Let $X_1, X_2, ..., X_n$ be i.i.d. with a common density function:
$$f(x;\theta)=\theta x^{\theta-1}$$ for $0<x<1$ and $\theta>0$. So this is $BETA(\theta,1)$ distribution.
For the following three estimators for $\theta$:
Method of Moments (MOM) Estimator: $\tilde \theta=\frac{\bar X}{1-\bar X}$
Maximum Likelihood Estimator (MLE): $\hat \theta=\frac{-n}{\sum_{i=1}^n \ln(X_i)}$
Bayes' Estimator, where the prior distribution of $\theta$ is exponential with mean 2: $\check \theta=\frac{2(n+1)}{1-2\sum_{i=1}^n \ln(X_i)}$
Without doing any calculations, which estimator among these three estimators will you prfer? Explain your preference.
I honestly have no idea; how do you make an inuitive choice of an estimator?