Method of Moments: GPD

Basic Approach

Suppose that the location parameter μ is known and we are able to calculate the mean and variance of a sample that we suspect may be from a population that follows a Generalized Pareto (GPD) distribution. We show how to estimate the other two parameters using the method of moments, as follows.

Estimates if μ is known

Since

GPD mean formula

solving for ξ, we obtain

Solving for xi

It now follows that

1-xi

1-2*xi

We also know that

Variance of GPD formula

Thus

Revised variance expression

Solving for σ, we get

Sigma expression

Thus

New xi expression

which provides us with the estimates for σ and ξ if μ is known. In summary, we have the following property.

Property 1: The following formulas provide estimates for the GPD parameters σ and ξ if μ is known.

xi mom estimate

sigma mom estimate

Estimates if μ is unknown

If μ is not known, then let γ = the skewness of the sample data. Then for ξ < 1/3 (and so 1–3ξ > 0), we know that

skewness estimate for GPD

Then

skewness squared

Polynomial version

Polynomial in xi

The roots of this equation can be obtained using Bairstow’s method. You can use any real root ξ < 1/3.

Since

Variance of GPD formula

it follows that

Sigma estimate

Finally, since

GPD mean formula

it follows that

Mu estiamte

Example

Example 1: Using the method of moments, estimate the values of the σ and ξ parameters for the GPD that best fits the data in column D of Figure 1 assuming that the location parameter μ = 2.

GPD fit example

Figure 1 – Fitting data to a GPD

Cells G6 and G7 of the figure show the estimates of the shape and scale parameters using Property 1.

Note that the data in column D was created by inserting the formula =GPD_INV(B3,B4,B5) in cells D3 through D22. The resulting estimates of σ = 1.117874 and ξ = -0.53897 in cells G7 and G6 are pretty close to the population parameters σ = 1.12 and ξ = -0.5 shown in cells G7 and G6.

Log-likelihood

For a sample x1, …, xn that follows a Generalized Pareto distribution the likelihood function is

Likelihood function

which results in the log-likelihood function

Log-likelihood

This is how the formula in cell G10 (and displayed in I10) was obtained.

Examples Workbook

Click here to download the Excel workbook with the example described on this webpage.

References

Mathworks (2022) Generalized Pareto distribution
https://it.mathworks.com/help/stats/generalized-pareto-distribution.html

Wikipedia (2022) Generalized Pareto distribution
https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

Chu, J., Dickin, O., & Nadarajah, S. (2019). A review of goodness of fit tests for Pareto distributions. Journal of Computational and Applied Mathematics, 361, 13-41. https://doi.org/10.1016/j.cam.2019.04.018

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