Bias of the Quasi Score Estimator of a Measurement Error Model Under Misspecification of the Regressor Distribution

In a structural error model the structural quasi score (SQS) estimator is based on the distribution of the latent regressor variable. If this distribution is misspecified the SQS estimator is (asymptotically) biased. Two types of misspecification are considered. Both assume that the statistician erroneously adopts a normal distribution as his model for the regressor distribution. In the first type of misspecification the true model consists of a mixture of normal distributions which cluster round a single normal distribution, in the second type the true distribution is a normal distribution admixed with a second normal distribution of low weight. In both cases of misspecification the bias, of course, tends to zero when the size of misspecification tends to zero. However, in the first case the bias goes to zero very fast so that small deviations from the true model lead only to a negligible bias, whereas in the second case the bias is noticeable even for small deviations from the true model

Nonparametric Inference via Bootstrapping the Debiased Estimator

In this paper, we propose to construct confidence bands by bootstrapping the debiased kernel density estimator (for density estimation) and the debiased local polynomial regression estimator (for regression analysis). The idea of using a debiased estimator was recently employed by Calonico et al. (2018b) to construct a confidence interval of the density function (and regression function) at a given point by explicitly estimating stochastic variations. We extend their ideas of using the debiased estimator and further propose a bootstrap approach for constructing simultaneous confidence bands. This modified method has an advantage that we can easily choose the smoothing bandwidth from conventional bandwidth selectors and the confidence band will be asymptotically valid. We prove the validity of the bootstrap confidence band and generalize it to density level sets and inverse regression problems. Simulation studies confirm the validity of the proposed confidence bands/sets. We apply our approach to an Astronomy dataset to show its applicabilityComment: Accepted to the Electronic Journal of Statistics. 64 pages, 6 tables, 11 figure

Polyelectrolyte Adsorption on Charged Substrate

The behavior of a polyelectrolyte adsorbed on a charged substrate of high-dielectric constant is studied by both Monte-Carlo simulation and analytical methods. It is found that in a low enough ionic strength medium, the adsorption transition is first-order where the substrate surface charge still keeps repulsive. The monomer density at the adsorbed surface is identified as the order parameter. It follows a linear relation with substrate surface charge density because of the electrostatic boundary condition at the charged surface. During the transition, the adsorption layer thickness remains finite. A new scaling law for the layer thickness is derived and verified by simulation.Comment: Proceedings of the 3rd Symposium on Slow Dynamics in Complex Systems, 3-8 November 2003, Sendai, Japa