Return to Answer

The essense of the nonparametric boostrap method is to use the EDF as if it were the true distribution and then perform monte carlo sampling/analysis to the EDF. Therefore, as long as the EDF is a good representation of the true CDF, the boostrap sampling distribution will be a good approximation of the true sampling distribution, and hence any statistics dervied from this approximate sampling distribution will be apprximately correct to the degree that your original data accurately depict the true underlying distribution. Hence, as the sample size gets bigger, both the EDF and the associated sample statistics converge to the true values. The convergence therorem I cited is useful for continuous functions, but its really not needed. All we need to know is that as the sample size gets bigger, the statistical error in treating the EDF as the CDF approaches zero by the law of large numbers. Therefore, simulation from this estimated distribution converges to simulation from the true distribution as the sample size grows.

The major caveat to this is when your sampling statsistic does not uniformly converge to the true value, with maximial order statsitics being a classic example.

Therefore, the answer to your question is that any quantity of your sampling distribution can be estimated from the boostrap sampling distribution as long as the bootstrap sampling distribution itself converges uniformly to the true sampling distribution. Below are some links, some of which contain further links to very interesting papers, on when this condition is not met.

This paper will also be interesting for you. the encyclopedia of mathematics has a good entry on boostrap failure as well. This has also been discussed previously on Cross validated: What are examples where a "naive bootstrap" fails?

The essense of the nonparametric boostrap method is to use the EDF as if it were the true distribution and then perform monte carlo sampling/analysis to the EDF. Therefore, as long as the EDF is a good representation of the true CDF, the boostrap sampling distribution will be a good approximation of the true sampling distribution, and hence any statistics dervied from this approximate sampling distribution will be apprximately correct to the degree that your original data accurately depict the true underlying distribution. Hence, as the sample size gets bigger, both the EDF and the associated sample statistics converge to the true values. The convergence therorem I cited is useful for continuous functions, but its really not needed. All we need to know is that as the sample size gets bigger, the statistical error in treating the EDF as the CDF approaches zero by the law of large numbers. Therefore, simulation from this estimated distribution converges to simulation from the true distribution as the sample size grows.

The major caveat to this is when your sampling statsistic does not uniformly converge to the true value, with maximial order statsitics being a classic example.

Therefore, the answer to your question is that any quantity of your sampling distribution can be estimated from the boostrap sampling distribution as long as the bootstrap sampling distribution itself converges uniformly to the true sampling distribution. Below are some links, some of which contain further links to very interesting papers, on when this condition is not met.

This paper will also be interesting for you. the encyclopedia of mathematics has a good entry on boostrap failure as well. This has also been discussed previously on Cross validated: What are examples where a "naive bootstrap" fails?

The essense of the nonparametric boostrap method is to use the EDF as if it were the true distribution and then perform monte carlo sampling/analysis to the EDF. Therefore, as long as the EDF is a good representation of the true CDF, the boostrap sampling distribution will be a good approximation of the true sampling distribution, and hence any statistics dervied from this approximate sampling distribution will be apprximately correct to the degree that your original data accurately depict the true underlying distribution. Hence, as the sample size gets bigger, both the EDF and the associated sample statistics converge to the true values. The convergence therorem I cited is useful for continuous functions, but its really not needed. All we need to know is that as the sample size gets bigger, the statistical error in treating the EDF as the CDF approaches zero by the law of large numbers. Therefore, simulation from this estimated distribution converges to simulation from the true distribution as the sample size grows.

The major caveat to this is when your sampling statsistic does not uniformly converge to the true value, with maximial order statsitics being a classic example.

This paper will also be interesting for you. the encyclopedia of mathematics has a good entry on boostrap failure as well. This has also been discussed previously on Cross validated: What are examples where a "naive bootstrap" fails?

The essense of the nonparametric boostrap method is to use the EDF as if it were the true distribution and then perform monte carlo sampling/analysis to the EDF. Therefore, as long as the EDF is a good representation of the true CDF, the boostrap sampling distribution will be a good approximation of the true sampling distribution, and hence any statistics dervied from this approximate sampling distribution will be apprximately correct to the degree that your original data accurately depict the true underlying distribution. Hence, as the sample size gets bigger, both the EDF and the associated sample statistics converge to the true values. The convergence therorem I cited is useful for continuous functions, but its really not needed. All we need to know is that as the sample size gets bigger, the statistical error in treating the EDF as the CDF approaches zero by the law of large numbers. Therefore, simulation from this estimated distribution converges to simulation from the true distribution as the sample size grows.

The major caveat to this is when your sampling statsistic does not uniformly converge to the true value, with maximial order statsitics being a classic example.

Therefore, the answer to your question is that any quantity of your sampling distribution can be estimated from the boostrap sampling distribution as long as the bootstrap sampling distribution itself converges uniformly to the true sampling distribution. Below are some links, some of which contain further links to very interesting papers, on when this condition is not met.

This paper will also be interesting for you. the encyclopedia of mathematics has a good entry on boostrap failure as well. This has also been discussed previously on Cross validated: What are examples where a "naive bootstrap" fails?

The essense of the nonparametric boostrap method is to use the EDF as if it were the true distribution and then perform monte carlo sampling/analysis to the EDF. Therefore, as long as the EDF is a good representation of the true CDF, the boostrap sampling distribution will be a good approximation of the true sampling distribution, and hence any statistics dervied from this approximate sampling distribution will be apprximately correct to the degree that your original data accurately depict the true underlying distribution. Hence, as the sample size gets bigger, both the EDF and the associated sample statistics converge to the true values. The convergence therorem I cited is useful for continuous functions, but its really not needed. All we need to know is that as the sample size gets bigger, the statistical error in treating the EDF as the CDF approaches zero by the law of large numbers. Therefore, simulation from this estimated distribution converges to simulation from the true distribution as the sample size grows.

The major caveat to this is when your sampling statsistic does not uniformly converge to the true value, with maximial order statsitics being a classic example.

This paper will also be interesting for you. the encyclopedia of mathematics has a good entry on boostrap failure as well

The essense of the nonparametric boostrap method is to use the EDF as if it were the true distribution and then perform monte carlo sampling/analysis to the EDF. Therefore, as long as the EDF is a good representation of the true CDF, the boostrap sampling distribution will be a good approximation of the true sampling distribution, and hence any statistics dervied from this approximate sampling distribution will be apprximately correct to the degree that your original data accurately depict the true underlying distribution. Hence, as the sample size gets bigger, both the EDF and the associated sample statistics converge to the true values. The convergence therorem I cited is useful for continuous functions, but its really not needed. All we need to know is that as the sample size gets bigger, the statistical error in treating the EDF as the CDF approaches zero by the law of large numbers. Therefore, simulation from this estimated distribution converges to simulation from the true distribution as the sample size grows.

The major caveat to this is when your sampling statsistic does not uniformly converge to the true value, with maximial order statsitics being a classic example.

This paper will also be interesting for you. the encyclopedia of mathematics has a good entry on boostrap failure as well. This has also been discussed previously on Cross validated: What are examples where a "naive bootstrap" fails?