Method of moments for the uniform distribution

Question

The method of moments (MoM) estimates parameters by matching theoretical moments to sample moments.

Derive the MoM estimators for both endpoints $a$ and $b$ of a Uniform $(a, b)$ distribution. Where can this estimator go wrong, and how does it compare to the MLE?

Accepted Answer

The recipe. With sample mean m1 = \frac{1}{n}\sum Xi and sample variance \hat v = \frac{1}{n}\sum Xi^2 - m1^2, match the first two theoretical moments to the data. Match mean and variance. \frac{a + b}{2} = m1, \qquad \frac{(b - a)^2}{12} = \hat v. From the second equation, b - a = \sqrt{12\,\hat v} = 2\sqrt{3\,\hat v}. Combining with the first (which gives a + b = 2m1): \hat a{	ext{MoM}} = m1 - \sqrt{3\,\hat v}, \qquad \hat b{	ext{MoM}} = m1 + \sqrt{3\,\hat v}. Where it goes wrong. MoM only uses the mean and variance, so it can produce endpoints that are inside the observed data: it is ent…

Method of moments for the uniform distribution

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