# Will ratio of two consistent estimator be consistent in general

If I have two statistic say $$T_1$$ and $$T_2$$ both of which are consistent for $$\theta_1$$ and $$\theta_2$$, then will the ratio $$\frac{T_1}{T_2}$$ be also consistent for $$\frac{\theta_1}{\theta_2}$$ in general.

We have an invariance property for a continuous function of a consistent estimator. Can that property be extended for the function of two different estimator.

• This looks like a one-step demonstration: provided $\theta_2\ne 0,$ isn't the function $(\theta_1,\theta_2)\to\theta_1/\theta_2$ continuous?
– whuber
Feb 17, 2021 at 14:14
• Yes, it is. I was just wondering if we can extend the invariance property for the two estimator case too. Feb 17, 2021 at 14:18
• Like if I have a normal distribution and I know $\bar{X}$ is consistent for $\mu$ and $S^2$ is consistent for $\sigma^2$, then can i say that $\frac{\bar{x}}{S^2}$ will be consistent for $\frac{\mu}{\sigma^2}$ Feb 17, 2021 at 14:20
• My point is that this is a one estimator case: the random variable $(T_1,T_2)$ estimates $(\theta_1,\theta_2).$
– whuber
Feb 17, 2021 at 14:20
• Ahh. I see what you are saying, we can look at this as a single estimator. I get it. Feb 17, 2021 at 14:21

What you need is the theorem that if $$T_n$$ is consistent for some parameter $$\theta$$, and $$g$$ is a continuous function, then $$g(T_n)$$ is consistent for $$g(\theta)$$. This is the continuous mapping theorem, see Two Different Proofs of Continuous Mapping Theorem.
You have $$g(\theta_1, \theta_2)=\theta_1/\theta_2$$ which is continuous on $$\theta_2 \not= 0$$. So, assuming that, $$g(T_1, T_2)=T_1/T_2$$ is consistent for $$\theta_1 / \theta_2$$.