How to verify the convergence rate in Monte Carlo simulation?

Given a iid random samples $$X\sim N(\theta,1)$$, we have a unknown parameter $$\theta$$ and its estimator $$T_n=T_n(X_1,\dots,X_n)$$. If we have strictly proved that the convergence rate is $$|T_n-\theta|=O_p(N^{-1/3})$$ It seems that $$T_n$$ is a consistent estimator.

I would like to verify this convergence rate in the Monte Carlo simulation. I'm a little unclear on which quantity should I plot the Monte Carlo estimate? Is it Mean square error MSE? Say I have 1000 samples and each sample is repeated 500 times. It can be drawn that the MSE gradually converges to the x-axis as the number of samples increases.

I try to simulate the for a fix $$\epsilon>0$$, $$P(|T_n-\theta|>\epsilon)$$

I fix $$\epsilon =0.01$$ and sample size $$n=1000$$. That is the mean of $$|T_n^{j}-\theta|>\epsilon$$ for replications $$j=1,\dots, 30$$. But I got the following result.

I'm not sure if it will work 100%, but to show that your estimator has the convergence rate of $$O_p(N^{-\frac{1}{3}})$$ and you want assess it, from the definition for any given $$\epsilon > 0$$: $$\lim_{n \rightarrow \infty} Pr(|T_n - \theta| > \epsilon) = 0$$ So to evaluate this visualize I would suggest fixing some small $$\epsilon$$ and for each $$n$$ simulate $$|T_n - \theta|$$ several times, and calculate the empirical $$Pr(|T_n - \theta| > \epsilon)$$ and plot it. And with this you can see how the convergence will behave against $$N^{-\frac{1}{3}}$$. BTW it isn't prove anything, you need to make the theoretical calculations to demonstrate any statement.
• Thank you! But what is the empirical of $P(|T_n-\theta|>\epsilon)$? For example, if we fix $\epsilon=0.01$ and $\theta=0$, that is $P(|T_n|>\epsilon)$. I am not sure what is the Monte Carlo estimate of this distribution． Commented Jul 20, 2022 at 1:56
• What I have done is about the simulation of the expectation. For example, the empirical of expectation is the average of $\frac{1}{N}\sum T_n^j$ where I draw $T_n^j$ from each samples, $N$ is the number of replications. Commented Jul 20, 2022 at 1:57
• To calculate the empirical $P(|T_n - \theta|) > \epsilon$ one way is fix $n$ and $\epsilon$ and simulate let's say for example 100 times $T_n$, and $i$ be the index of the simulation and then calculate $mean((|T_n^i - \theta|) > \epsilon)$ , Commented Jul 20, 2022 at 13:17
• Can you please see my edit? I simulate the mean of $|T_n^i-\theta|>\epsilon$. But the result is wired? Commented Jul 20, 2022 at 16:00