# Case Study: Parameter Estimation of Generalized Pareto Distribution

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Case study background and problem formulations

PROBLEM 1: Parameter estimation of Generalized Pareto Distribution
• Maximum Likelihood estimate
• Harmonic method
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Data and solution in MATLAB Environment

Problem Datasets # of Samples
Dataset1 Matlab Code Library Solution 1250
Dataset2 Matlab Code Library Solution 12500
PROBLEM 2: Estimation of Generalized Pareto Distribution for residuals of quantile regression
Minimize kb_err
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kb_err = Koenker and Basset error function
Parameter estimation for residuals:
• Maximum Likelihood estimate
• Harmonic method
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Data and solution in MATLAB Environment

Problem Datasets # of Variables # of Samples
Dataset1 Matlab Code Library Solution 5 1264
Dataset2 Matlab Code Library Solution 1 5000
Dataset3 Matlab Code Library Solution 1 50000
PROBLEM 3: Estimation of Generalized Pareto Distribution for residuals of CVaR regression
Minimize cvar2_err (Minimizing CVaR (Superquantile) error)
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cvar2_err = CVaR (Superquantile) error
Parameter estimation for residuals:
• Maximum Likelihood estimate
• Harmonic method
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Data and solution in MATLAB Environment

Problem Datasets # of Variables # of Samples
Dataset1 Matlab Code Library Solution 5 1264

CASE STUDY SUMMARY

This case study solves the problem of parameter estimation for generalized Pareto distribution. Two approaches for parameter estimating are implemented. The first one is maximum likelihood estimate (see Kotz and all 2000). The second one is known as the estimate by harmonic method (see Golodnikov and all. 2019) and is based on the maximum entropy principle with the Renyi entropy and moment constraints. Estimates were evaluated for artificial samples with different length (Problem 1) and for residuals of quantile regression (Problem 2, Dataset2 and Dataset3). Quantile (Problem 2, Dataset 1) and CVaR regression (Problem 3) were evaluated for return distribution of the Fidelity Magellan Fund on the Russell Value Index (RUJ), RUSSELL 1000 VALUE INDEX (RLV), Russell 2000 Growth Index (RUO) and Russell 1000 Growth Index (RLG). Parameters of GPD were estimated according to the described above techniques.

References

• Golodnikov, A., Grechuk, B., Zabarankin, M., Uryasev, S. Method of Moments and Renyi Entropy Maximization.
• Kotz, S., and S. Nadarajah. Extreme Value Distributions: Theory and Applications. London: Imperial College Press, 2000.