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A sequential design problem for rank aggregation is commonly encountered in psychology, politics, marketing, sports, etc. In this problem, a decision maker is responsible for ranking K items by sequentially collecting noisy pairwise comparisons from judges. The decision maker needs to choose a pair of items for comparison in each step, decide when to stop data collection, and make a final decision after stopping based on a sequential flow of information. Because of the complex ranking structure, existing sequential analysis methods are not suitable. In this paper, we formulate the problem under a Bayesian decision framework and propose sequential procedures that are asymptotically optimal. These procedures achieve asymptotic optimality by seeking a balance between exploration (i.e., finding the most indistinguishable pair of items) and exploitation (i.e., comparing the most indistinguishable pair based on the current information). New analytical tools are developed for proving the asymptotic results, combining advanced change of measure techniques for handling the level crossing of likelihood ratios and classic large deviation results for martingales, which are of separate theoretical interest in solving complex sequential design problems. A mirror-descent algorithm is developed for the computation of the proposed sequential procedures. 

Abstract We revisit a singular value decomposition (SVD) algorithm given in Chen et al. (Psychometrika 84:124–146, 2019b) for exploratory item factor analysis (IFA). This algorithm estimates a multidimensional IFA model by SVD and was used to obtain a starting point for joint maximum likelihood estimation in Chen et al. (2019b). Thanks to the analytic and computational properties of SVD, this algorithm guarantees a unique solution and has computational advantage over other exploratory IFA methods. Its computational advantage becomes significant when the numbers of respondents, items, and factors are all large. This algorithm can be viewed as a generalization of principal component analysis to binary data. In this note, we provide the statistical underpinning of the algorithm. In particular, we show its statistical consistency under the same double asymptotic setting as in Chen et al. (2019b). We also demonstrate how this algorithm provides a scree plot for investigating the number of factors and provide its asymptotic theory. Further extensions of the algorithm are discussed. Finally, simulation studies suggest that the algorithm has good finite sample performance.