# Article

**Keywords:**

nonlinear regression models; linearization domains; linearization conditions

**Summary:**

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.

References:

[1] D. M. Bates, D. G. Watts:

**Relative curvature measures of nonlinearity**. J. Roy. Statist. Soc. B 42 (1980), 1–25.

MR 0567196
[2] P. R. Halmos:

**Finite-dimensional Vector Spaces**. Springer-Verlag, New York-Heidelberg-Berlin, 1974.

MR 0409503 |

Zbl 0288.15002
[3] A. Jenčová:

**A choice of criterion parameters in linearization of regression models**. Acta Math. Univ. Comenianae, Vol LXIV, 2 (1995), 227–234.

MR 1391038
[4] L. Kubáček:

**On a linearization of regression models**. Appl. Math. 40 (1995), 61–78.

MR 1305650
[5] L. Kubáček:

**Models with a low nonlinearity**. Tatra Mountains Math. Publ. 7 (1996), 149–155.

MR 1408464
[6] A. Pázman:

**Nonlinear Statistical Models**. Kluwer Acad. Publishers, Dordrecht-Boston-London, and Ister Science Press, Bratislava, 1993.

MR 1254661