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.

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1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$
    – Hermi
    Commented Jul 20, 2022 at 1:56
  • $\begingroup$ 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. $\endgroup$
    – Hermi
    Commented Jul 20, 2022 at 1:57
  • $\begingroup$ 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 $mean((|T_n^i - \theta|) > \epsilon)$ , $\endgroup$
    – Allan
    Commented Jul 20, 2022 at 13:17
  • $\begingroup$ Can you please see my edit? I simulate the mean of $|T_n^i-\theta|>\epsilon$. But the result is wired? $\endgroup$
    – Hermi
    Commented Jul 20, 2022 at 16:00
  • $\begingroup$ Yes, is the mean of this expression. I don't know what result being wired mean. $\endgroup$
    – Allan
    Commented Jul 20, 2022 at 16:05

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