Covariate shift occurs when the distribution of input features differs between the training and testing phases. In covariate shift, estimating an unknown function's moment is a classical problem that remains under-explored, despite its common occurrence in real-world scenarios. In this paper, we investigate the minimax lower bound of the problem when the source and target distributions are known. To achieve the minimax optimal bound (up to a logarithmic factor), we propose a two-stage algorithm. Specifically, it first trains an optimal estimator for the function under the source distribution, and then uses a likelihood ratio reweighting procedure to calibrate the moment estimator. In practice, the source and target distributions are typically unknown, and estimating the likelihood ratio may be unstable. To solve this problem, we propose a truncated version of the estimator that ensures double robustness and provide the corresponding upper bound. Extensive numerical studies on synthetic examples confirm our theoretical findings and further illustrate the effectiveness of our proposed method.

MOTIVATION: The development of next generation sequencing technology provides an efficient and powerful approach to rare variant detection. To identify genetic variations, the essential question is how to quantity the sequencing error rate in the data. Because of the advantage of easy implementation and the ability to integrate data from different sources, the empirical Bayes method is popularly employed to estimate the sequencing error rate for SNP detection. RESULTS: We propose a novel statistical model to fit the observed non-reference allele frequency data, and utilize the empirical Bayes method for both genotyping and SNP detection, where an ECM algorithm is implemented to estimate the model parameters. The performance of our proposed method is investigated via simulations and real data analysis. It is shown that our method makes less genotype-call errors, and with the parameter estimates from the ECM algorithm, it attains high detection power with FDR being well controlled. AVAILABILITY AND IMPLEMENTATION: The proposed algorithm is wrapped in the R package ebGenotyping, which can be downloaded from http://cran.r-project.org/web/packages/ebGenotyping/ CONTACT: [email protected] information: Supplementary data are available at Bioinformatics online.