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Answer: Both the halving error and the standard error of a quantile estimator decrease as the number of quantiles used in the estimation process increases.
**Correct Answer: D** **Explanation:** D is correct because both the halving error and the standard error of a quantile estimator decrease as the number of quantiles (n) used in the estimation process increases. This is due to the law of large numbers and improved sampling precision with larger sample sizes. **Why other options are incorrect:** - **A is incorrect:** Coherent risk measures can be created using sophisticated weighting functions and do not need to be equally weighted. Equal weighting is not a requirement for coherence. - **B is incorrect:** The building blocks of quantile estimation are essentially the same as those needed to estimate coherent risk measures. Both rely on similar statistical foundations and data processes. - **C is incorrect:** QQ plots (Quantile-Quantile plots) are useful diagnostic tools for assessing distributional fit, but they are not used to analyze or evaluate the precision of quantile estimators themselves.
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A risk analyst at an investment bank is examining how quantile estimators can be incorporated into the bank's risk measures. The analyst focuses on how estimators are constructed and how their precision and usefulness are determined. Which of the following statements about quantile estimators is correct?
A
Each quantile in the loss distribution must have an equal weight when used to create a coherent risk measure.
B
The data and processes involved in estimating quantiles are different from those used to estimate coherent risk measures.
C
QQ plots are a useful tool to evaluate the precision of a quantile estimator.
D
Both the halving error and the standard error of a quantile estimator decrease as the number of quantiles used in the estimation process increases.