Sample design for analysis using high-influence probability sampling

Sample designs are typically developed to estimate summary statistics such as means, proportions and prevalences. Analytical outputs may also be a priority but there are fewer methods and results on how to efficiently design samples for the fitting and estimation of statistical models. This paper develops a general approach for determining efficient sampling designs for probability-weighted maximum likelihood estimators and considers application to generalized linear models. We allow for non-ignorable sampling, including outcome-dependent sampling. The new designs have probabilities of selection closely related to influence statistics such as dfbeta and Cook's distance. The new approach is shown to perform well in a simulation based on data from the New Zealand Health Survey.