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Summary
A somewhat general class of situations, that include Kolmogorov-Smirnov type results as special cases, is considered. These situations, which are described in the following sections, are required to have uniquely determined probability properties when the sample values used are from continuous populations of any nature. If the populations sampled are discrete, however, these probability values are not uniquely determined. This paper shows that the values for the continuous case represent bounds for the values that occur in any discrete case. The method used to show that these bound relations hold consists in noting that any discrete data situation can be interpreted as a situation involving the grouping of continuous data. Then bound relationships are established between the values of probabilities for the grouped data situations and the corresponding ungrouped data situations, which are the situations considered for the case of the continuous data. These bounds on probabilities for discrete data cases should be useful for practical applications. In practice, all data are discrete (due to limitations in measurement accuracy).
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- John E. Walsh
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Walsh, J.E. Bounded probability properties of Kolmogorov-Smirnov and similar statistics for discrete data.Ann Inst Stat Math15, 153–158 (1963). https://doi.org/10.1007/BF02865912
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