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FRM Part II · FRM Exam Part II · Beyond Exceedance-Based Backtesting of Value-at-Risk Models

An analyst applies the Kolmogorov-Smirnov (KS) test to 250 PIT values to check uniformity. The test statistic is defined as:

The Kolmogorov-Smirnov statistic is the largest absolute vertical gap between the empirical CDF of the PIT values and the theoretical uniform CDF. Because it focuses on a single maximum distance, it is most sensitive near the center of the distribution rather than the tails.

  1. AThe sum of squared differences between the empirical and theoretical CDFs, weighted toward the tails
  2. BThe maximum absolute vertical distance between the empirical CDF and the theoretical CDFCorrect
  3. CThe number of observations falling below the 1% quantile of the forecast distribution
  4. DThe likelihood ratio between a restricted and an unrestricted normal model for the transformed series

Explanation

The KS statistic is the supremum of the absolute difference between the empirical and hypothesized CDFs. The first option describes a tail-weighted quadratic measure like Anderson-Darling. The last describes the Berkowitz likelihood ratio.

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