Is the estimator $\hat{r} = \frac{\overline{X}}{1-\overline{X}}$ consistent?

I do not know what to do with the following exercise:

Consider an i.i.d. sequence $X_1,\ldots,X_n$ of $Bernoulli(p)$ distributed RVs for $0<p<1$. Use the method of moments to find an estimator $\hat{r}$ for the odds $r = \frac{p}{1-p}$. Is $\hat{r}$ consistent?

I understand that the method of moments gives us $\hat{r} = \frac{\overline{X}}{1-\overline{X}}$, but I do not see how to prove or disprove consistency here. I know that a sufficient condition for consistency ist that $$\lim_{n \rightarrow \infty} \mathbb{E}[\hat{r}] = r \quad \text{ and } \quad \lim_{n \rightarrow \infty} \mathbb{V}[\hat{r}] = 0$$

, but I do not see how to compute the Expected Value or the Variance of a quotient of RVs. Could you please help me?

Solution 1:

By weak Law of Large Numbers, $\overline X \xrightarrow{p} \mathbb E[X_1]=p$. Next, the function $g(x)=\frac{x}{1-x}$ is continuous at $x=p\neq 1$. So by continuous mapping theorem for convergence in probability, $$ g(\overline X)=\dfrac{\overline X}{1-\overline X} \xrightarrow{p} g(p) =\dfrac{p}{1-p}. $$ This means that the estimator $\hat r$ is consistent.

Solution 2:

If your problem is the quotient, using the fact that $\frac{\bar{X}}{1-\bar{X}} = \frac{1}{1-\bar{X}}-1$ might be enough.