Coupling methods for multistage sampling

Multistage sampling is commonly used for household surveys when there exists no sampling frame, or when the population is scattered over a wide area. Multistage sampling usually introduces a complex dependence in the selection of the final units, which makes asymptotic results quite difficult to prove. In this work, we consider multistage sampling with simple random without replacement sampling at the first stage, and with an arbitrary sampling design for further stages. We consider coupling methods to link this sampling design to sampling designs where the primary sampling units are selected independently. We first generalize a method introduced by [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 5 (1960) 361-374] to get a coupling with multistage sampling and Bernoulli sampling at the first stage, which leads to a central limit theorem for the Horvitz--Thompson estimator. We then introduce a new coupling method with multistage sampling and simple random with replacement sampling at the first stage. When the first-stage sampling fraction tends to zero, this method is used to prove consistency of a with-replacement bootstrap for simple random without replacement sampling at the first stage, and consistency of bootstrap variance estimators for smooth functions of totals.Comment: Published at http://dx.doi.org/10.1214/15-AOS1348 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Exact balanced random imputation for sample survey data

Surveys usually suffer from non-response, which decreases the effective sample size. Item non-response is typically handled by means of some form of random imputation if we wish to preserve the distribution of the imputed variable. This leads to an increased variability due to the imputation variance, and several approaches have been proposed for reducing this variability. Balanced imputation consists in selecting residuals at random at the imputation stage, in such a way that the imputation variance of the estimated total is eliminated or at least significantly reduced. In this work, we propose an implementation of balanced random imputation which enables to fully eliminate the imputation variance. Following the approach in Cardot et al. (2013), we consider a regularized imputed estimator of a total and of a distribution function, and we prove that they are consistent under the proposed imputation method. Some simulation results support our findings

Large sample properties of the Midzuno sampling scheme with probabilities proportional to size

Midzuno sampling enables to estimate ratios unbiasedly. We prove the asymptotic normality for estimators of totals and ratios under Midzuno sampling. We also propose consistent variance estimators

Wave-mixing origin and optimization in single and compact aluminum nanoantennas

The outstanding optical properties for plasmon resonances in noble metal nanoparticles enable the observation of non-linear optical processes such as second-harmonic generation (SHG) at the nanoscale. Here, we investigate the SHG process in single rectangular aluminum nanoantennas and demonstrate that i) a doubly resonant regime can be achieved in very compact nanostructures, yielding a 7.5 enhancement compared to singly resonant structures and ii) the $\chi_{\perp\perp\perp}$ local surface and $\gamma_{bulk}$ nonlocal bulk contributions can be separated while imaging resonant nanostructures excited by a tightly focused beam, provided the $\chi_{\perp\parallel\parallel}$ local surface is assumed to be zero, as it is the case in all existing models for metals. Thanks to the quantitative agreement between experimental and simulated far-field SHG maps, taking into account the real experimental configuration (focusing and substrate), we identify the physical origin of the SHG in aluminum nanoantennas as arising mainly from $\chi_{\perp\perp\perp}$ local surface sources