Nonlinear regression models with increasing numbers of unknown parameters

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Regression models with an increasing number of unknown parameters and with unknown and different observation error variances are of interest in important applications. The reason is that, with an increased number of unknown parameters, the unknown function can be approximated more accurately in experiments. Moreover, in some applications, repeated tests at a single point are costly (financially and technically), which hampers the estimation of the unknown error variance, which is different at different observation points. Regression models have been addressed in numerous publications [1‐3], but models with an increasing number of unknown parameters have received little attention, which motivates our interest in this subject. In [4, 5] various approaches were suggested for estimating the covariance matrix of the deviation vector of the unknown parameter for linear models of finite dimension. Linear regression models of an increasing dimension were studied in [6], where necessary and sufficient conditions were found for the consistency of least squares estimators and the covariance matrix of the deviation vector. In this paper, an approach is suggested for determining least squares estimators and covariance matrix elements. The results can be used to construct a confidence interval for the unknown function in nonlinear regression models with an increasing number of unknown parameters.