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nonlinear regression model; linearization; quadratization
In nonlinear regression models an approximate value of an unknown parameter is frequently at our disposal. Then the linearization of the model is used and a linear estimate of the parameter can be calculated. Some criteria how to recognize whether a linearization is possible are developed. In the case that they are not satisfied, it is necessary to take into account either some quadratic corrections or to use the nonlinear least squares method. The aim of the paper is to find some criteria for an ordering linear and quadratic estimators.
[1] D. M.  Bates, D. G.  Watts: Relative curvature measures of nonlinearity. J.  Roy. Statist. Soc. Ser.  B 42 (1980), 1–25. MR 0567196
[2] J. P.  Imhof: Computing the distribution of quadratic forms in normal variables. Biometrika 48 (1961), 419–426. DOI 10.1093/biomet/48.3-4.419 | MR 0137199 | Zbl 0136.41103
[3] A.  Jenčová: A comparison of linearization and quadratization domains. Appl. Math. 42 (1997), 279–291. DOI 10.1023/A:1023064412279 | MR 1453933
[4] L.  Kubáček: On a linearization of regression models. Appl. Math. 40 (1995), 61–78. MR 1305650
[5] L.  Kubáček, L.  Kubáčková, J.  Volaufová: Statistical Models with Linear Structures. Veda, Bratislava, 1995.
[6] L.  Kubáček: Models with a low nonlinearity. Tatra Mt. Math. Publ. 7 (1996), 149–155. MR 1408464
[7] L.  Kubáček: Quadratic regression models. Math. Slovaca 46 (1996), 111–126. MR 1414414
[8] L.  Kubáček: Corrections of estimators in linearized models. Acta Univ. Palack. Olomuc., Fac. Rerum Math. 37 (1998), 69–80. MR 1690475
[9] L. Kubáček, L.  Kubáčková: Regression models with a weak nonlinearity. Technical Reports. (1998), University of Stuttgart, 1–64.
[10] P. B.  Patnaik: The non-central $\chi ^2$ and $F$-distributions and their applications. Biometrika 36 (1949), 202–232. MR 0034564
[11] A.  Pázman: Nonlinear Statistical Models. Kluwer Academic Publishers, DordrechtBoston-London and Ister Science Press, Bratislava, 1993. MR 1254661
[12] R.  Potocký, To Van Ban: Confidence regions in nonlinear regression models. Appl. Math. 37 (1992), 29–39. MR 1152155
[13] F. E. Satterthwaite: An approximate distribution of estimates of variance components. Biometrics Bulletin 2 (1946), 110-114. DOI 10.2307/3002019
[14] E.  Tesaříková, L. Kubáček: How to deal with regression models with a weak nonlinearity. Discuss. Math. Probab. Stat. 21 (2001), 21–48. DOI 10.7151/dmps.1018 | MR 1868926
[15] B. L.  Welch: The generalization of Student’s problem when several different population variances are involved. Biometrika 34 (1947), 28–35. MR 0019277 | Zbl 0029.40802
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