## Simex estimator for polynomial errors-in-variables model.(English)Zbl 1153.62051

The authors consider the polynomial errors-in-variables model of order $$m\geq 1$$, $\begin{cases}{y_{i}=\beta_{0}+\beta_{1}\xi_{i}+\dots+\beta_{m}\xi_{i}^ {m}+\varepsilon_{i},}\\ {x_{i}=\xi_{i}+\delta_{i},}\quad i=\overline{1,n}.\end{cases}$ Here $$\{\xi_{i}, i\geq 1\}$$, $$\{\varepsilon_{i}, i\geq 1\}$$, $$\{\delta_{i}, i\geq 1\}$$ are i.i.d. and mutually independent sequences. It is assumed that $$E|\xi_{1}|^{m}<\infty$$, $$\delta_1\sim N(0,\sigma^2_{\delta})$$, $$\sigma^{2}_{\delta}$$ is known, $$E[\varepsilon_1]=0$$, $$E[\varepsilon_1^2]<\infty$$. The variances of $$\xi_1$$, $$\varepsilon_1$$, and $$\delta_1$$ are supposed to be positive. The key idea underlying Simex is the fact that the effect of measurement errors on an estimator can be determined experimentally via simulations. This is achieved by studying the naive regression estimator as a function $$f$$ of the measurement error variance in the regressors.
The purpose of this paper is to construct a consistent Simex estimator of the regression parameters. The observed variables are used for modelling the function $$f$$. Simulation studies show that for finite samples the Simex estimator in polynomial regression can sometimes produce extremely large estimation errors as well as the OLS estimator. It is proposed how to modify this estimator for small samples still preserving its asymptotic properties.

### MSC:

 62J02 General nonlinear regression 62F12 Asymptotic properties of parametric estimators 65C60 Computational problems in statistics (MSC2010)

### Keywords:

Hermite polynomials