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Statistical Field Theory An Introduction To Exactly Solved Models In Statistical Physics 2nd Edition Giuseppe Mussardo - Solutions

You are using OLS to fit a regression equation. True or false, and explain:(a) If you exclude a variable from the equation, but the excluded variable is orthogonal to the other variables in the equation, you won’t bias the estimated coefficients of the remaining variables.(b) If you exclude a
True or false, and explain: as long as the design matrix has full rank, the computer can find the OLS estimator βˆ. If so, what are the assumptions good for? Discuss briefly.
Does R2 measure the degree to which a regression equation fits the data?Or does it measure the validity of the model? Discuss briefly.
Suppose Yi = aui + bvi + i for i = 1,..., 100. The i are IID with mean 0 and variance 1. The u’s and v’s are fixed not random; these two data variables have mean 0 and variance 1: the correlation between them is r. If r = ±1, show that the design matrix has rank 1. Otherwise,let aˆ, bˆ be
True or false, and explain:(a) Collinearity leads to bias in the OLS estimates.(b) Collinearity leads to bias in the estimated standard errors for the OLS estimates.(c) Collinearity leads to big standard errors for some estimates.
Suppose (Xi, Wi, i) are IID as triplets across subjects i = 1,..., n, where n is large; E(Xi) = E(Wi) = E(i) = 0, and i is independent of (Xi, Wi). Happily, Xi and Wi have positive variance; they are not perfectly correlated. The response variable Yi is in truth this:Yi = a Xi + bWi + i .We can
(This continues question 10.) Tom elects to run a regression of Yi on Xi , omitting Wi . He will use the coefficient of Xi to estimate a.(a) What happens to Tom if Xi and Wi are independent?(b) What happens to Tom if Xi and Wi are dependent?Hint: see exercise 3B15.
Suppose (Xi, δi, i) are IID as triplets across subjects i = 1,..., n, where n is large; and Xi , δi , i are mutually independent. Furthermore, E(Xi) = E(δi) = E(i) = 0 while E(Xi 2) = E(δi 2) = 1 and E(i 2) =σ2 > 0. The response variable Yi is in truth this:Yi = a Xi + i .We can recovera, up
(Continues question 12.) Letc, d, e be real numbers and let Wi =cXi + dδi + ei. Dick elects to run a regression of Yi on Xi and Wi, again without an intercept. Dick will use the coefficient of Xi in his regression to estimatea. If e = 0, Dick still getsa, up to random error—as long as d = 0.
(Continues questions 12 and 13.) Suppose, however, that e = 0. Then Dick has a problem. To see the problem more clearly, assume that n is large. Let Q =X W be the design matrix, i.e., the first column is the Xi and the second column is the Wi. Show that QQ/n .= E(X2 i ) E (XiWi)E(XiWi) E (W2 i
There is a population consisting of N subjects, with data variables x and y. A simple regression equation can in principle be fitted by OLS to the population data: yi = a + bxi + ui, where N i=1 ui = N i=1 xiui = 0.Although Harry does not have access to data on the full population, he can take a
Over the period 1950–99, the correlation between the size of the population in the United States and the death rate from lung cancer was 0.92.Does population density cause lung cancer? Discuss briefly 22. In the HIP trial (chapter 1), women in the treatment group who refused screening were
Let βˆ be the OLS estimator in (1), where the design matrix X has full rank p
Verify that theorems 4.1–4 continue to hold, if we replace conditions(4.4–5) with condition (2) above.
If the n×p matrix X has rank p
Let βˆOLS be the OLS estimator in (1), where the design matrix X has full rank p
Let βˆGLS be the GLS estimator in (1), where the design matrix X has full rank p
Suppose Y = Xβ+. The design matrix X is n×p with rank p
(Hard.) There are three observations on a variable Y for each individual i = 1, 2,..., 800. There is an explanatory variable Z, which is scalar.Maria thinks that each subject i has a “fixed effect” ai and there is a parameter b common to all 800 subjects. Her model can be stated this way:Yij =
But she is afraid that var(ij ) = σ2 i depends on the subject i. Can you get this into the GLS framework? What would you use for the response vector Y in (1)? The design matrix? (This will get ugly.) With her model, what can you say about G in (7)? How would you estimate her model?
We have an OLS model with p = 1, and X is a column of 1’s. Findβˆ and σˆ 2 in terms of Y and n. If the errors are IID N (0, σ2), find the distribution of βˆ − β, σˆ 2, and √n(βˆ − β)/σˆ . Hint: see exercise 3B16.
Lei is a PhD student in sociology. She has a regression equation Yi =a + bXi + Ziγ + i. Here, Xi is a scalar, while Zi is a 1×5 vector of control variables, and γ is a 5×1 vector of parameters. Her theory is that b = 0. She is willing to assume that the i are IID N (0, σ2), independent of X
A philosopher of science writes,“Suppose we toss a fair coin 10,000 times, the first 5000 tosses being done under a red light, and the last 5000 under a green light.The color of the light does not affect the coin. However, we would expect the statistical null hypothesis—that exactly as many
An archeologist fits a regression model, rejecting the null hypothesis thatβ2 = 0, with P < 0.005. True or false and explain:(a) β2 must be large.(b) βˆ2 must be large.
Suppose Ui = α+δi for i = 1,...,n. The δi are independent N (0, σ2).The parameters α and σ2 are unknown. How would you test the null hypothesis that α = 0 against the alternative that α = 0?
Suppose Ui are independentN (α, σ2)for i = 1,...,n. The parametersα and σ2 are unknown. How would you test the null hypothesis thatα = 0 against the alternative that α = 0?
In exercise 1, what happens if the δi are IID with mean 0, but are not normally distributed? if n is small? large?
InYule’s model (1.1), how would you test the null hypothesis c = d = 0 against the alternative c = 0 or d = 0? Be explicit. You can use the metropolitan unions, 1871–81, for an example. What assumptions would be needed on the errors in the equation? (See lab 6 at the back of the book.)
There is another way to define the numerator of the F-statistic. Let e(s)be the vector of residuals from the small model. Show thatXβˆ2 − Xβˆ(s)2 = e(s)2 − e2.Hint: what is Xβˆ(s)2 + e(s)2?
(Hard.) George uses OLS to fit a regression equation with an intercept, and computes R2. Georgia wants to test the null hypothesis that all the coefficients are 0, except for the intercept. Can she compute F from R2, n, and p? If so, what is the formula? If not, why not?
Suppose Xi are independent normal random variables with variance 1, for i = 1, 2, 3. The means are α +β, α +2β , and 2α +β, respectively.How would you estimate the parameters α and β?
The F-test, like the t-test, assumes something in order to demonstrate something. What needs to be assumed, and what can be demonstrated?To what extent can the model itself be tested using F? Discuss briefly.
Suppose Y = Xβ + where(i) X is n× p of rank p, and(ii) E(|X) = γ , a non-random n×1 vector, and(iii) cov(|X) = G, a non-random positive definite n×n matrix.Let βˆ = (X X)−1X Y . True or false and explain:(a) E(βˆ|X) = β.(b) cov(βˆ|X) = σ2(X X)−1.In (a), the exceptional case γ
(This continues question 3.) Suppose p > 1, the first column of X is all 1’s, and γ1 =···= γn.(a) Is βˆ1 biased or unbiased given X?(b) What about βˆ2?
Suppose Y = Xβ + where(i) X is fixed not random, n× p of rank p, and(ii) the i are IID with mean 0 and variance σ2, but(iii) the i need not be normal.Let βˆ = (X X)−1X Y . True or false and explain:(a) E(β)ˆ = β.(b) cov(β)ˆ = σ2(X X)−1.(c) If n = 100 and p = 6, it is probably OK
Suppose that X1, X2,...,Xn, δ1, δ2,...,δn are independent N (0, 1)variables, and Yi = X2 i − 1 + δi. However, Julia regresses Yi on Xi.What will she conclude about the relationship between Yi and Xi?
Suppose U and V1,...,Vn are IID N (0, 1) variables; µ is a real number. Let Xi = µ + U + Vi. Let X = n−1 n i=1 Xi and s2 =(n − 1)−1 n i=1(Xi − X)2.(a) What is the distribution of Xi?(b) Do the Xi have a common distribution?(c) Are the Xi independent?(d) What is the distribution of X? of
Suppose Xi are N (µ, σ2) for i = 1,...,n, where n is large. We use X to estimate µ. True or false and explain:(a) If the Xi are independent, then X will be around µ, being off by something like s/√n; the chance that |X − µ| < s/√n is about 68%.(b) Even if the Xi are dependent, X will be
Suppose Xi has mean µ and variance σ2 for i = 1,...,n, where n is large. These random variables have a common distribution, which is not normal. We use X to estimate µ. True or false and explain:(a) If the Xi are IID, then X will be around µ, being off by something like s/√n; the chance that
Discussing an application like example 2 in section 4, a social scientist says “one-step GLS is very problematic because it simply downweights observations that do not fit the OLS model.”(a) Does one-step GLS downweight observations that do not fit the OLS model?
You are thinking about a regression model Y = Xβ + , with the usual assumptions. A friend suggests adding a column Z to the design matrix.If you do it, the bigger design matrix still has full rank. What are the arguments for putting Z into the equation? Against putting it in?
A random sample of size 25 is taken from a population with mean µ.The sample mean is 105.8 and the sample variance is 110. The computer makes a t-test of the null hypothesis that µ = 100. It doesn’t reject the null. Comment briefly
Suppose x1,...,xn and y1,...,yn have means x, y; the standard deviations are sx > 0, sy > 0; and the correlation is r. Let cov(x, y) = 1 nn i=1 (xi − x)(yi − y).(“cov” is shorthand for covariance.) Show that—(a) cov(x, y) = rsx sy .(b) The slope of the regression line for predicting y
Suppose x1,...,xn and y1,...,yn are real numbers, with x = y = 0 and sx = sy = 1. Show that 1 nn i=1(xi + yi)2 = 2(1 + r) and 1nn i=1(xi − yi)2 = 2(1 − r), where r = r(x, y). Show that−1 ≤ r ≤ 1.
A die is rolled 250 times. The fraction of times it lands ace will be around , give or take or so.
One hundred draws are made at random with replacement from the box 1 2 2 5 . The draws come out as follows: 17 1 ’s, 54 2 ’s, and 29 5 ’s. Fill in the blanks.(a) For the , the observed value is 0.8 SEs above the expected value. (Reminder: SE = standard error.)(b) For the , the observed value
Equation (7) is a . Options:model parameter random variable
In equation (7), a is . Options (more than one may be right):observable unobservable a parameter a random variable Repeat forb. For i. For Yi.
According to equation (7), the 439.00 in table 1 is . Options:a parameter a random variable the observed value of a random variable
Suppose x1,...,xn and y1,...,yn are real numbers, with sx > 0 and sy > 0. Let x∗ be x in standard units; similarly for y. Show that r(x, y) = r(x∗, y∗).
Suppose x1,...,xn are real numbers. Let x = (x1 +···+ xn)/n. Let c be a real number.(a) Show that n i=1 (xi − x) = 0.(b) Show that n i=1 (xi − c)2 = n i=1 (xi − x)2+ n(x − c)2.Hint: (xi −c) = (xi − x) + (x − c).(c) Show thatn i=1 (xi −c)2, as a function ofc, has a unique
A statistician has a sample, and is computing the sum of the squared deviations of the sample numbers from a number q. The sum of the squared deviations will be smallest when q is the . Fill in the blank (25 words or less) and explain.
In the HIP trial (table 1), what is the evidence confirming that treatment has no effect on death from other causes?
A die is rolled 180 times. Find the expected number of aces, and the variance for the number of aces. The number of aces will be around, give or take or so. (A die has six faces, all equally likely; the face with one spot is the “ace.”)
In example 2, is 35/12 the variance of a random variable? of data?maybe both? Discuss briefly.
Someone wants to analyze the HIP data by comparing the women who accept screening to the controls. Is this a good idea?
Was Snow’s study of the epidemic of 1853–54 (table 2) a randomized controlled experiment or a natural experiment? Why does it matter that the Lambeth company moved its intake point in 1852? Explain briefly
WasYule’s study a randomized controlled experiment or an observational study?
In equation (2), suppose the coefficient of Out had been −0.755. What would Yule have had to conclude? If the coefficient had been +0.005?Exercises 6–8 prepare for the next chapter. If the material is unfamiliar, you might want to read chapters 16–18 in Freedman-Pisani-Purves (2007), or
Suppose X1, X2,...,Xn are independent random variables, with common expectation µ and variance σ2. Let Sn = X1 + X2 +···+ Xn.Find the expectation and variance of Sn. Repeat for Sn/n.
Suppose X1, X2,...,Xn are independent random variables, with a common distribution: P (Xi = 1) = p and P (Xi = 0) = 1 − p, where 0
Keefe et al (2001) summarize their data as follows:“Thirty-five patients with rheumatoid arthritis kept a diary for 30 days. The participants reported having spiritual experiences, such as a desire to be in union with God, on a frequent basis. On days that participants rated their ability to
According to many textbooks, association is not causation. To what extent do you agree? Discuss briefly.
In equation (1), variance applies to data, or random variables? What about correlation in (4)?
On page 22, below table 1, you will find the number 439.01. Is this a parameter or an estimate? What about the 0.05?
Suppose we didn’t have the last line in table 1. Find the regression of length on weight, based on the data in the first 5 lines of the table.
In example 1, is 900 square pounds the variance of a random variable?or of data? Discuss briefly.
A die is rolled twice. Let Xi be the number of spots on the ith roll for i = 1, 2.(a) Find P (X1 = 3 | X1 + X2 = 8), the conditional probability of a 3 on the first roll given a total of 8 spots.(b) Find P (X1 + X2 = 7 | X1 = 3).(c) Find E(X1 | X1+X2 = 6), the conditional expectation of X1 given
(Hard.) Suppose x1,...,xn are real numbers. Suppose n is odd and the xi are all distinct. There is a unique median µ: the middle number when the x’s are arranged in increasing order. Let c be a real number. Show that f (c) = n i=1 |xi −c|, as a function ofc, is minimized when c = µ.Hints.
Find adj(B). This is just to get on top of the definitions; later, we do all this sort of thing on the computer.
In exercise 14, suppose p = 1 and X is a column of 1’s. Show thatβˆ is the mean of the Y ’s. How is this related to exercise 2B12(c), i.e., part (c), exercise 12, set B, chapter 2?
This exercise explains a stepwise procedure for computing βˆ in exercise 14. There are hints, but there is also some work to do. Let M be the first p − 1 columns of X, so M is n×(p − 1). Let N be the last column of X, so N is n×1.(i) Let γˆ1 = (MM)−1MY and f = Y − Mγˆ1 .(ii) Let
Show that the 1,1 element of cov(U ) equals var(U1); the 2,3 element equals cov(U2, U3).
Show that cov(U ) is symmetric.
If A is a fixed (i.e., non-random) matrix of size n×3 and B is a fixed matrix of size 1×m, show that E(AUB) = AE(U )B.
Show that cov(AU ) = Acov(U )A.
If c is a fixed vector of size 3×1, show that var(cU ) = ccov(U )c and cov(U +c) = cov(U ).Comment. If V is an n×1 random vector, C is a fixed m×n matrix, and D is a fixed m×1 vector, then cov(CV + D) = Ccov(V )C.
What’s the difference between U = (U1 + U2 + U3)/3 and E(U )?
Suppose ξ and ζ are two random vectors of size 7×1. If ξ ζ = 0, are ξand ζ independent? What about the converse: if ξ and ζ are independent, is ξζ = 0?
Suppose ξ and ζ are two random variables with E(ξ ) = E(ζ ) = 0.Show that var(ξ ) = E(ξ 2) and cov(ξ , ζ ) = E(ξ ζ ).Notes. More generally, var(ξ ) = E(ξ 2) − [E(ξ )]2 and cov(ξ , ζ ) =E(ξ ζ ) − E(ξ )E(ζ ).
Suppose ξ is an n×1 random vector with E(ξ ) = 0. Show that cov(ξ ) =E(ξ ξ ).Notes. Generally, cov(ξ ) = E(ξ ξ ) − E(ξ )E(ξ ) and E(ξ ) = [E(ξ )].
Suppose ξi, ζi are random variables for i = 1,...,n. As pairs, they are independent and identically distributed in i. Let ξ = 1 nn i=1 ξi, and likewise for ζ . True or false, and explain:(a) cov(ξi, ζi) is the same for every i.(b) cov(ξi, ζi) = 1 nn i=1(ξi − ξ )(ζi − ζ ).
Suppose u,v are n×1; neither is identically 0. What is the rank of u×v?
In exercise 14, suppose p = 1, so X is a column vector. Show thatβˆ = X·Y/X2.
Show that A×adjA = adjA×A = det(A)×In. Repeat, for B. What is n in each case?
Find the rank and the trace of A. Repeat, for B.
Find the rank of C.
If possible, find the trace and determinant of C. If not, why not?
If possible, find A2. If not, why not? (Hint: A2 = A×A.)
If possible, find C2. If not, why not?
Suppose M is m×n and N is n×p.(a) Show that (MN ) = NM.(b) Suppose m = n = p, and M,N are both invertible. Show that(MN )−1 = N−1M−1 and (M)−1 = (M−1).
Suppose X is n×p with p ≤ n. If X has rank p, show that XX has rank p, and conversely. Hints. Suppose X has rank p and c is p×1. Then XXc = 0p×1 ⇒ cXXc = 0 ⇒ Xc2 = 0 ⇒ Xc = 0n×1.Notes. The matrix XX is p×p. The rank is p if and only if XX is invertible. The ⇒ is shorthand for
If A is m×n and B is n×m, show that trace(AB) = trace(BA). Hint:the iith element of AB is j AijBj i, while the jj th element of BA is i Bj iAij .
If u and v are n×1, show that u + v2 = u2 + v2 + 2u · v.
If u and v are n×1, show that u + v2 = u2 + v2 if and only if u ⊥ v. (This is Pythagoras’ theorem in n dimensions.)
Suppose X is n×p with rank p
The random variable X has density f on the line; σ and µ are real numbers. What is the density of σX + µ? of X2? Reminder: if X has density f , then P (X < x) = x−∞ f (u)du.
If M is m×n and N is m×n, show that (M + N ) = M + N.
10.9. Entanglement entropy Consider a system of two spin–12 degrees of freedom and a pure state given by| = cosθ | ↑↓+sin θ | ↓↑.The observer A measures the first spin while B measures the second one.a. Compute the entanglement entropy SA.b. Check that is takes its larger value SA =
7.5. Dimensional regularization An alternative way to regularize the integrals encountered in perturbative series of QFT consists of the dimensional regularization. The main idea behind this approach is to consider the integrals as functions of the dimensionality d of the system, regarded as a

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