the autocorrelation problem when compared to OLS and 2SLS.

### MSC:

62J02 | General nonlinear regression |

62J12 | Generalized linear models (logistic models) |

62L12 | Sequential estimation |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

### Keywords:

nonlinear regression; autocorrelation; autoregressive process; two-stage least squares; polynomial taper### Software:

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\textit{B. Aşıkgil} and \textit{A. Erar}, Appl. Math. Comput. 219, No. 18, 9743--9754 (2013; Zbl 1290.62048)

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### References:

[1] | Gallant, A. R., Nonlinear statistical models, (1987), John Wiley and Sons New York · Zbl 0611.62071 |

[2] | Glasbey, C. A., Correlated residuals in nonlinear regression applied to growth data, Applied Statistics, 28, 251-259, (1979) |

[3] | Glasbey, C. A., Nonlinear regression with autoregressive time series errors, Biometrics, 36, 135-139, (1980) · Zbl 0424.62063 |

[4] | Glasbey, C. A., Examples of regression with serially correlated errors, The Statistician, 37, 277-291, (1988) |

[5] | Bender, R.; Heinemann, L., Fitting nonlinear regression models with correlated errors to individual pharmacodynamic data using SAS software, Journal of Pharmacokinetics and Biopharmaceutics, 23, 87-100, (1995) |

[6] | Gallant, A. R.; Goebel, J. J., Nonlinear regression with autocorrelated errors, Journal of the American Statistical Association, 71, 961-967, (1976) · Zbl 0337.62046 |

[7] | Seber, G. A.F.; Wild, C. J., Nonlinear regression, (1989), John Wiley and Sons New York · Zbl 0721.62062 |

[8] | Hartley, H. O., The modified Gauss-Newton method for the Fitting of nonlinear regression functions by least squares, Technometrics, 3, 269-280, (1961) · Zbl 0096.34603 |

[9] | Marquardt, D. W., An algorithm for least squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics, 11, 431-441, (1963) · Zbl 0112.10505 |

[10] | Enders, W., Applied econometric time series, (1995), John Wiley and Sons New York |

[11] | B. Aşıkgil, A. Erar, Modified two-stage least squares method, in: Proceedings of the XIIIth International Conference ASMDA, Vilnius, 2009, pp. 124−128. |

[12] | Haykin, S., Adaptive filter theory, (1991), Prentice-Hall New Jersey · Zbl 0723.93070 |

[13] | Dahlhaus, R., Small sample effects in time series analysis: a new asymptotic theory and a new estimate, The Annals of Statistics, 16, 808-841, (1988) · Zbl 0662.62100 |

[14] | Brockwell, P. J.; Dahlhaus, R., Generalized Levinson-durbin and burg algorithms, Journal of Econometrics, 118, 129-149, (2004) · Zbl 1033.62091 |

[15] | B. Aşıkgil, Nonlinear regression in the presence of autocorrelated disturbances, unpublished Ph.D. thesis, Mimar Sinan F.A. University, 2009. |

[16] | Özer, İ.; Çağlar, A., Protein-mediated nonphotochemical bleaching of malachite Green in aqueous solution, Dyes and Pigments, 54, 11-16, (2002) |

[17] | Bates, D. M.; Watts, D. G., Nonlinear regression analysis and its applications, (1988), John Wiley and Sons New York · Zbl 0728.62062 |

[18] | Newman, M. E.J., Power laws Pareto distributions and zipf’s law, Contemporary Physics, 46, 323-351, (2005) |

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