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Chapter 12: Q12.1-3E (page 791)
If \({\bf{X}}\) has the Cauchy distribution, the mean \({\bf{X}}\)does not exist. What would you expect to happen if you simulated a large number of Cauchy random variables and computed their average?
The average would increase or decrease depending on the observations.
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Let \({\bf{Y}}\) be a random variable with some distribution. Suppose that you have available as many pseudo-random variables as you want with the same distribution as \({\bf{Y}}\). Describe a simulation method for estimating the skewness of the distribution of \({\bf{Y}}\). (See Definition 4.4.1.)
If \({\bf{X}}\)has the \({\bf{p}}.{\bf{d}}.{\bf{f}}.\)\({\bf{1/}}{{\bf{x}}^{\bf{2}}}\)for\({\bf{x > 1}}\), the mean of \({\bf{X}}\) is infinite. What would you expect to happen if you simulated a large number of random variables with this \({\bf{p}}.{\bf{d}}.{\bf{f}}.\) and computed their average?
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 data consisting of 30 lactic acid concentrations in cheese,10 from example 8.5.4 and 20 from Exercise 16 in sec.8,6, Fit the same model used in Example 8.6.2 with the same prior distribution, but this time use the Gibbs sampling algorithm in Example 12.5.1. simulate 10,000 pairs of \(\left( {{\bf{\mu ,\tau }}} \right)\) parameters. Estimate the posterior mean of \({\left( {\sqrt {{\bf{\tau \mu }}} } \right)^{ - {\bf{1}}}}\), and compute the standard simulation error of the estimator.
Suppose that \(\left( {{X_1},{Y_1}} \right),...,\left( {{X_n},{Y_n}} \right)\) form a random sample from a bivariate normal distribution with means \({\mu _x} and {\mu _y},variances \sigma _x^2and \sigma _y^2,and correlation \rho .\) Let R be the sample correlation. Prove that the distribution of R depends only on \(\rho ,not on {\mu _x},{\mu _y},\sigma _x^2,or \sigma _y^2.\)
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