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Chapter 12: Q3SE (page 850)

Test the t pseudo-random number generator on your computer. Simulate 10,000 t pseudo-random variables with m degrees of freedom for m=1,2,5,10,20. Then draw t quantile plots

Short Answer

Expert verified

Generate using \(R\) with \(rt(10000,df),\) use the code for \(Q - Q plots. \)

Step by step solution

01

Pseudo-random generator

The procedures widely used to produce random values are not random. When random variables are produced by a predetermined method but pass the predefined test statistic for random chance, they are called pseudo-random numbers

02

To draw t quantile plots.

In RStudio, you may generate a random sample of size \(n = 10000\) from a Student t distribution with the built-in function r t (10000, d f), df is the degrees of freedom of the student t distribution.

Then, plot the Q-Q plot using the code given below. From the figures given below for different choices of degrees of freedom, one may see that the theoretical quantiles versus the sample quantiles create a reasonably straight line, meaning that the sample comes from the t distribution.

The noticeable fact is that there are extreme values for what is known as the Cauchy distribution. This is expected \(d f = 1,\) as the sample for such distribution has a significant variance. The other plots do not have such huge tails.

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Most popular questions from this chapter

Use the data in Table 10.6 on page 640. We are interested in the bias of the sample median as an estimator of the median of the distribution.

a. Use the non-parametric bootstrap to estimate this bias.

b. How many bootstrap samples does it appear that you need in order to estimate the bias to within .05 with a probability of 0.99?

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.

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.

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.

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.
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