Consistency of an estimator of the parameters of polynomial regression with known variance relation for the errors in the measurement of the regressor and the response. (Ukrainian, English) Zbl 1197.62027

Shklyar, S. V.

Teor. Jmovirn. Mat. Stat. 76, 160-175 (2007); translation in Theory Probab. Math. Stat. 76, 181-197 (2008).

The author deals with a functional model of the polynomial regression with Gaussian measurement errors [see C.-L. Cheng and J.W. Van Ness, Statistical regression with measurement error. London: Arnold (1999; Zbl 0947.62046) for more details.] Let \(\{\xi_i,\;i=1,2,\dots\}\) be a sequence of non-random numbers, and let \(\eta_i= \bar{\beta}_0+\bar{\beta}_1\xi_i+\bar{\beta}_2\xi_i^2+\cdots+\bar{\beta}_k\xi_i^k,\;i=1,2,\dots\) Here \(\bar{\beta}=(\bar{\beta}_0,\bar{\beta}_1,\dots,\bar{\beta}_k)\) is a parameter and \(k\geq1\) is a fixed known number. Assume that the pairs \((\xi_i,\eta_i)\) are observed with errors, that is, the pairs \((x_i,y_i),\;i=1,\dots,n\), are observed, where \(x_i=\xi_i+\delta_i,y_i=\eta_i+\varepsilon_i\). The measurement errors are independent and have the same Gaussian distribution: \((\delta_i,\varepsilon_i)^{\top}\sim N(0,\bar{\kappa}\Omega_0)\). The covariance matrix \(\bar{\kappa}\Omega_0\) is known up to the factor \(\bar{\kappa}\geq0\). The problem is to estimate the parameter \(\bar{\beta}\) from the observations.
C.-L. Cheng and H. Schneeweiss [J. R. Stat. Soc., Ser. B 60, No. 1, 189–199 (1998; Zbl 0909.62064); S. van Huffel et al. (eds.), Total least squares and errors-in-variables modeling. Analysis, algorithms and applications. Dordrecht: Kluwer, 131–143 (2002; Zbl 0995.65013)] proposed an estimator for the regression parameters for the case where the covariance matrix of the measurement errors is known up to a scalar factor. The estimator is constructed by using the method of moments. However the conditions for the consistency of this estimator are not given in these publications. In this article, sufficient conditions for the strong consistency of the proposed estimator are given and the rate of convergence is estimated.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

62F12	Asymptotic properties of parametric estimators
62J02	General nonlinear regression
62F10	Point estimation

Keywords:

polynomial regression; Gaussian error; covariance matrix; strong consistency; rate of convergence

Citations:

Zbl 0947.62046; Zbl 0909.62064; Zbl 0995.65013