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Answer: One limitation of the Kolmogorov-Smirnov test is its lack of sensitivity at the center of the distribution, so that it is ineffective in detecting differences in central masses of the two distributions.
## Explanation Let's analyze each option: **Option A: CORRECT** - The Kolmogorov-Smirnov (KS) test is indeed less sensitive to differences in the central part of distributions and more sensitive to differences in the tails. - This is because the KS test statistic is based on the maximum vertical distance between the empirical and theoretical cumulative distribution functions, which often occurs in the tails rather than the center. **Option B: INCORRECT** - The Cramer-von Mises test is not a variation of the KS test that relies on mean absolute deviation. - It actually uses the squared differences between the empirical and theoretical CDFs integrated over the entire distribution, making it more sensitive to differences throughout the distribution. **Option C: INCORRECT** - In VaR backtesting, the null hypothesis for goodness-of-fit tests (including Anderson-Darling) is typically that the PITs (Probability Integral Transforms) follow a uniform distribution. - The statement incorrectly states the null hypothesis as "empirical CDF is not cumulative standard uniform." **Option D: INCORRECT** - When the null hypothesis is true, the test statistics of these goodness-of-fit tests are not necessarily equal to zero. - They follow specific distributions (like the Kolmogorov distribution for KS test) and are typically small but not necessarily zero. The correct answer is **A** because it accurately describes the limitation of the Kolmogorov-Smirnov test regarding its sensitivity to central versus tail differences in distributions.
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Cindy, a post-graduate student in risk management, is learning how to use various goodness-of-fit tests to assess the uniformity of the distribution of PITs. Which of the following statements correctly describes these tests?
A
One limitation of the Kolmogorov-Smirnov test is its lack of sensitivity at the center of the distribution, so that it is ineffective in detecting differences in central masses of the two distributions.
B
The Cramer-von Mises test is a variation of the Kolmogorov-Smirnov test that relies on the mean absolute deviation of the distribution.
C
In the context of VaR backtesting, the null hypothesis of Anderson-Darling test is that the empirical CDF is not cumulative standard uniform.
D
When the null hypothesis is true, the test statistic of all three goodness-of-fit tests is equal to zero.