Best estimator for geometric distribution

Question

I need an estimator for geometric distribution $\text{Geom}(p)$ that best fits my data $X_1, X_2,\ldots$

Is $\widehat{p} = \dfrac{1}{\overline{X}}$ the answer? Both MLE and method of moments yield this estimator. However, I don't really know how to argue that $\dfrac{1}{\overline{X}}$ best fits my data, perhaps in a sense of ordinary least squares, but I dont know how to work with OLS with non linear distribution.

Thanks for the help.

Answer 1

Estimator $\widehat{p} = \dfrac{1}{\overline{X}}$ is the only estimator for $p$ of $\text{Geom}(p)$. Thus it is the best fitting.