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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{6 but these are not to be submitted for grading.

1. Suppose that S is an n n matrix where n may be very large and the elements of S may not be explicitly de ned. We are interested in approximating the trace of S, that is, the sum

of its diagonal elements. For example, if S is a smoothing matrix in regression (yb = Sy) then the trace of S gives a measure of the e ective number of parameters using in the smoothing method. (In multiple regression models, the smoothing matrix is the projection matrix X(XT X) 1XT whose trace is the number of columns of X.)

1. Show that if A and B are m n and n m matrices, respectively, then tr(AB) = tr(BA). (This is a well-known fact but humour me with a proof!)

1. Suppose that V is a random vector of length n such that E[V V T ] = I. If S is an n n

non-random matrix, show that

h i h i h i

E V T SV = E tr SV V T = tr SE V V T = tr(S)

and so tr(S) can be estimated by

1 m

X T

tr(dS) = m i=1 V i SV i

where V 1; ; V m are independent random vectors with E[V iV Ti ] = I.

1. Suppose that the elements of each V i are independent, identically distribution random variables with mean 0 and variance 1. Show that Var(tr(dS) is minimized by taking the elements of V i to be 1 each with probability 1=2.

Hint: This is easier than it looks { Var(V T SV ) = E[(V T SV )2] tr(S)2 so it su ces to

 minimize n n n n E[(V T SV )2] = sijsk`E(ViVjVkV`):

X X X X

i=1 j=1 k=1 `=1

Given our conditions on the elements of V i, V1; ; Vn, most of E(ViVjVkV`) are either 0 or 1. You should be able to show that

n

E[(V T SV )2] = X s2iiE(Vi4) + constant

i=1

1

and nd Vi to minimize E(Vi4) subject to E(Vi2) = 1.

(d) Suppose we estimate the function g in the non-parametric regression model

yi = g(xi) + “i for i = 1; ; n

using loess (i.e. the R function loess) where the smoothness is determined by the parameter span lying between 0 and 1. Given a set of predictors fxig and a value of span, write an R function to approximate the e ective number of parameters.

2. Suppose that X1; ; Xn are independent Gamma random variables with common density

 f(x; ; ) = x 1 exp( x) for x > 0 ( )

where > 0 and > 0 are unknown parameters.

1. The mean and variance of the Gamma distribution are = and = 2, respectively. Use these to de ne method of moments estimates of and based on the sample mean and variance of the data x1; ; xn

1. Derive the likelihood equations for the MLEs of and and derive a Newton-Raphson

algorithm for computing the MLEs based on x1; ; xn. Implement this algorithm in R and test on data generated from a Gamma distribution (using the R function rgamma). Your function should also output an estimate of the variance-covariance matrix of the MLEs { this can be obtained from the Hessian of the log-likelihood function.

Important note: To implement the Newton-Raphson algorithm, you will need to compute the rst and second derivatives of ln ( ). These two derivatives are called (respectively) the digamma and trigamma functions, and these functions are available in R as digamma and trigamma; for example,

• gamma(2) # gamma function evaluated at 2 [1] 1

• digamma(2) # digamma function evaluated at 2 [1] 0.4227843

• trigamma(2) # trigamma function evaluated at 2 [1] 0.6449341

2

Supplemental problems:

3. Consider LASSO estimation in linear regression where we de ne b to minimize

 n p Xi (yi y xiT )2 +j jj X =1 j=1

for some > 0. (We assume that the predictors are centred and scaled to have mean 0 and variance 1, in which case y is the estimate of the intercept.) Suppose that the least squares estimate (i.e. for = 0) is non-unique | this may occur, for example, if there is some exact linear dependence in the predictors or if p > n. De ne

n

= min X(yi y xTi )2

i=1

and the set

 C = n ) : ( : (yi y xiT )2 = Xi =1

We want to look at what happens to the LASSO estimate b as # 0.

(a) Show that b minimizes

 ( n (yi y xiT 1 Xi =1 (b) Find the limit of ( 1 n (yi y =1 Xi
• p

)2 + X j jj:

j=1

)

xTi )2

as # 0 as a function of . (What happens when 62 ?)C Use this to deduce that as # 0,

 b b b p minimizes jX ! 0 where 0 j jj on the set C. =1
1. Show that b0 is the solution of a linear programming problem. (Hint: Note that C can be expressed in terms of satisfying p linear equations.)

4. Consider minimizing the function

g(x) = x2 2 x + jxj

where > 0 and 0 < < 1. (This problem arises, in a somewhat more complicated form, in shrinkage estimation in regression.) The function jxj has a \cusp” at 0, which mean that if is su cient large then g is minimized at x = 0.

3

(a) g is minimized at x = 0 if, and only if,

 2 “ 2 2 #1 j j2 : (1) 2 2

Otherwise, g is minimized at x satisfying g0(x ) = 0. Using R, compare the following two iterative algorithms for computing x (when condition (1) does not hold):

(i) Set x0 = and de ne

 x = jxk 1j k = 1; 2; 3; k 2 xk 1

(ii) The Newton-Raphson algorithm with x0 = .

Use di erent values of , , and to test these algorithms. Which algorithm is faster?

1. Functions like g arise in so-called bridge estimation in linear regression (which are gener-alizations of the LASSO) { such estimation combines the features of ridge regression (which

shrinks least squares estimates towards 0) and model selection methods (which produce ex-act 0 estimates for some or all parameters). Bridge estimates b minimize (for some > 0

and > 0),

 n p (yi xiT )2 + j jj : (2) Xi X =1 j=1

See the paper by Huang, Horowitz and Ma (2008) (\Asymptotic properties of bridge esti-mators in sparse high-dimensional regression models” Annals of Statistics. 36, 587{613) for details. Describe how the algorithms in part (a) could be used to de ne a coordinate descent algorithm to nd b minimizing (2) iteratively one parameter at a time.

(c) Prove that g is minimized at 0 if, and only if, condition (1) in part (a) holds.

1. Suppose that A is a symmetric non-negative de nite matrix with eigenvalues 12

n 0. Consider the following algorithm for computing the maximum eigenvalue 1:

 Given x0, de ne for k = 0; 1; 2; , xk+1 = Axk and k+1 = xkT+1Axk+1 . kAxkk2 xkT+1xk+1

Under certain conditions, k ! 1, the maximum eigenvalue of A; this algorithm is known as the power method and is particularly useful when A is sparse.

(a) Suppose that v1; ; vn are the eigenvectors of A corresponding to the eigenvalues 1; ; n. Show that k ! 1 if xT0 v1 6= 0 and 1 > 2.

1. What happens to the algorithm if if the maximum eigenvalue is not unique, that is, 1 = 2 = = k?

4

1. Consider the estimation procedure in problem 2 of Assignment #2 (where we used the

Gauss-Seidel algorithm to estimate f ig). Use both gradient descent and accelerated gradient descent to estimate f ig. To nd an appropriate value of , it is useful to approximate the maximum eigenvalue of the Hessian matrix of the objective function { the algorithm in problem 5 is useful in this regard.

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