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

An analyst backtests a VaR model with 250 PIT values. The Kolmogorov-Smirnov statistic is computed as the maximum absolute gap between the empirical CDF and the uniform CDF. The largest gap is found at PIT value 0.50, where the empirical CDF equals 0.58. Which statement is correct?

The KS statistic is 0.08. It is the maximum absolute vertical distance between the empirical CDF (0.58) and the uniform CDF (0.50) at the point of largest deviation. It is not the empirical CDF value itself and is not doubled.

  1. AThe KS statistic is 0.08, the largest vertical distance between the two CDFsCorrect
  2. BThe KS statistic is 0.58, the empirical CDF at that point
  3. CThe KS statistic is 0.16, twice the gap
  4. DThe KS statistic is 0.08 multiplied by the square root of 250, which is always the reported statistic for any sample

Explanation

The uniform CDF at 0.50 is 0.50, so the gap is 0.58 - 0.50 = 0.08. KS is the supremum of the absolute difference, so it is 0.08. Scaling by the square root of n is used only when comparing to some critical-value tables, and does not change the raw statistic definition.

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