# Intuitive Choice of Estimator for Parameter of Beta Distribution

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?

• Very closely related to this recent Question. – BruceET Feb 15 '18 at 3:10
• Perhaps MOM is the simplest to grasp. MOM estimators are often unbiased, but not this one. // The Bayes' estimator may have some intuitive appeal if you really believe the prior. If you know the posterior distribution, maybe you should include that in your Question. // MLEs have the nice property that they are asymptotically norm. Here $n=50$ is not large enough for a good normal fit. There are several recent papers on bias corr of the MLE; they are not freely available online, but may be freely available at your local univ library. // Perhaps you got this question because it's not intuitive. – BruceET Feb 15 '18 at 3:15