Question

5. Using the Kolmogorov-Smirnov two-sample test, show that the t statistic with large degrees of freedom can be approximated by the standard normal distribution. How large should the degrees of freedom be? Hint: Do simulations on a large sample.

Answer 1

The Kolmogorov-Smirnov test is defined by: H0: The data follow a specified distribution Ha: The data do not follow the specified distribution Test Statistics: The Kolmogorov-Smirnov test statistic is defined as D=max1≤i≤N(F(Yi)−i−1N,iN−F(Yi)) where F is the theoretical cumulative distribution of the distribution being tested which must be a continuous distribution (i.e., no discrete distributions such as the binomial or Poisson), and it must be fully specified (i.e., the location,scale, and shape parameters cannot be estimated from the data). Significant values: α The hypothesis regarding the distributional form is rejected if the test statistic, D, is greater than the critical value obtained from a table. There are several variations of these tables in the literature that use somewhat different scalings for the K-S test statistic and critical regions. These alternative formulations should be equivalent, but it is necessary to ensure that the test statistic is calculated in a way that is consistent with how the critical values were tabulated.

formula for the computation of the Kolmogorov-Smirnov goodness of fit statistic: D=max1≤i≤N∣∣∣F(Yi)−iN∣∣∣ this formula can be rewritten as: D=max1≤i≤N(F(Yi)−iN,iN−F(Yi)) This form makes it clear that an upper bound on the difference between these two formulas is i/N. For actual data, the difference is likely to be less than the upper bound.