# Show that $\hat\theta=\frac{2 \bar Y- 1}{1- \bar Y}$ is an unbiased estimator for $\theta$

Let $Y_1,Y_2,...,Y_n$ denote a random sample from the probability density function $$f(y| \theta)= \begin{cases} ( \theta +1)y^{ \theta}, & 0 < y<1 , \theta> -1 \\ 0, & \mbox{elsewhere}, \end{cases}$$

Find an Estimator for $\theta$ by using the method of moments and show that it is consistent.

I have found the estimator but unsure how to show that it is consistent estimator.

E(Y)=$\frac{ \theta +1}{ \theta +2}$ and $m_1'(u)= \frac{1}{n} \sum_{i=1}^{n}Y_i= \bar Y$

Now, E(Y)=$\frac{ \theta +1}{ \theta +2}$=$m_1'(u)= \frac{1}{n} \sum_{i=1}^{n}Y_i= \bar Y$

So $\bar Y=\frac{ \theta +1}{ \theta +2} \to \hat\theta=\frac{2 \bar Y- 1}{1- \bar Y}$

Now I am unsure how to show that $\hat\theta=\frac{2 \bar Y- 1}{1- \bar Y}$ is an consistent estimator for $\theta$

• Do you know the definition of an unbiased estimator? Feb 14, 2013 at 5:01
• So the problem is you don't know how to work out the expectation for this estimator? Do you know how to work out the distribution of $1 - \bar{Y}$? Feb 14, 2013 at 5:06
• Don't post questions to two different SE's at the same time, and don't ask a question you already have an answer to. Feb 14, 2013 at 5:21
• From the FAQ: "Please note, however, that cross-posting is not encouraged on SE sites. Choose one best location to post your question."; on the other part, okay, yeah, I see that you asked before getting an answer. Feb 14, 2013 at 5:30

This is your statistical model: you have random variables $$Y_1,\dots,Y_n$$, which are independent and identically distributed, with $$Y_i\sim\mathrm{Beta}(\theta +1,1)$$, for $$\theta>-1$$.
An unbiased estimator $$\hat{\theta}_n=\hat{\theta}_n(Y_1,\dots,Y_n)$$ of the parameter $$\theta$$, by definition, must satisfy $$\mathrm{E}_\theta[\hat{\theta}_n]=\theta$$, for every $$\theta$$. The original question is wrong, because the estimator obtained by applying the method of moments to this problem is not unbiased.
An estimator $$\hat{\theta}_n$$ of the parameter $$\theta$$ is (weakly) consistent if $$\hat{\theta}_n \stackrel{P_\theta}{\longrightarrow} \theta$$, for every $$\theta$$, and strongly consistent if $$\hat{\theta}_n \longrightarrow \theta$$, a.s. $$[P_\theta]$$, for every $$\theta$$. In what follows, we compute the method of moments estimator for $$\theta$$ and prove that it is strongly consistent (yielding that it is weakly consistent).
To obtain the estimator, note that first population moment is $$\mathrm{E}_\theta[Y_1] =(\theta+1)/(\theta+2)$$, and the first sample moment is $$\bar{Y}_n=(Y_1+\dots+Y_n)/n$$. Equating both moments we find the estimator $$\hat{\theta}_n = \frac{2\bar{Y}_n-1}{1-\bar{Y}_n} \, .$$ Since the function $$t\mapsto (2t-1)/(1-t)$$ is continuous in the appropriate domain, by the Strong Law of Large Numbers we have $$\hat{\theta}_n \longrightarrow \frac{2\mathrm{E}[Y_1]-1}{1-\mathrm{E}[Y_1]} = \theta \, ,$$ almost surely $$[P_\theta]$$, for every $$\theta$$. Therefore, the method of moments estimator $$\hat{\theta}_n$$ is a consistent estimator of the parameter $$\theta$$.