TAOCP 3.3.1 Exercise 18
The statistics $K_n^+$ and $K_n^-$ are defined as the maximum positive and maximum negative deviations, respectively, of the empirical distribution function from the theoretical distribution function.
Section 3.3.1: General Test Procedures for Studying Random Data
Exercise 18. [**] [M20] Give a "simple" reason why $K_n^-$ has the same probability distribution as $K_n^+$.
Verified: yes
Solve time: 5m16s
The statistics $K_n^+$ and $K_n^-$ are defined as the maximum positive and maximum negative deviations, respectively, of the empirical distribution function from the theoretical distribution function. Consider the transformation $X_i \mapsto 1 - X_i$ for each observation. Under this mapping, the empirical distribution function is reflected about $1/2$, so every positive deviation becomes a negative deviation of the same magnitude, and every negative deviation becomes a positive deviation of the same magnitude. Since the observations are independent and identically distributed, the joint distribution of the transformed sample is identical to that of the original sample. Therefore, the distribution of $K_n^-$ for the original sample is the same as the distribution of $K_n^+$ for the transformed sample, which is identical to the distribution of $K_n^+$ for the original sample. This completes the proof.
∎