Use the nonparametric bootstrap to obtain standard errors of the estimates so obtained for the parameters pi1,\mu1,\mu2,\sigma1^2, and\sigma2^2

To compute standard errors for the parameters $\pi_1$, $\mu_1$, $\mu_2$, $\sigma_1^2$, and $\sigma_2^2$ using the nonparametric bootstrap, follow these steps:

1. Break Down the Problem

• Understand the parameters to estimate:
  1. $\pi_1$: Proportion of Group 1
  2. $\mu_1$: Mean of Group 1
  3. $\mu_2$: Mean of Group 2
  4. $\sigma_1^2$: Variance of Group 1
  5. $\sigma_2^2$: Variance of Group 2
• Collect original dataset $D$.
• Define the bootstrapping process.

2. Relevant Concepts

• Nonparametric bootstrap involves resampling the dataset with replacement.
• After obtaining resampled datasets, recompute the parameters.

3. Analysis and Detail

1. Obtain the Original Estimates:

   • Compute original estimates $\hat{\pi}_1, \hat{\mu}_1, \hat{\mu}_2, \hat{\sigma}_1^2, \hat{\sigma}_2^2$ from the dataset $D$.

2. Perform Bootstrapping: a. Set the number of bootstrap resamples $B$ (commonly, $B = 1000$). b. For $b = 1$ to $B$:

   • Generate a bootstrap sample $D^*_b$ by resampling $D$ with replacement.
   • Compute the estimates \hat{\pi}_1^*_b, \hat{\mu}_1^*_b, \hat{\mu}_2^*_b, \hat{\sigma}_1^{2*}_b, \hat{\sigma}_2^{2*}_b from $D^*_b$.

3. Calculate Standard Errors:

   • Compute the standard errors for each parameter: \text{SE}(\hat{\pi}_1) = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\pi}_1^*_b - \bar{\hat{\pi}}_1^*)^2} \text{SE}(\hat{\mu}_1) = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\mu}_1^*_b - \bar{\hat{\mu}}_1^*)^2} \text{SE}(\hat{\mu}_2) = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\mu}_2^*_b - \bar{\hat{\mu}}_2^*)^2} \text{SE}(\hat{\sigma}_1^2) = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\sigma}_1^{2*}_b - \bar{\hat{\sigma}}_1^{2*})^2} \text{SE}(\hat{\sigma}_2^2) = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\sigma}_2^{2*}_b - \bar{\hat{\sigma}}_2^{2*})^2}
   • Where $\bar{\hat{\pi}}_1^*, \bar{\hat{\mu}}_1^*, \bar{\hat{\mu}}_2^*, \bar{\hat{\sigma}}_1^{2*}, \bar{\hat{\sigma}}_2^{2*}$ are the means of the bootstrap estimates.

4. Verify and Summarize

• Validate the method by checking if the distribution of bootstrap estimates is centered around the original estimates.
• Summarize the obtained standard errors for each parameter to understand their variability.

Final Answer

The standard errors of the estimates $\pi_1$, $\mu_1$, $\mu_2$, $\sigma_1^2$, and $\sigma_2^2$ are computed via the nonparametric bootstrap method as described, yielding estimates that reflect the variability via resampling.