Two unknowns, two moments for a normal fit

Question

You model the size of a bid-ask spread (in ticks) as a normal distribution with unknown mean $\mu$ and unknown variance $\sigma^2$ . From a large sample you compute the average $\frac{1}{n}\sum X_i = 2$ and the average of the squares $\frac{1}{n}\sum X_i^2 = 13$ .

Use the method of moments to estimate $\mu$ and $\sigma^2$ .

Accepted Answer

Two unknowns, two equations. The method of moments matches as many theoretical moments as there are parameters. With two unknowns, use the first two: the mean and the average of the squares. Write m1 = \frac1n\sum Xi and m2 = \frac1n\sum Xi^2. The model moments. For a normal distribution, \mathbb{E}[X] = \mu, \qquad \mathbb{E}[X^2] = \mu^2 + \sigma^2. The second line is just the identity "average of squares equals mean squared plus variance." Match and solve. The first equation gives \hat\mu = m1 = 2. Rearranging the second, \hat{\sigma}^2 = m2 - m1^2 = 13 - 2^2 = 9, so \hat\sigma = 3 ticks. …

Two unknowns, two moments for a normal fit

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