Chapter 12: Q12.3-4E (page 815)
Show how to simulate Cauchy random variables using the probability integral transformation.
\(X = \tan \left( {\pi U - \frac{\pi }{2}} \right)\)
Thus, simulate \(U\) to obtain the simulation of \(r.v.\)with \(p.d.f.\) from Cauchy distribution.
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Suppose that \({x_1},...,{x_n}\) from a random sample from an exponential distribution with parameter\(\theta \).Explain how to use the parametric bootstrap to estimate the variance of the sample average\(\overline X \).(No simulation is required.)
In this problem, we shall outline a Bayesian solution to the problem described in Example 7.5.10 on page 423. Let \(\tau \)= 1/ฯ2 and use a proper normal-gamma prior to the form described in Sec. 8.6. In addition to the two parameters, ฮผ and \(\tau \), introduce n additional parameters. For i = 1, n, let Yi = 1 if Xi came from the normal distribution with mean ฮผ and precision \(\tau \), and let Yi = 0 if Xi came from the standard normal distribution
a. Find the conditional distribution of ฮผ given ฯ; Y1, ..., Yn; and X1, Xn.
b. Find the conditional distribution of ฯ given ฮผ; Y1, ..., Yn; and X1, Xn.
c. Find the conditional distribution of Yi given ฮผ; ฯ; X1, Xn; and the other Yj's.
d. Describe how to find the posterior distribution of ฮผ \(\tau \)using Gibbs sampling.
e. Prove that the posterior mean of Yi is the posterior probability that Xi came from the normal distribution with unknown mean and variance
The skewness of a random variable was defined in Definition 4.4.1. Suppose that \({X_1},...,{X_n}\) form a random sample from a distribution \(F\). The sample skewness is defined as
\({M_3} = \frac{{\frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^3}} }}{{{{\left( {\frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} } \right)}^{3/2}}}}\)
One might use \({M_3}\) as an estimator of the skewness \(\theta \) of the distribution F. The bootstrap can estimate the bias and standard deviation of the sample skewness as an estimator \(\theta \).
a. Prove that \({M_3}\) is the skewness of the sample distribution \({F_{{n^*}}}\)
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.
In Example 12.6.7, let \(\left( {{X^*},{Y^*}} \right)\) be a random draw from the sample distribution\({F_n}\) . Prove that the correlation \({X^*} and {Y^*}\) is R in Eq. (12.6.2).
The \({\chi ^2}\) goodness-of-fit test (see Chapter 10) is based on an asymptotic approximation to the distribution of the test statistic. For small to medium samples, the asymptotic approximation might not be very good. Simulation can be used to assess how good the approximation is. Simulation can also be used to estimate the power function of a goodness-of-fit test. For this exercise, assume that we are performing the test that was done in Example 10.1.6. The idea illustrated in this exercise applies to all such problems.
a. Simulate \(v = 10,000\) samples of size \(n = 23\) from the normal distribution with a mean of 3.912 and variance of 0.25. For each sample, compute the \({\chi ^2}\) goodness of fit statistic Q using the same four intervals that were used in Example 10.1.6. Use the simulations to estimate the probability that Q is greater than or equal to the 0.9,0.95 and 0.99 quantiles of the \({\chi ^2}\) distribution with three degrees of freedom.
b. Suppose that we are interested in the power function of a \({\chi ^2}\) goodness-of-fit test when the actual distribution of the data is the normal distribution with a mean of 4.2 and variance of 0.8. Use simulation to estimate the power function of the level 0.1,0.05 and 0.01 tests at the alternative specified.
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