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Chapter 12: Q12.2-14E (page 803)

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.)

Short Answer

Expert verified

Average\({Z_i} = \frac{{{{\left( {{Y_i} - \bar Y} \right)}^3}}}{{{S^2}}}\)

Step by step solution

01

Definition for skewness of the distribution:

Skewness is a measure of a distribution's symmetry. The mode of a distribution is its highest point. The mode denotes the\(x - \)axis response value with the highest probability. A skewed distribution is one in which the tail on one side of the mode is fatter or longer than the tail on the other: it is asymmetrical.

02

Find a simulation method for estimating the skewness of the distribution of \(Y\):

Definition. The skewness of a random variable\(X\)with finite third moment is defined as

\(E\left[ {\frac{{{{(X - \mu )}^3}}}{{{\sigma ^3}}}} \right]\)

Let\(\nu \)be a large number, and simulate\({Y_1},{Y_2}, \ldots ,{Y_\nu }\)random variables from the specific distribution. By averaging simulated values

\({Z_i} = \frac{{{{\left( {{Y_i} - \bar Y} \right)}^3}}}{{{S^2}}}\)

Where\(\bar Y\)is the average and\({S^2}\)sample average of the simulated values\(\nu \)

The estimate of skewness is obtained.

Hence, the estimate of the skewness is

\(Z = \frac{1}{\nu }{Z_i}\)

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Most popular questions from this chapter

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?

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.

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.)

Use the data in table 11.19 on page 762.This time fits the model developed in Example 12.5.6.use the prior hyperparameters \(\,{{\bf{\lambda }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = }}{{\bf{\alpha }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}\,{\bf{ = 1,}}\,\,{{\bf{\beta }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = 0}}{\bf{.1}},{{\bf{\mu }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = 0}}{\bf{.001}}\)and \({{\bf{\psi }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = 800}}\)obtain a sample of 10,000 from the posterior joint distribution of the parameters. Estimate the posterior mean of the three parameters \({{\bf{\mu }}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{\mu }}_{\scriptstyle{\bf{2}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{\mu }}_{\scriptstyle{\bf{3}}\atop\scriptstyle\,}}\)

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?
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