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Estimation in a semiparametric partially linear errors-in-variables model. (English) Zbl 0977.62036

Summary: We consider the partially linear model relating a response $Y$ to predictors $\left(X,T\right)$ with mean function ${X}^{T}\beta +g\left(T\right)$ when the $X$’s are measured with additive error. The semiparametric likelihood estimate of T.A. Severini and J.G. Staniswalis [J. Am. Stat. Assoc. 89, No. 426, 501-511 (1994; Zbl 0798.62046)] leads to biased estimates of both the parameter $\beta$ and the function $g\left(·\right)$ when measurement error is ignored.

We derive a simple modification of their estimator whieh is a semiparametric version of the usual parametric correction for attenuation. The resulting estimator of $\beta$ is shown to be consistent and its asymptotic distribution theory is derived. Consistent standard error estimates using sandwich-type ideas are also developed.

##### MSC:
 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics 62G08 Nonparametric regression 62G20 Nonparametric asymptotic efficiency
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