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Chapter 12: Q4SE (page 850)

Let X and Y be independent random variables with \(X\) having the t distribution with five degrees of freedom and Y having the t distribution with three degrees of freedom. We are interested in \(E\left( {|X - Y|} \right).\)

a. Simulate 1000 pairs of \(\left( {{X_i},{Y_i}} \right)\) each with the above joint distribution and estimate \(E\left( {|X - Y|} \right).\)

b. Use your 1000 simulated pairs to estimate the variance of \(|X - Y|\) also.

c. Based on your estimated variance, how many simulations would you need to be 99 percent confident that your estimator is within the actual mean?

Short Answer

Expert verified

(a) Close to\(1.3755\)for\(n = 10000.\)

(b) Close to\(1.3656\)for\(n = 10000.\)

((c)) n=10071.

Step by step solution

01

(a) To find the estimated expected value varies around.

The code below gives an estimate of the\(E\left( {|X - Y|} \right).\)To obtain the estimate, and one should generate random numbers from the joint distribution\(\left( {X, Y} \right).\)As the random variables are independent, one may generate random numbers from the distribution of X and distribution of Y.

For the simulation, the sample size of \(n = 10000\) is used. The estimate of the expected value varies around \(1.3755.\)

02

(b) To find the estimated variance around

The same code gives the estimate of the variance of\(\left( {X, Y} \right).\)using the sample variance of generated sample for\(\left( {X, Y} \right).\)The estimate is around 1.3656.

03

(c) To find the simulation and mean.

Lastly, the necessary simulation sample size may be found from

As \(\sigma \) is unknown in this case, use the estimate obtained in (b), which is \(\sqrt {1.3656} .\) Here, \(\gamma = 0.99\) and . Remember that \(\Phi \) is the cdf of the standard normal distribution. Then, the necessary sample size which satisfies the requirements is

\(\begin{aligned}{c}n = {\Phi ^{ - 1}}\frac{{1 + \gamma }}{2}\frac{{{s^2}}}{ \in }\\ = {\Phi ^{ - 1}}\frac{{1 + 0.99}}{2}\;\;\;{\kern 1pt} \frac{{{{\sqrt {1.367} }^2}}}{{0.01}}\\ = 10071\end{aligned}\)

However, the simulation sample size will differ every time you run the code (unless you place "seed" before).

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Most popular questions from this chapter

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).

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 to estimate the variance to within .005 with a probability of 0.95?

Consider the power calculation done in Example 9.5.5.

a. Simulate \({v_0} = 1000\) i.i.d. noncentral t pseudo-random variables with 14 degrees of freedom and noncentrality parameter \(1.936.\)

b. Estimate the probability that a noncentral t random variable with 14 degrees of freedom and noncentrality parameter \(1.936\) is at least \(1.761.\) Also, compute the standard simulation error.

c. Suppose that we want our estimator of the noncentral t probability in part (b) to be closer than \(0.01\) the true value with probability \(0.99.\) How many noncentral t random variables do we need to simulate?

Let \({\bf{f}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\) be a joint p.d.f. Suppose that \(\left( {{{\bf{x}}_{{\bf{1}}\,}}^{\left( {\bf{i}} \right)}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}^{\left( {\bf{i}} \right)}} \right)\)has the joint p.d.f. Let \(\left( {{{\bf{x}}_{{\bf{1}}\,}}^{\left( {{\bf{i + 1}}} \right)}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}^{\left( {{\bf{i + 1}}} \right)}} \right)\)be the result of applying steps \(2\,\,and\,\,3\) of the Gibbs sampling algorithm on-page \({\bf{824}}\). Prove that \(\left( {{{\bf{x}}_{{\bf{1}}\,}}^{\left( {{\bf{i + 1}}} \right)}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}^{\left( {\bf{i}} \right)}} \right)\) and \(\left( {{{\bf{x}}_{{\bf{1}}\,}}^{\left( {{\bf{i + 1}}} \right)}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}^{\left( {{\bf{i + 1}}} \right)}} \right)\)also have the joint p.d.f. f.

Let \({{\bf{z}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{z}}_{\scriptstyle{\bf{2}}\atop\scriptstyle\,}}....\) from a Markov chain, and assume that distribution of \({z_{\scriptstyle1\atop\scriptstyle\,}}\)is the stationary distribution. Show that the joint distribution \(\left( {{{\bf{z}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{z}}_{\scriptstyle{\bf{2}}\atop\scriptstyle\,}}} \right)\)of is the same as the joint distribution of \(\left( {{{\bf{z}}_{\scriptstyle{\bf{i}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{z}}_{\scriptstyle{\bf{i + 1}}\atop\scriptstyle\,}}} \right)\) for all\(i > 1\) convenience, you may assume that the Markov chain has finite state space, but the result holds in general.
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