Let Y have the F distribution with m and n degrees of

Prove that the formula in Eq. (12.4.1) is the same as importance sampling in which the importance function is the p.d.f. of the uniform distribution on the interval [a, b]

Let g be a function, and suppose that we wish to compute the mean of g(X) where X has the p.d.f. f . Suppose that we can simulate pseudo-random values with the p.d.f. f . Prove that the following are the same: Simulate X(i) values with the p.d.f. f , and average the values of g(X(i)) to obtain the estimator. Do importance sampling with importance function f to estimate the integral g(x)f (x) dx

Let Y have the F distribution with m and n degrees of freedom. We wish to estimate Pr(Y > c). Consider the p.d.f. f (x) = (n/2)cn/2 xn/2+1 if x>c, 0 otherwise. a. Explain how to simulate pseudo-random numbers with the p.d.f. f . b. Explain how to estimate Pr(Y > c) using importance sampling with the importance function f . c. Look at the form of the p.d.f. of Y , Eq. (9.7.2), and explain why importance sampling might be more ef- ficient than sampling i.i.d. F random variables with m and n degrees of freedom if c is not small.

We would like to calculate the integral  0 log(1 + x) exp(x) dx. a. Simulate 10,000 exponential random variables with parameter 1 and use these to estimate the integral. Also, find the simulation standard error of your estimator. b. Simulate 10,000 gamma random variables with parameters 1.5 and 1 and use these to estimate the integral (importance sampling). Find the simulation standard error of the estimator. (In case you do not have the gamma function available, (1.5) = /2.) c. Which of the two methods appears to be more effi- cient? Can you explain why?

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).

. Suppose that we wish to estimate the integral  1 x2 2 e0.5x2 dx. In parts (a) and (b) below, use simulation sizes of 1000. a. Estimate the integral by importance sampling using random variables having a truncated normal distribution. That is, the importance function is 1 2[1 (1)] e0.5x2 , for x > 1. b. Estimate the integral by importance sampling using random variables with the p.d.f. x exp(0.5[1 x2]), for x > 1. Hint: Prove that such random variables can be obtained as follows: Start with a random variable that has the exponential distribution with parameter 0.5, add 1, then take the square root. c. Compute and compare simulation standard errors for the two estimators in parts (a) and (b). Can you explain why one is so much smaller than the other?

Let (X1, X2) have the bivariate normal distribution with both means equal to 0, both variances equal to 1, and the correlation equal to 0.5. We wish to estimate = Pr(X1 2, X2 1) using simulation. a. Simulate a sample of 10,000 bivariate normal vectors with the above distribution. Use the proportion of vectors satisfying the two inequalities X1 2 and X2 1 as the estimator Z of . Also compute the simulation standard error of Z. b. Use the method described in Example 12.4.3 with 10,000 simulations to produce an alternative estimator Zof . Compute the simulation standard error of Zand compare Zto the estimate in part (a).

. Suppose that we wish to approximate the integral g(x) dx. Suppose that we have a p.d.f. f that we shall use as an importance function. Suppose that g(x)/f (x) is bounded. Prove that the importance sampling estimator has finite variance.

. Let F be a continuous strictly increasing c.d.f. with p.d.f. f . Let V have the uniform distribution on the interval [a, b] with 0 a

For the situation described in Exercise 6, use stratified importance sampling as follows: Divide the interval (1, )into five intervals that each have probability 0.2 under the importance distribution. Sample 200 observations from each interval. Compute the simulation standard error. Compare this simulation to the simulation in Exercise 6 for each of parts (a) and (b).

. In the notation used to develop stratified importance sampling, prove that X = XJ and X have the same distribution. Hint: The conditional p.d.f. of X given J = j is fj . Use the law of total probability.

Consider again the situation described in Exercise 15 of Sec. 12.2. Suppose that Wu has the Laplace distribution with parameters = 0 and = 0.1u1/2. See Eq. (10.7.5) for the p.d.f. a. Prove that the m.g.f. of Wu is (t) =  1 t2u 100 1 , for 10u1/2 S0. How much smaller is the simulation standard error

The method of control variates is a technique for reducing the variance of a simulation estimator. Suppose that we wish to estimate = E(W ). A control variate is another random variable V that is positively correlated with W and whose mean we know. Then, for every constant k > 0, E(W kV + k) = . Also, if k is chosen carefully, Var(W kV + k) < Var(W ). In this exercise, we shall see how to use control variates for importance sampling, but the method is very general. Suppose that we wish to compute g(x) dx, and we wish to use the importance function f . Suppose that there is a function h such that h is similar to g but h(x) dx is known to equal the value c. Let k be a constant. Simulate X(1) ,...,X(v) with the p.d.f. f , and define W(i) = g(X(i)) f (X(i)) , V (i) = h(X(i)) f (X(i)) , Y (i) = W(i) kV (i), for all i. Our estimator of g(x) dx is then Z = 1 v v i=1 Y (i) + kc. a. Prove that E(Z) = g(x) dx. b. Let Var(W(i)) = 2 W and Var(V (i)) = 2 V . Let be the correlation between W(i) andV (i). Prove that the value of k that makes Var(Z) the smallest is k = W /V .

Suppose that we wish to integrate the same function g(x) as in Example 12.4.1. a. Use the method of control variates that was described in Exercise 13 to estimate g(x) dx. Let h(x) = 1/(1 + x2) for 0

The method of antithetic variates is a technique for reducing the variance of simulation estimators. Antithetic variates are negatively correlated random variables that share a common mean and common 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 g(x) 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 probability integral transformation. That is, fori = 1,...,v, let X(i) = F 1(U(i)), where U(i) 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 i = 1,...,v, define T (i) = F 1 (1 U(i)), W(i) = g(X(i)) f (X(i)) , V (i) = g(T (i)) f (T (i)) , Y (i) = 0.5 # W(i) + V (i)$ . Our estimator of g(x) dx is then Z = 1 v v i=1 Y (i). a. Prove that T (i) has the same distribution as X(i). b. Prove that E(Z) = g(x) dx. c. If g(x)/f (x) is a monotone function, explain why we would expect W(i) and V (i) to be negatively correlated. d. If W(i) and V (i) are negatively correlated, show that Var(Z) is less than the variance one would get with 2v simulations without antithetic variates.

Use the method of antithetic variates that was described in Exercise 15. Let g(x) be the function that we tried to integrate in Example 12.4.1. Let f (x) be the function f3 in Example 12.4.1. Estimate Var(Y (i)), and compare it to 2 3 from Example 12.4.1

For each of the exercises in this section that requires a simulation, see if you can think of a way to use control variates or antithetic variates to reduce the variance of the simulation estimator.