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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

In Sec. 10.2, we discussed \({\chi ^2}\) goodness-of-fit tests for composite hypotheses. These tests required computing M.L.E.'s based on the numbers of observations that fell into the different intervals used for the test. Suppose instead that we use the M.L.E.'s based on the original observations. In this case, we claimed that the asymptotic distribution of the \({x^2}\) test statistic was somewhere between two different \({\chi ^2}\) distributions. We can use simulation to better approximate the distribution of the test statistic. In this exercise, assume that we are trying to test the same hypotheses as in Example 10.2.5, although the methods will apply in all such cases.

a. Simulate \(v = 1000\) samples of size \(n = 23\) from each of 10 different normal distributions. Let the normal distributions have means of \(3.8,3.9,4.0,4.1,\) and \(4.2\) Let the distributions have variances of 0.25 and 0.8. Use all 10 combinations of mean and variance. For each simulated sample, compute the \({\chi ^2}\) statistic Q using the usual M.L.E.'s of \(\mu \) , and \({\sigma ^2}.\) For each of the 10 normal distributions, estimate the 0.9,0.95, and 0.99 quantiles of the distribution of Q.

b. Do the quantiles change much as the distribution of the data changes?

c. Consider the test that rejects the null hypothesis if \(Q \ge 5.2.\) Use simulation to estimate the power function of this test at the following alternative: For each \(i,\left( {{X_i} - 3.912} \right)/0.5\) has the t distribution with five degrees of freedom.

The method of antithetic variates is a technique for reducing the variance of simulation estimators. Antithetic variates are negatively correlated random variables with an expected mean and variance. The variance of the average of two antithetic variates is smaller than the variance of the average of two i.i.d. variables. In this exercise, we shall see how to use antithetic variates for importance sampling, but the method is very general. Suppose that we wish to compute \(\smallint \,g\left( x \right)\,\,dx\), and we wish to use the importance function f. Suppose that we generate pseudo-random variables with the p.d.f. f using the integral probability transformation. For \(\,{\bf{i = 1,2,}}...{\bf{,\nu ,}}\,\)let \({{\bf{X}}^{\left( {\bf{i}} \right)}}{\bf{ = }}{{\bf{F}}^{{\bf{ - 1}}}}\left( {{\bf{1 - }}{{\bf{U}}^{\left( {\bf{i}} \right)}}} \right)\), where \({{\bf{U}}^{\left( {\bf{i}} \right)}}\)has the uniform distribution on the interval (0, 1) and F is the c.d.f. Corresponding to the p.d.f. f . For each \(\,{\bf{i = 1,2,}}...{\bf{,\nu ,}}\,\) define

\(\begin{aligned}{l}{{\bf{T}}^{\left( {\bf{i}} \right)}}{\bf{ = }}{{\bf{F}}^{ - {\bf{1}}}}\left( {{\bf{1}} - {{\bf{U}}^{\left( {\bf{i}} \right)}}} \right)\,\,{\bf{.}}\\{{\bf{W}}^{\left( {\bf{i}} \right)}}{\bf{ = }}\frac{{{\bf{g}}\left( {{{\bf{X}}^{\left( {\bf{i}} \right)}}} \right)}}{{{\bf{f}}\left( {{{\bf{X}}^{\left( {\bf{i}} \right)}}} \right)}}\\{{\bf{V}}^{\left( {\bf{i}} \right)}}{\bf{ = }}\frac{{{\bf{g}}\left( {{{\bf{T}}^{\left( {\bf{i}} \right)}}} \right)}}{{{\bf{f}}\left( {{{\bf{T}}^{\left( {\bf{i}} \right)}}} \right)}}\\{{\bf{Y}}^{\left( {\bf{i}} \right)}}{\bf{ = 0}}{\bf{.5}}\left( {{{\bf{W}}^{\left( {\bf{i}} \right)}}{\bf{ + k}}{{\bf{V}}^{\left( {\bf{i}} \right)}}} \right){\bf{.}}\end{aligned}\)

Our estimator of\(\smallint \,{\bf{g}}\left( {\bf{x}} \right)\,\,{\bf{dx}}\)is then\({\bf{Z = }}\frac{{\bf{I}}}{{\bf{\nu }}}\sum\nolimits_{{\bf{i = 1}}}^{\bf{\nu }} {{{\bf{Y}}^{\left( {\bf{i}} \right)}}{\bf{.}}} \).

a. Prove that\({T^{\left( i \right)}}\)has the same distribution as\({X^{\left( i \right)}}\).

b. Prove that\({\bf{E}}\left( {\bf{Z}} \right){\bf{ = }}\smallint \,\,{\bf{g}}\left( {\bf{x}} \right)\,\,{\bf{dx}}\).

c. If\({\bf{g}}\left( {\bf{x}} \right)\,{\bf{/f}}\left( {\bf{x}} \right)\)it is a monotone function, explain why we expect it \({{\bf{V}}^{\left( {\bf{i}} \right)}}\)to be negatively correlated.

d. If \({{\bf{W}}^{\left( {\bf{i}} \right)}}\) and \({{\bf{V}}^{\left( {\bf{i}} \right)}}\)are negatively correlated, show that Var(Z) is less than the variance one would get with 2v simulations without antithetic variates.

Let \(U\) have the uniform distribution on the interval\((0,1)\). Show that the random variable \(W\)defined in Eq. (12.4.6) has the p.d.f. \(h\)defined in Eq. (12.4.5).

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?

Let \({{\bf{X}}_{\bf{1}}},...,{{\bf{X}}_n}\) be i.i.d. with the normal distribution having mean \(\mu \) and precision \(\tau \).Gibbs sampling allows one to use a prior distribution for \(\left( {\mu ,\tau } \right)\) in which \(\mu \) and\(\tau \) are independent. With mean \({\mu _0}\) and variance, \({\gamma _0}\) Let the prior distribution of \(\tau \)being the gamma distribution with parameters \({\alpha _0}\) and \({\beta _0}\) .

a. Show that the Table \({\bf{12}}{\bf{.8}}\) specifies the appropriate conditional distribution for each parameter given the other.

b. Use the new Mexico nursing home data(Examples \({\bf{12}}{\bf{.5}}{\bf{.2}}\,{\bf{and}}\,{\bf{12}}{\bf{.5}}{\bf{.3}}\) ). Let the prior hyperparameters be \({{\bf{\alpha }}_{\bf{0}}}{\bf{ = 2,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 6300,}}{{\bf{\mu }}_{\bf{0}}}{\bf{ = 200}}\), and \({\gamma _0} = 6.35 \times {10^{ - 4}}.\) Implement a Gibbs sampler to find the posterior distribution \(\left( {\mu ,\tau } \right).\,\) . In particular, calculate an interval containing \(95\) percent of the posterior distribution of \(\mu \)
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