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Instructions: Solutions to problems 1 and 2 are to be submitted on Blackboard (PDF les strongly preferred). You are strongly encouraged to do problems 3{5 but these are not to be submitted for grading.

1. An interesting variation of rejection sampling is the ratio of uniforms method. We start Z 1

by taking a bounded function h with h(x) 0 for all x and h(x) dx < 1. We then

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de ne the region

q

Ch = (u; v) : 0 u h(v=u)

and generate (U; V ) uniformly distributed on Ch. We then de ne the random variable X = V =U.

 (a) The joint density of (U; V ) is f(u; v) = 1 for (u; v) 2 Ch jChj

where jChj is the area of Ch. Show that the joint density of (U; X) is

 u for 0 u q g(u; x) = h(x) h jC j

and that the density of X is h(x) for some > 0.

1. The implementation of this method is somewhat complicated by the fact that it is

typically di cult to sample (U; V ) from a uniform distribution on Ch. However, it is usually possible to nd a rectangle of the form Dh = f(u; v) : 0 u u+; v v v+g such that

Ch is contained within Dh. Thus to draw (U; V ) from a uniform distribution on Ch, we can use rejection sampling where we draw proposals (U ; V ) from a uniform distribution on the rectangle Dh; note that the proposals U and V are independent random variables with Unif(0; u+) and Unif(v ; v+) distributions, respectively. Show that we can de ne u+, v and v+ as follows:

 + = x q x q + x q u h(x) v h(x) v h(x): max = min x = max x

(Hint: It su ces to show that if (u; v) 2 Ch then (u; v) 2 Dh where Dh is de ned using u+, v , and v+ above.)

(c) Implement (in R) the method above for the standard normal distribution taking h(x) =

q q

exp( x2=2). In this case, u+ = 1, v = 2=e = 0:8577639, and v+ = 2=e = 0:8577639. What is the probability that proposals are accepted?

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2. Suppose we observe y1; ; yn where

yi = i + “i (i = 1; ; n)

where f”ig is a sequence of random variables with mean 0 and nite variance representing noise. We will assume that 1; ; n are dependent in the sense that j i i 1j is small for most values of i. The least squares estimates of 1; ; n are trivial | bi = yi for all i

• but we can modify least squares in a number of ways to accommodate the \smoothness” assumption on f ig. In this problem, we will consider estimating f ig by minimizing

 n n Xi (yi i)2 +( i i 1)2 X =1 i=2 n

where > 0 is a tuning parameter. (The term X( i i 1)2 is a \roughness” penalty. In

 n i=2 practice, it is more common to use j i i 1j as it better estimates jumps in f ig.) =2 Xi (a) Show the estimates b1; ; bn satisfy the equations y1 = (1 + ) 12 j y j = j 1 b+ (1 b j+1 (j = 2; ; n 1) + 2 ) yn bn 1 b b = + (1 + ) n: b b

(Note that if we write this in matrix form y = A b, the matrix A is sparse.)

1. Show (using results from class) that both the Jacobi and Gauss-Seidel algorithms can be used to compute the estimates de ned in part (a).

1. Write a function in R to implement the Gauss-Seidel algorithm above. The inputs for this function are a vector of responses y and the tuning parameter lambda. Test your function (for various tuning parameters) on data generated by the following command:

> y <- c(rep(0,250),rep(1,250),rep(0,50),rep(1,450)) + rnorm(1000,0,0.1)

How does varying change the estimates, particularly the estimates of the transitions from 0 to 1 in the step function?

Supplemental problems (not to hand in):

1. To generate random variables from some distributions, we need to generate two indepen-dent two independent random variables Y and V where Y has a uniform distribution over some nite set and V has a uniform distribution on [0; 1]. It turns out that Y and V can be generated from a single Unif(0; 1) random variable U.

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(a) Suppose for simplicity that the nite set is f0; 1; ; n 1g for some integer n 2. For U Unif(0; 1), de ne

Y = bnUc and V = nU Y

where bxc is the integer part of x. Show that Y has a uniform distribution on the set f0; 1; ; n 1g, V has a uniform distribution on [0; 1], and Y and V are independent.

(b) What happens to the precision of V de ned in part (a) as n increases? (For example, if

• has 16 decimal digits and n = 106, how many decimal digits will V have?) Is the method in part (a) particularly feasible if n is very large?

1. One issue with rejection sampling is a lack of e ciency due to the rejection of random variables generated from the proposal density. An alternative is the acceptance-complement (A-C) method described below.

Suppose we want to generate a continuous random variable from a density f(x) and that

f(x) = f1(x) + f2(x) (where both f1 and f2 are non-negative) where f1(x) g(x) for some density function g. Then the A-C method works as follows:

1. Generate two independent random variables U Unif(0; 1) and X with density g.

1. If U f1(X)=g(X) then return X.

1. Otherwise (that is, if U > f1(X)=g(X)) generate X from the density

 f2 (x) = f2 (x) : Z 1

f2(t) dt

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Note that we must be able to easily sample from g and f2 in order for the A-C method to be e cient; in some cases, they can both be taken to be uniform distributions.

1. Show that the A-C method generates a random variable with a density f. What is the probability that the X generated in step 1 (from g) is \rejected” in step 2?

1. Suppose we want to sample from the truncated Cauchy density

 f(x) = 2 ( 1 x 1) (1 + x2)

using the A-C method with f2(x) = k, a constant, for 1 x 1 (so that f2 (x) = 1=2 is a uniform density on [ 1; 1]) with

f1(x) = f(x) f2(x) = f(x) k ( 1 x 1):

If g(x) is also a uniform density on [ 1; 1] for what range of values of k can the A-C method be applied?

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(c) De ning f1, f2, and g as in part (b), what value of k minimizes the probability that X generated in step 1 of the A-C algorithm is rejected?

1. Suppose we want to generate a random variable X from the tail of a standard normal distribution, that is, a normal distribution conditioned to be greater than some constant b. The density in question is

 f(x) = exp( x2 =2) for x b p

2 (1 (b))

with f(x) = 0 for x < b where (x) is the standard normal distribution function. Consider rejection sampling using the shifted exponential proposal density

g(x) = b exp( b(x b)) for x b.

1. De ne Y be an exponential random variable with mean 1 and U be a uniform random variable on [0; 1] independent of Y . Show that the rejection sampling scheme de nes X =

b + Y =b if

Y 2

2 ln(U) b2 :

(Hint: Note that b + Y =b has density g.)

(b) Show the probability of acceptance is given by

p

2 b(1 (b)):

exp( b2=2)

What happens to this probability for large values of b? (Hint: You need to evaluate M = max f(x)=g(x).)

(c) Suppose we replace the proposal density g de ned above by

g (x) = exp( (x b)) for x b.

(Note that g is also a shifted exponential density.) What value of maximizes the proba-bility of acceptance? (Hint: Note that you are trying to solve the problem

min max f(x)

>0 x b g (x)

for . Because the density g (x) has heavier tails, the minimax problem above will have the same solution as the maximin problem

max min f(x)

x b >0 g (x)

which may be easier to solve.)

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