Abstract
In this article, we develop estimation approaches for nonparametric multiple regression measurement error models when both independent validation data on covariables and primary data on the response variable and surrogate covariables are available. An estimator which integrates Fourier series estimation and truncated series approximation methods is derived without any error model structure assumption between the true covariables and surrogate variables. Most importantly, our proposed methodology can be readily extended to the case that only some of covariates are measured with errors with the assistance of validation data. Under mild conditions, we derive the convergence rates of the proposed estimators. The finite-sample properties of the estimators are investigated through simulation studies.
Highlights
We can consider the following nonparametric regression model of a scaler response Y on an explanatory variable X =Y g ( X ) + ε, (1)where g (⋅) is assumed to be a smooth, continuous but unknown nonparametric regression function and ε is ( ) a noise variable with E (ε | X ) = 0 and E ε 2 < ∞
It is not uncommon that the explanatory variable X is measured with error and instead only its surrogate variable W can be observed
The missing information for the statistical inference will be taken from a sample Wj, X j, N +1 ≤ j ≤ N + n, of so-call validation data independent of the primary sample
Summary
[17] develops a kernel-based approach for nonparametric regression function estimation with surrogate data and validation sampling. His method is not applicable for model (1) since it assumes that the response but not the covariable is measured with error. [20] [21] consider kernel-based estimators while [22] and [23] develop series or sieve estimators Their methods require an instrumental variable, and assume that the explanatory variable X is directly observable without errors. We replace the infinite-dimensional operator T by the finite-dimensional approximation to avoid higher-order coefficient estimation, and it develops an estimator of g We extend this method to the case that only some of covariates are measured with errors.
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