Hodges–Lehmann asymptotic efficiency of the Kolmogorov and Smirnov goodness-of-fit tests

One considers the Hodges–Lehmann asymptotic efficiency of the Kolmogorov and Smirnov goodness-of-fit tests, which measures the rate of the exponential decrease of the errors of the second kind, under a fixed significance level. It is shown that the Kolmogorov test is always asymptotically optimal in this sense, while the one-sided Smirnov test is asymptotically optimal under additional conditions imposed on the parametric family of distributions.