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Chapter 12: Q5E (page 822)

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).

Short Answer

Expert verified

This is true because the p.d.f. of random variable with uniform distribution on \((0,1)\) is equal to\(l\).

Step by step solution

01

Definition for importance sampling

  • Many integrals can be advantageously recast as random variable functions.
  • We can estimate integrals that would otherwise be impossible to compute in closed form if we can simulate a large number of random variables with proper distributions.
02

Determine the inverse function and its derivative 

Random variable \(W\)is defined as

\(W = {\mu _2} + {\sigma _2}{\Phi ^{ - 1}}\left( {U\Phi \left( {\frac{{{c_2} - {\mu _2}}}{{{\sigma _2}}}} \right)} \right)\)And function \(h\) is defined as

\(h\left( {{x_2}} \right) = \frac{{{{\left( {2\pi \sigma _2^2} \right)}^{ - 1/2}}\exp \left( {{{\left( {{x_2} - {\mu _2}} \right)}^2}/\left( {2\sigma _2^2} \right)} \right)}}{{\Phi \left( {\left( {{c_2} - {\mu _2}} \right)/{\sigma _2}} \right)}},\;\;\; - \infty < {x_2} \le {c_2}\)

From

\(w = {\mu _2} + {\sigma _2}{\Phi ^{ - 1}}\left( {u\Phi \left( {\frac{{{c_2} - {\mu _2}}}{{{\sigma _2}}}} \right)} \right)\)

It follows that

\(u = \frac{{\Phi \left( {\left( {w - {\mu _2}} \right)/{\sigma _2}} \right)}}{{\Phi \left( {\left( {{c_2} - {\mu _2}} \right)/{\sigma _2}} \right)}}\)Is the inverse transformation.

The derivative of \(\Phi \)is the p.d.f. of a standard normal distribution, hence, the derivative of the inverse function is

\(\frac{\partial }{{\partial w}}u = \frac{1}{{\Phi \left( {\left( {{c_2} - {\mu _2}} \right)/{\sigma _2}} \right)}} \cdot {\left( {2\pi \sigma _2^2} \right)^{ - 1/2}}\exp \left( {\frac{{{{\left( {{x_2} - {\mu _2}} \right)}^2}}}{{2\sigma _2^2}}} \right) = h\left( {{x_2}} \right)\)

Which is p.d.f. ofW.

This is true because the p.d.f. of random variable with uniform distribution on \((0,1)\) is equal to \(1.\)

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Most popular questions from this chapter

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 \(\mu \)

Describe how to convert a random sample \({{\bf{U}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{U}}_{\bf{n}}}\) from the uniform distribution on the interval \({\bf{[0,1]}}\) to a random sample of size \({\bf{n}}\) from the uniform distribution on the interval\({\bf{[a,b]}}\).

Test the gamma pseudo-random number generator on your computer. Simulate 10,000 gamma pseudo-random variables with parameters a and 1 for \(a = 0.5,1,1.5,2,5,\) 10. Then draw gamma quantile plots

Let \({\bf{f}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right){\bf{ = cg}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\) be a joint p.d.f for \(\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right){\bf{,}}\)each \({x_{2\,}}\)let\({{\bf{h}}_{{\bf{2}}\,}}\left( {{{\bf{x}}_{{\bf{1}}\,}}} \right){\bf{ = g}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\) that is what we get by considering \({\bf{g}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\) as a function of \({{\bf{x}}_{{\bf{1}}\,}}\)for fixed \({{\bf{x}}_{2\,}}\)show that there is a multiplicative factor \({{\bf{c}}_{{\bf{2}}\,}}\)that does not depend on such that is the conditional p.d.f of \({{\bf{x}}_{{\bf{1}}\,}}\) given \(\left( {{{\bf{x}}_{{\bf{2}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\)

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.
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