# Variance of bootstrap sample of size n

• ghostyc
In summary, the conversation is about finding the bias in a sample and population. The result in the notes is for the sample mean, but the task is to find the bias in the population variance. The person is working on the bias in theta star hat, but is confused about what they are supposed to be looking for. They are currently manipulating an equation to find the bias, but have gotten stuck when taking the expectation.
ghostyc
http://img138.imageshack.us/img138/6060/98259799.jpg

I have done part one and found that the bias is $$\frac{\theta}{n}$$

then i don't know how to proceed.

in my notes, have the following result,

http://img189.imageshack.us/img189/9699/resultc.jpg

i am thinking, this time i have to find the bias in $$\hat{\theta^*}$$
then, if i work through, i got it's unbiased...

am i doing something wrong?

I am confused with that, the result in my notes, is it for the mean of the sample?

but here, we are aksed for the population variance?

Thanks

Last edited by a moderator:

EnumaElish said:

i will look into it again
sometimes, i really get confused what i am looking for

EnumaElish said:

I am now working on the bias is

$$\bar{\theta}^{*} - \hat{\theta} = \frac{1}{n} \sum_{i=1}^n (y_i^* - \bar{y} )^2 - \frac{1}{n} \sum_{i=1}^n (y_i - \bar{y} )^2$$
after some munipulation , i got to

$$\frac{1}{n} \sum_{i=1}^n \left( {y_i^*}^2 - y_i^2 -2\bar{y} (y_i^*+y_i) \right)$$

then i take expectation, i got stuck again...

## 1. What is the purpose of calculating the variance of a bootstrap sample of size n?

The variance of a bootstrap sample helps to estimate the variability of a population parameter, such as the mean or standard deviation. It allows us to understand how much the sample statistic may vary from sample to sample, and therefore provides insight into the accuracy and stability of our estimates.

## 2. How is the variance of a bootstrap sample calculated?

The variance of a bootstrap sample is calculated by taking the average of the squared differences between each data point and the sample mean. This is known as the sample variance. The process is repeated for multiple bootstrap samples, and the variance of these sample variances is then calculated to estimate the variance of the population.

## 3. Can the variance of a bootstrap sample be used to make inferences about the population variance?

Yes, the variance of a bootstrap sample can be used to estimate the population variance. However, it is important to note that this is an estimate and may not be exact. The accuracy of the estimate depends on the size and representativeness of the original sample, as well as the number of bootstrap samples taken.

## 4. How does the size of the bootstrap sample affect the variance calculation?

The larger the bootstrap sample size, the more accurate the estimate of the population variance will be. This is because larger sample sizes provide more information about the population and reduce the variability of the estimates. However, increasing the sample size also increases the computational time and resources needed for the calculation.

## 5. What are the limitations of using the variance of a bootstrap sample?

One limitation is that the bootstrap method assumes that the original sample is representative of the population. If this is not the case, the estimates obtained from the bootstrap sample may not accurately reflect the population. Additionally, the bootstrap method relies on random sampling, so it may not be appropriate for datasets with non-random patterns or dependencies.

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