Estimating a Laplace scale from the second moment

Question

You model a heavy-tailed but symmetric shock as a Laplace (double-exponential) distribution centered at $0$ with unknown scale $b$ . Its variance is $2b^2$ . From your data the average of the squares is $\frac{1}{n}\sum X_i^2 = 8$ .

Use the method of moments to estimate $b$ .

Accepted Answer

Skip the mean. The Laplace here is symmetric about 0, so its mean is 0 regardless of b. The first moment equation is empty. Match the second moment instead. Match the second moment. With a zero mean, the second moment equals the variance, which for this Laplace is 2b^2: \mathbb{E}[X^2] = 2b^2. Set that equal to the sample average of the squares m2 = 8: 2b^2 = 8 \implies b^2 = 4 \implies \hat b = 2. Check. A Laplace with scale 2 has variance 2 \cdot 2^2 = 8, matching the data. Takeaway. For a zero-centered Laplace the second moment is 2b^2, so \hat b = \sqrt{m2/2}. As always with a symmetric s…

Estimating a Laplace scale from the second moment

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