Estimation for the nonlinear errors-in-variables model

Estimation of the parameters of the functional nonlinear measurement error model is considered. A simulation bias adjusted (SIMBA) estimation procedure is presented. In the SIMBA procedure, internal Monte Carlo simulation based on the sample data is used to adjust a naive estimator, such as the ordinary least squares estimator, for bias. Let the measurement error variance s2un be a sequence depending on the sample size n, and assume s2un → 0 as n → infinity. Under some regularity conditions, the order in probability convergence rate for the SIMBA estimator is max s4un , n-1/2, while the order in probability convergence rate for the ordinary least squares estimator is max s2un , n-1/2. Monte Carlo simulation is conducted to test the performance of SIMBA for four models: linear model, quadratic model, cosine model and logistic model. Monte Carlo simulation shows that the SIMBA estimation procedure outperforms or is comparable to methods such as simulation extrapolation, regression calibration and adjusted least squares. An example application of SIMBA estimation for the logistic regression model with errors in variables is given. In the example, the relation between minerals from dietary intake and the supplement use for people over 50 is studied. The data are from the two surveys: the Third National Health and Nutrition Examination Survey and the related Supplemental Nutrition Survey. One interesting result is that people whose dietary intake of minerals is high are more likely to take supplements