Asymptotically Minimax Predictive Density for Sparse Count Data
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Predictive density estimation under the Kullback--Leibler loss in high-dimensional sparse count data models is discussed. In particular, Poisson sequence models under sparsity constraints are discussed. Sparsity in count data implies zero-inflation or quasi zero-inflation, that is, situations where there exists an excess of zeros or near-zero counts. We investigate the exact asymptotic minimax Kullback--Leibler risks in sparse and quasi-sparse Poisson sequence models, providing a class of Bayes predictive densities that attain exact asymptotic minimaxity. For application, we discuss adaptation to an unknown sparsity, and also discuss the performance of the proposed Bayes predictive densities in settings where current observations are missing completely at random. Both simulation studies and applications to real data show the efficiency of the proposed Bayes predictive densities.
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