1 Some Basic Theory 1 1.1 Consistency and Unbiasedness at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Kolmogorov–Smirnov Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Proof of the Kolmogorov–Smirnov Theorem . . ....

3 Statistical methods 4 3.1 Wald-Wolfowitz test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Kolmogorov-Smirnov tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Mean deviation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Aggregated F-test . . . . . . ....

We construct a Kolmogorov-Smirnov test for the null hypothesis that the average treatment effect is non-negative conditional on all possible values of the covariates. The null hypothesis of our interest can be characterized as a conditional moment inequality under the unconfoundedness assumption, and we employ the instrumental variable method to convert the conditional moment inequality into un...

The Pareto distribution model assumption in the peaks over threshold method, will be tested by making using of the Kolmogorov-Smirnov goodness of fit method. Pareto distributed variables can be transformed to exponential, and the test will be for exponentiality. It was found that the statistic can be used as an indication of where to choose the threshold and to check the Pareto model assumption.