Estimating a symmetric uniform half-width

Question

You model a symmetric fluctuation as Uniform $(-\theta, \theta)$ with unknown half-width $\theta$ . From your data the average of the squares is $\frac{1}{n}\sum X_i^2 = 12$ .

Use the method of moments to estimate $\theta$ .

Accepted Answer

The mean is useless. A Uniform(-	heta, 	heta) is symmetric about 0, so its mean is 0 for every 	heta. Matching the mean gives "0 = 0" and reveals nothing. Move to the second moment. Match the second moment. With mean 0, the second moment equals the variance. A uniform on an interval of total width 2	heta has variance (width squared over 12): \mathbb{E}[X^2] = \operatorname{Var}(X) = \frac{(2	heta)^2}{12} = \frac{	heta^2}{3}. Set that equal to the sample average of the squares m2 = 12: \frac{	heta^2}{3} = 12 \implies 	heta^2 = 36 \implies \hat	heta = 6. (We take the positive root sinc…

Estimating a symmetric uniform half-width

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