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We propose a kernel density based estimation for nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. For the hypothesis testing of parametric components, restricted estimators under the null hypothesis and test statistics are proposed. The asymptotic properties for the estimators and test statistics are established. A bootstrap procedure is proposed to calculate critical values. Lastly, we make some comparisons between the proposed estimator and nonlinear least squares estimator through the simulation. Our simulation studies indicate that the kernel density based estimator provides a better alternative under some settings, particularly for multimodal distribution of the error term. [ABSTRACT FROM AUTHOR]

We consider the estimation and hypothesis test for single-index model by proposing a profile mean squares relative error estimation method, coupling with a leave-one-component-out method. The asymptotic normality for the PMSRE estimator is studied. For the hypothesis test of the single-index parameter, a restricted estimator under the null hypothesis and test statistics are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Lastly, we make some comparisons between the proposed estimator and the profile least squares estimator and the profile least product relative estimator through the simulation. Our simulation studies indicate that the profile mean squares relative error estimator provides a better alternative under some settings, particularly for multimodal or asymmetric distributions of the error term. [ABSTRACT FROM AUTHOR]