A random sample of size $n$, $Y_1, Y_2, \ldots, Y_n$ is drawn from a population. The population distribution has the following probability density function
$f(y;\theta)= \begin{cases} \sqrt{\theta}y^{\sqrt{\theta}-1}, & \text{when 0 $\le$ y $\le$ 1} \\ 0, & \text{elsewhere} \end{cases}$
Find an estimator of $\theta$ using the method of moments; call it $\hat \theta$. Prove $\hat \theta$ is a consistent estimator of $\theta$
What I have done so far:
First I opted to find $\mu$. So I set up my integral and solved: $$\mu = \int_0^1 \sqrt{\theta}y^{\sqrt{\theta}}dy = \frac{\sqrt{\theta}}{\sqrt{\theta}+1}$$
Then I solved the equation for $\theta$ to get $\theta = \frac{\mu^2}{(\mu-1)^2}$. So I let the estimator be $\hat{\theta} = \frac{\bar{Y^2}}{(\bar{Y}-1)^2}$
My trouble is coming from how I will prove $\hat{\theta}$ is consistent. I know two ways to show something is consistent, but I am not entirely sure how to apply that in this case. I know if:
1) $\lim \limits_{n \to \infty}E(\hat{\theta})= \theta$ and $\lim \limits_{n \to \infty}V({\hat{\theta}}) = 0$
or
2) $\lim \limits_{n \to \infty}P(|\hat{\theta_n} - \theta| \le \epsilon)=1$ or $\lim \limits_{n \to \infty}P(|\hat{\theta_n} - \theta| \gt \epsilon)= 0$
Then $\hat{\theta}$ is a consistent estimator for $\theta$. I'm just struggling to make use of those definitions. I'm not sure if my estimator is incorrect, or if I'm forgetting something, but any help would be greatly appreciated.
You can combine the (weak) law of large numbers and the continuous mapping theorem. Since $\bar{Y}_n$ converges to $\mu = \dfrac{\sqrt{\theta}}{\sqrt{\theta}+1}$ and the function $g(y) = \dfrac{y^2}{(y-1)^2}$ is continuous on the unit interval $(0,1)$, what can you conclude about $\hat{\theta}_n = g(\bar{Y}_n)$?
Note: The above convergence results are meant to take place in probability, which is what you need to show consistency.