Question

Frequently, the bootstrap percentile confidence interval can be improved if the estimator \(\widehat{\theta}\) is standardized by an estimate of scale. To illustrate this, consider a bootstrap for a confidence interval for the mean. Let \(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*}\) be a bootstrap sample drawn from the sample \(x_{1}, x_{2}, \ldots, x_{n} .\) Consider the bootstrap pivot [analog of \((4.9 .13)]:\) $$ t^{*}=\frac{\bar{x}^{*}-\bar{x}}{s^{*} / \sqrt{n}} $$ where \(\bar{x}^{*}=n^{-1} \sum_{i=1}^{n} x_{i}^{*}\) and $$ s^{* 2}=(n-1)^{-1} \sum_{i=1}^{n}\left(x_{i}^{*}-\bar{x}^{*}\right)^{2} . $$ (a) Rewrite the percentile bootstrap confidence interval algorithm using the mean and collecting \(t_{j}^{*}\) for \(j=1,2, \ldots, B\). Form the interval $$ \left(\bar{x}-t^{*(1-\alpha / 2)} \frac{s}{\sqrt{n}}, \bar{x}-t^{*(\alpha / 2)} \frac{s}{\sqrt{n}}\right) $$ where \(t^{*(\gamma)}=t_{([\gamma * B])}^{*} ;\) that is, order the \(t_{j}^{*} \mathrm{~s}\) and pick off the quantiles. (b) Rewrite the \(\mathrm{R}\) program percentciboot.s and then use it to find a \(90 \%\) confidence interval for \(\mu\) for the data in Example 4.9.3. Use 3000 bootstraps. (c) Compare your confidence interval in the last part with the nonstandardized bootstrap confidence interval based on the program percentciboot.s.

Short answer

The task involves rewriting of percentile bootstrap confidence interval algorithm, sample collection, confidence interval formulation, and comparison of standardized and nonstandardized confidence intervals using R program. The detailed solutions are provided in the steps above.

Step-by-step solution

Problem analysis

To solve this problem, firstly, the percentile bootstrap confidence interval algorithm using the mean needs to be rewritten. Then, samples are collected to form the interval. Quantiles of the ordered sample are obtained using the formula: \(t^{*(\gamma)}=t_{([\gamma * B])}^{*}\).

Rewrite the bootstrap formula

Rewrite the percentile bootstrap confidence interval algorithm using the mean and collecting \(t_{j}^{*}\) for \(j=1,2, \ldots, B\) as follows:\n- Draw a bootstrap sample \(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*}\) from the observed data.\n- Calculate \(t^{*}=\frac{\bar{x}^{*}-\bar{x}}{s^{*} / \sqrt{n}}\) \n- Replace \(t^{*}\) by its percentile \(t^{*(\gamma)}\)\n- Form the confidence interval as \(\left(\bar{x}-t^{*(1-\alpha / 2)} \frac{s}{\sqrt{n}}, \bar{x}-t^{*(\alpha / 2)} \frac{s}{\sqrt{n}}\right)\)

R program

In R, use the bootstrap resampling method to generate 3000 bootstrap samples. The percentile bootstrap confidence interval for each bootstrap replication is computed and stored. The function percentciboot.s is used to compute the interval.

Compare confidence intervals

After obtaining the confidence interval via R program, compare this interval with the one obtained through nonstandardized bootstrap confidence interval from the same percentciboot.s program. This comparison is to check how much the intervals vary and if the standardization helps bring a significant improvement in the confidence interval estimation.