Solved: Use the blood pressure data in Table 9.2 that was

Suppose that X1,...,Xn form a random sample from an exponential distribution with parameter . Explain how to use the parametric bootstrap to estimate the variance of the sample average X. (No simulation is required.)

Let x1,...,xn be the observed values of a random sample X = (X1,...,Xn). Let Fn be the sample c.d.f. Let J1,...,Jn be a random sample with replacement from the numbers {1,...,n}. Define X i = xJi for i = 1,...,n. Show that X = (X 1,...,X n) is an i.i.d. sample from the distribution F

Let n be odd, and let X = (X1,...,Xn) be a sample of size n from some distribution. Suppose that we wish to use the nonparametric bootstrap to estimate some feature of the sample median. Compute the probability that the sample median of a nonparametric bootstrap sample will be the smallest observation from the original data X

Use the data in the first column of Table 11.5 on page 699. These data give the boiling points of water at 17 different locations from Forbes experiment. Let F be the distribution from which these boiling points were drawn. We might not be willing to make many assumptions about F. Suppose that we are interested in the bias of the sample median as an estimator of the median of the distribution F. Use the nonparametric bootstrap to estimate this bias. First, do a pilot run to compute the simulation standard error of the simulation approximation, and then see how many bootstrap samples you need in order for your bias estimate (for distribution F) to be within 0.02 of the true bias (for distribution F) with probability at least 0.9.

Use the data in Table 10.6 on page 640. We are interested in the bias of the sample median as an estimator of the median of the distribution. a. Use the nonparametric bootstrap to estimate this bias. b. How many bootstrap samples does it appear that you need in order to estimate the bias to within .05 with probability 0.99?

Use the data in Exercise 16 of Sec. 10.7. a. Use the nonparametric bootstrap to estimate the variance of the sample median. b. How many bootstrap samples does it appear that you need in order to estimate the variance to within .005 with probability 0.95?

Use the blood pressure data in Table 9.2 that was described in Exercise 10 of Sec. 9.6. Suppose now that we are not confident that the variances are the same for the two treatment groups. Perform a parametric bootstrap analysis of the sort done in Example 12.6.10. Use v =10,000 bootstrap simulations. a. Estimate the probability of type I error for a twosample t test whose nominal level is 0 = 0.1. b. Correct the level of the two-sample t test by computing the appropriate quantile of the bootstrap distribution of |U(i)|. c. Compute the simulation standard error for the quantile in part (b).

In Example 12.6.7, let (X, Y ) be a random draw from the sample distribution Fn. Prove that the correlation between X and Y is R in Eq. (12.6.2).

Use the data on fish prices in Table 11.6 on page 707. Suppose that we assume only that the distribution of fish prices in 1970 and 1980 is a continuous joint distribution with finite variances. We are interested in the properties of the sample correlation coefficient. Construct 1000 nonparametric bootstrap samples for solving this exercise. a. Approximate the bootstrap estimate of the variance of the sample correlation. b. Approximate the bootstrap estimate of the bias of the sample correlation. c. Compute simulation standard errors of each of the above bootstrap estimates.

Use the beef hot dog data in Exercise 7 of Sec. 8.5. Form 10,000 nonparametric bootstrap samples to solve this exercise. a. Approximate a 90 percent percentile bootstrap con- fidence interval for the median calorie count in beef hot dogs. b. Approximate a 90 percent percentile-t bootstrap confidence interval for the median calorie count in beef hot dogs. c. Compare these intervals to the 90 percent interval formed using the assumption that the data came from a normal distribution.

The skewness of a random variable was defined in Definition 4.4.1. Suppose that X1,...,Xn form a random sample from a distribution F. The sample skewness is defined as M3 = 1 n n i=1(Xi X)3 # 1 n n i=1(Xi X)2 $3/2 . One might use M3 as an estimator of the skewness of the distribution F. The bootstrap can be used to estimate the bias and standard deviation of the sample skewness as an estimator of . a. Prove that M3 is the skewness of the sample distribution Fn. b. Use the 1970 fish price data in Table 11.6 on page 707. Compute the sample skewness, and then simulate 1000 bootstrap samples. Use the bootstrap samples to estimate the bias and standard deviation of the sample skewness.

Suppose that (X1, Y1), . . . , (Xn, Yn) form a random sample from a bivariate normal distribution with means x and y, variances 2 x and 2 y, and correlation . Let R be the sample correlation. Prove that the distribution of R depends only on , not on x, y, 2 x, or 2 y