I'm trying to find the moment estimator for the density function $$f(y)=\theta y^{\theta-1}$$ and check whether this is biased.
I know this is a $Beta(\theta,1)$ distribution and it looks like I only need to find the first moment in order to find the estimator.
For $Y\sim \mathrm{Beta}(\theta,1)\implies\bar{Y}=\frac{\theta}{\theta+1}$
$$\theta\bar{Y}+\bar{Y}=\theta \implies \hat{\theta}=\frac{\bar{Y}}{1-\bar{Y}}$$
I'm pretty sure I've got that correct but I have no idea how to check whether this is biased or not. Finding the convolution isn't panning out too well and I can't seem to figure out a way to manipulate the expression in $\bar{Y}$ to give me anything to work with.
Any pointers towards answering this would be greatly appreciated.
Assuming $X_1,X_2,\ldots,X_n$ are i.i.d $\mathsf{Beta}(\theta,1)$, we have $E(\overline X)=\frac{\theta}{1+\theta}$, the population mean.
Let $g(x)=\frac{x}{1-x}$ be a function on $(0,1)$, so that $g''>0$ for all $x\in(0,1)$.
That is, as @Henry says, $g$ is convex.
By Jensen's inequality,
$$E( g(\overline X))>g (E(\overline X))$$
Or, $$E\left[\frac{\overline X}{1-\overline X}\right]>\frac{E(\overline X)}{1-E(\overline X)}=\theta$$
Note that equality does not hold in the inequality above as $g$ is not a constant or an affine function.
Finding the actual value of $E\left[\frac{\overline X}{1-\overline X}\right]$ is non-trivial as the sampling distribution of $\overline X$ is not a standard one. So I would not worry about the exact bias of the estimator here.