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The purpose of this paper is to examine the properties of the histogram when the cells are allowed to be arbitrary. Given a random sample from an unknown probability density f on I, we wish to construct a histogram. Any partition of I can be used as cells. The optimal partition minimizes the mean integrated squared error (MISE) of the histogram from f. An expression is found for the infimum of MISE over all partitions. It is proved that the infimum is attained asymptotically by minimizing MISE over a class of partitions of locally equisized cells.
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Atsuyuki Kogure (1987) studied this question.
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