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Inference from probability and nonprobability samples
Published in Uwe Engel, Anabel Quan-Haase, Sunny Xun Liu, Lars Lyberg, Handbook of Computational Social Science, Volume 2, 2021
Rebecca Andridge, Richard Valliant
Estimation of model parameters often requires solving a set of estimating equations for the parameter estimates. The estimating equations can be linear in the parameters, as for linear regression or nonlinear, as for generalized linear models. In design-based finite population estimation, the estimating equations include survey weights and are estimators of types of finite population totals (Binder & Roberts, 2009). If weights are constructed for a nonprobability sample that are appropriate for estimating totals, then those weights can also be used in the estimating equations. Consequently, weight construction for nonprobability samples can play the same role in estimation as in probability sampling.
Inference in High-Dimensional Graphical Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
Jana Janková, Sara van de Geer
The idea of using regularized estimators for construction of asymptotically normal estimators is based on removing the bias associated with the penalty. Consider a real-valued loss function ρΘ $ \rho _{\Theta } $ and let Rn(Θ):=∑i=1nρΘ(Xi)/n $ R_n(\Theta ):=\sum _{i=1}^n \rho _{\Theta }(X^i)/n $ denote the average risk, given an independent sample X1,⋯,Xn. $ X^1,\dots ,X^n. $ Under differentiability conditions, a regularized M-estimator Θ^ $ \hat{\Theta } $ based on the risk function Rn $ R_n $ can often be characterized by estimating equations R˙n(Θ^)+ξ(Θ^)=0, $$ \begin{aligned} \dot{R}_n({\hat{\Theta }}) + \xi (\hat{\Theta })=0, \end{aligned} $$
Inference for Errors-in-Variables Models in the Presence of Systematic Errors with an Application to a Satellite Remote Sensing Campaign
Published in Technometrics, 2019
Bohai Zhang, Noel Cressie, Debra Wunch
Our strategy is as follows. We propose a two-stage estimation procedure: In the first stage (Section 3.2), the variances of random errors in Xi and Yi are estimated from datasets of individual observations. In the second stage (Section 3.3), regression coefficients a and b and the systematic-error variance τ2y are estimated from the regression data {(Xi, Yi): i = 1, …, N}. At each estimation stage, we substitute parameter estimates obtained from the previous stage into the estimating equations as if they were known. Thus, estimation at the second stage can be seen as pseudo maximum likelihood estimation (Gong and Samaniego 1981).