# Comparison of confidence intervals: bootstrap & exact resampling

Consider data $$X_1,...X_n$$ generated from a probability distribution $$F$$ with density $$f$$.

I'm interested in constructing confidence intervals for a parameter say, $$\theta(F)$$. Via Monte Carlo simulations, I want to compare the expected lengths and coverage probabilities of $$X\%$$ confidence intervals based on

• exact resampling (i.e., can generate multiple independent samples of size $$n$$ from $$F$$) and

• bootstrap.

This explains the fundamental difference between Bootstrapping and Monte Carlo procedure. While I understand what it means to use Monte Carlo simulations to generate confidence intervals for bootstrap resampling method, I'm not sure how to proceed with this.

Theoretically, what I know is if $$X_1^{*},X_2^{*},\ldots,X_n^{*}$$ is a single resample then, the bootstrap estimate for $$G_n(t)=P_F(\sqrt{n}[\hat{\theta}-\theta]\le t)$$ is given by $$\hat G_n(t)=P_{F_n}(\sqrt{n}[\hat\theta^*-\hat\theta]\le t)$$ where $$F_n$$ is the resampled distribution. Using Monte Carlo, if we have generated $$B$$ resamples, the bootstrap estimate is $$\tilde G_n(t)=\frac{1}{B}\sum^B \hat G_{n,i}(t)$$.

Also, I know that the pivotal quantity $$\sqrt{n}[\hat{\theta}-\theta]$$ can be a little different for some parameters, just using it as a placeholder.

Confidence interval from a one resample $$X_1^*,\ldots,X_n^*$$ would be for $$\hat\theta$$: $$\left(\hat\theta^*\pm\frac{q_{\alpha/2}}{\sqrt{n}}\right)$$ where, $$q_{\alpha/2}$$ is the $$\alpha/2$$th quantile from $$F_n$$. But where does the Monte Carlo estimate prove useful?

From a single bootstrapped sample I get a single estimate of the mean. To use Monte Carlo, I get $$B$$ estimates of the mean using which, I need to create a confidence interval but, how? Moreover, a single confidence interval is not enough to get the expected length and coverage probability. Do I need to get $$N$$ different confidence intervals? I don't want the complete procedure written, I just want to have the gaps in my understanding cleared.

• What is the specific question? Feb 20 at 13:11
• @SextusEmpiricus "Via Monte Carlo simulations, compare the expected lengths and coverage probabilities of 95% confidence intervals for the parameter i) mean $\mu$ based on a) exact resampling b) bootstrap for the family of distributions $N(\mu,1)$." I do know the theoretical aspects of it to some extrent but I don't understand what the correct procedure should be. Feb 20 at 13:15
• This sounds like a homework question and the task could be entirely done for you, but could you describe more specifically which aspect/part you do not understand and what you have tried untill now. Feb 20 at 13:22
• Are you able to create the confidence intervals based on a bootstrap approach and based on a resampling from the exact distribution? Feb 20 at 13:24
• $$\left(\hat\theta^*\pm\frac{q_{\alpha/2}}{\sqrt{n}}\right)$$ I believe that you do not need the $\sqrt{n}$ factor there. Feb 20 at 13:54

So you create $$N$$ simulations of an observed sample. For each observed sample you resample $$M$$ times to create estimates of a bootstrap distribution and estimates of a confidence interval. You will end up with $$N$$ confidence intervals, one for each simulated observation.