Carving model-free inference
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Many scientific studies are modeled as hierarchical procedures where the starting point of data-analysis is based on pilot samples that are employed to determine parameters of interest. With the availability of more data, the scientist is tasked with conducting a meta-analysis based on the augmented data-sets, that combines explorations from the pilot stage with a confirmatory study in the second stage. Casting these two-staged procedures into a conditional framework, inference is based on a carved likelihood. Such a likelihood is obtained in Fithian et al. (2014) by conditioning the law of the augmented data upon the selection carried out on the first stage data. Though the concept of carving is more general, the theory for valid inference in this previous work is strongly tied to parametric models for the data, an example being the ubiquitous Gaussian model. Our focus in the current paper is to take a step towards model-free inference while integrating explorations with fresh samples in a data-efficient manner. Towards this goal, we provide results that validate carved inference in an asymptotic regime for a broad class of parameters.
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