Linearization conditions for regression models with unknown variance parameter

Anna Jenčová

Applications of Mathematics (2000)

  • Volume: 45, Issue: 2, page 145-160
  • ISSN: 0862-7940

Abstract

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In the case of the nonlinear regression model, methods and procedures have been developed to obtain estimates of the parameters. These methods are much more complicated than the procedures used if the model considered is linear. Moreover, unlike the linear case, the properties of the resulting estimators are unknown and usually depend on the true values of the estimated parameters. It is sometimes possible to approximate the nonlinear model by a linear one and use the much more developed linear methods, but some procedure is needed to recognize such situations. One attempt to find such a procedure, taking into account the requirements of the user, is given in , , , where the existence of an a priori information on the parameters is assumed. Here some linearization criteria are proposed and the linearization domains, i.e. domains in the parameter space where these criteria are fulfilled, are defined. The aim of the present paper is to use a similar approach to find simple conditions for linearization of the model in the case of a locally quadratic model with unknown variance parameter σ 2 . Also a test of intrinsic nonlinearity of the model and an unbiased estimator of this parameter are derived.

How to cite

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Jenčová, Anna. "Linearization conditions for regression models with unknown variance parameter." Applications of Mathematics 45.2 (2000): 145-160. <http://eudml.org/doc/33053>.

@article{Jenčová2000,
abstract = {In the case of the nonlinear regression model, methods and procedures have been developed to obtain estimates of the parameters. These methods are much more complicated than the procedures used if the model considered is linear. Moreover, unlike the linear case, the properties of the resulting estimators are unknown and usually depend on the true values of the estimated parameters. It is sometimes possible to approximate the nonlinear model by a linear one and use the much more developed linear methods, but some procedure is needed to recognize such situations. One attempt to find such a procedure, taking into account the requirements of the user, is given in , , , where the existence of an a priori information on the parameters is assumed. Here some linearization criteria are proposed and the linearization domains, i.e. domains in the parameter space where these criteria are fulfilled, are defined. The aim of the present paper is to use a similar approach to find simple conditions for linearization of the model in the case of a locally quadratic model with unknown variance parameter $\sigma ^2$. Also a test of intrinsic nonlinearity of the model and an unbiased estimator of this parameter are derived.},
author = {Jenčová, Anna},
journal = {Applications of Mathematics},
keywords = {nonlinear regression models; linearization domains; linearization conditions; nonlinear regression models; linearization domains; linearization conditions},
language = {eng},
number = {2},
pages = {145-160},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linearization conditions for regression models with unknown variance parameter},
url = {http://eudml.org/doc/33053},
volume = {45},
year = {2000},
}

TY - JOUR
AU - Jenčová, Anna
TI - Linearization conditions for regression models with unknown variance parameter
JO - Applications of Mathematics
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 45
IS - 2
SP - 145
EP - 160
AB - In the case of the nonlinear regression model, methods and procedures have been developed to obtain estimates of the parameters. These methods are much more complicated than the procedures used if the model considered is linear. Moreover, unlike the linear case, the properties of the resulting estimators are unknown and usually depend on the true values of the estimated parameters. It is sometimes possible to approximate the nonlinear model by a linear one and use the much more developed linear methods, but some procedure is needed to recognize such situations. One attempt to find such a procedure, taking into account the requirements of the user, is given in , , , where the existence of an a priori information on the parameters is assumed. Here some linearization criteria are proposed and the linearization domains, i.e. domains in the parameter space where these criteria are fulfilled, are defined. The aim of the present paper is to use a similar approach to find simple conditions for linearization of the model in the case of a locally quadratic model with unknown variance parameter $\sigma ^2$. Also a test of intrinsic nonlinearity of the model and an unbiased estimator of this parameter are derived.
LA - eng
KW - nonlinear regression models; linearization domains; linearization conditions; nonlinear regression models; linearization domains; linearization conditions
UR - http://eudml.org/doc/33053
ER -

References

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  1. Relative curvature measures of nonlinearity, J. Roy. Statist. Soc. B 42 (1980), 1–25. (1980) MR0567196
  2. Finite-dimensional Vector Spaces, Springer-Verlag, New York-Heidelberg-Berlin, 1974. (1974) Zbl0288.15002MR0409503
  3. A choice of criterion parameters in linearization of regression models, Acta Math. Univ. Comenianae, Vol LXIV, 2 (1995), 227–234. (1995) MR1391038
  4. On a linearization of regression models, Appl. Math. 40 (1995), 61–78. (1995) MR1305650
  5. Models with a low nonlinearity, Tatra Mountains Math. Publ. 7 (1996), 149–155. (1996) MR1408464
  6. Nonlinear Statistical Models, Kluwer Acad. Publishers, Dordrecht-Boston-London, and Ister Science Press, Bratislava, 1993. (1993) MR1254661

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