TAOCP 3.3.1 Exercise 8
Each of the 20 values of $K_{10}^+$ was itself computed from 10 observations.
Section 3.3.1: General Test Procedures for Studying Random Data
Exercise 8. [00] The text describes an experiment in which 20 values of the statistic $K_{10}^+$ were obtained in the study of a random sequence. These values were plotted, to obtain
Fig. 4, and a KS statistic was computed from the resulting graph. Why were the table entries for $n = 20$ used to study the resulting statistic, instead of the table entries for $n = 10$?
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Each of the 20 values of $K_{10}^+$ was itself computed from 10 observations. When these 20 values are treated as a new sequence for computing a Kolmogorov-Smirnov statistic, the effective sample size is $n = 20$, because we now have 20 independent values in the sequence. Therefore the appropriate table entries correspond to $n = 20$, not $n = 10$, since the KS test depends on the number of observations in the sequence to which it is applied. Using $n = 10$ would underestimate the variability of the empirical distribution of the 20 values and give incorrect significance levels.