Predicting sets of outcomes-instead of unique outcomes-is a promising solution to uncertainty quantification in statistical learning. Despite a rich literature on constructing prediction sets with statistical guarantees, adapting to unknown covariate shift-a prevalent issue in practice-poses a serious unsolved challenge. In this article, we show that prediction sets with finite-sample coverage guarantee are uninformative and propose a novel flexible distribution-free method, PredSet-1Step, to efficiently construct prediction sets with an asymptotic coverage guarantee under unknown covariate shift. We formally show that our method is asymptotically probably approximately correct, having well-calibrated coverage error with high confidence for large samples. We illustrate that it achieves nominal coverage in a number of experiments and a data set concerning HIV risk prediction in a South African cohort study. Our theory hinges on a new bound for the convergence rate of the coverage of Wald confidence intervals based on general asymptotically linear estimators.

Chemical compositions of mushrooms are greatly dependent on the geographical region, and also the different parts of the same mushroom have different chemical constitutions. Several chemical methods are employed for quality control of mushrooms. However, these methods are destructive, require skilled personnel and are time consuming. To overcome these limitations researchers are aiming for vibrational spectroscopic techniques. This review is focused on various studies related to the application of vibrational spectroscopy for classification, authentication and quality analysis of mushrooms. It was concluded that vibrational spectroscopy could be efficiently employed for assessing the quality, authenticity and geographical origin of the mushrooms. Fourier-transform infrared (FTIR) and near infrared (NIR) spectroscopy were the most explored, whereas, Raman spectroscopy is the least explored technique in this field. Compact and cost-effective spectrometers based on the selective wavelengths have to be designed and installed at commercial and industrial level for rapid quality control of mushrooms.