How heavy is the tail, and how sure am I?
A tail estimate is one number from a choice you made: how many of the largest observations to use. This lab asks you to look at the whole plot before you believe any single value.
Model contract
- Objective
- Estimate how fast the tail decays, and how confident that estimate deserves to be.
- Observed
- 2,000 positive tail magnitudes, from a process you do not get to inspect.
- Hidden
- Which process generated the sample. Revealed at stage 3.
- Assumptions
- Observations are iid. Real loss data is not, and this lab does not pretend otherwise.
- Parameters
- k, the number of largest observations used. The estimator has no other tuning.
- Units
- Tail magnitudes, in the same units as the input. ξ and α are dimensionless.
- Benchmark
- The plot across k. A single k is never the result.
- Failure
- The estimate drifts across the window, or the sample is too small to hold the window.
1. Predict the shape
You have 2,000 positive tail magnitudes and a Hill estimate for each choice of k. Across k from 100 to 400 (inclusive, in steps of 10), which description fits the plot better?
Synthetic iid samples only. The estimator assumes a power-law tail above the threshold, and real financial data is neither iid nor guaranteed to have one.