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Suppose $X_i$ is uniformly distributed on $[v;c]$, where $v$ is the parameter of interest and $c$ is some constant. The task is to find a GMM estimator of $v$.

I know that to derive a GMM estimator we need to specify moment conditions that contain our parameters of interest and then solve for those parameters by replacing population moments with their sample versions.

My best guess at this particular problem is that we have to use moment condition

$E[X_i - \mu] = 0$

with $\mu$ being population mean. Since $X_i$ is uniformly distributed on $[v;c]$, we can say that $\mu = \frac{1}{2}(c+v)$. This gives us

$E[X_i - \frac{1}{2}(c+v)] = 0$

Then we simply estimate using our sample

$\frac{1}{n}\sum\limits_{i=1}^n (X_i - \frac{1}{2}(c+v)) = 0$

Solving for $v$ gives us the desired estimator

$v=2\frac{1}{n}\sum\limits_{i=1}^n (X_i) - c$

Is this solution correct?

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  • $\begingroup$ You might want to double check your claim that $\mu = c-v$. $\endgroup$
    – Mark L. Stone
    Dec 17, 2015 at 2:44
  • $\begingroup$ Thanks for pointing out. Is everything ok now? $\endgroup$
    – user94383
    Dec 17, 2015 at 2:54

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