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I’m trying to understand what the method of moments is, exactly. I haven’t seen a perfectly clear (to me) statement of this, so here’s my current belief about what it is: You want to estimate some parameter as some formula using the $r$th central sample moments. Sometimes this is trivial because the parameter you’re estimating just is one of the central moments. But in a case where it’s not trivial because the parameter is not exactly one of the central moments, you may need to find a way to relate the parameter to a formula using central moments.

So for example suppose we want the method of moments estimate of $\lambda$ for $f(x|\lambda)=\lambda e^{-\lambda(x-1)}$ when $x >1$ and $f(x|\lambda)=0$ for $x\leq 1$. This is close to an exponential but not quite, so we can’t rely on the mean being the parameter. We compute the mean in order to see if we can relate it to the parameter:

$$E[X] = \int_1^\infty x\lambda e^{-\lambda(x-1)} \ dx $$ $$ = \lambda\left(x\cdot \frac{e^{-\lambda(x-1)}}{-\lambda} \Bigg|_1^\infty - \int_1^\infty \frac{e^{-\lambda(x-1)}}{-\lambda}\ dx \right) $$ $$ = \lambda \left(\frac{1}{-\lambda}-\frac{1}{-\lambda}\int_1^\infty e^{-\lambda(x-1)}\ dx\right) $$ $$ =\lambda \left(-\frac{1}{\lambda}+1\right) $$ $$ = \lambda-1 $$

So in this case, we can estimate $\hat\mu_k \approx E[X] = \lambda -1$ and therefore $\lambda \approx \hat\mu_k+1$?

Addem
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1 Answers1

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That's right. Basically the method of moments says, solve the equations $E[X \mid \vec{\lambda}]=\overline{X},E[X^2 \mid \vec{\lambda}]=\overline{X^2},\dots,E[X^n \mid \vec{\lambda}]=\overline{X^n}$ for $\vec{\lambda}$ and call that result your estimator.

These equations will usually be independent, so generally there will be as many of them as there are parameters, assuming $X$ is itself scalar.

Ian
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