The weighted ridge estimator in stochastic restricted linear measurement error models
The linear measurement error model \(y=Z\beta+\epsilon,~ X=Z+\Delta\) is considered, where \(y\) is a vector of response variables, \(Z\) is a matrix of unobservable values of regressors which can be observed through the matrix \(X\) with the measurement error \(\Delta,~ \beta\) is a vector of regression coefficients, and \(\epsilon\) is a vector of random errors. The covariance structure of the measurement errors is known, and the covariance matrix of \(\epsilon\) is known up to a scalar factor \(\sigma^2\). The vector \(\beta\) is subject to linear restrictions \(r=R\beta+e,\) where \(r\) is an observable random vector, \(R\) is a known matrix and \(e\) is an error vector, with covariance matrix known up to the scalar factor \(\sigma^2\). The random matrix \(\Delta\) and the random vectors \(\epsilon,~e\) are mutually independent. \textit{B. Schaffrin} and \textit{H. Toutenburg} [Z. Angew. Math. Mech. 70, No. 6, 735--738 (1990; Zbl 0714.62061)] developed the weighted mixed estimator (WME) by assigning not necessarily equal weights in the sample and prior information. In the paper under review, the weighted mixed ridge estimator (WMRE) is introduced. The performance of the WMRE against the WME is examined in terms of the asymptotic mean squared error matrix related to asymptotic normal distributions of the two estimators. A simulation study and a numerical example are given to illustrate the theoretical results.
scientific article; zbMATH DE number 6912431
scientific article