Questions and Answers of Linear State Space Systems

Exercise 6.9.11. Show that the spectral density f (ν) =π −|ν |π2 determines the covariance functionσ (k) =σ 2 if k = 0 4σ 2/(π|k|)2 if k is odd 0 otherwise.
Exercise 6.9.12. If yt and wt are independent second-order stationary processes with spectral densities fy and fw, find the spectral distribution of the stationary process zt = yt +wt .
Exercise 6.9.13. Suppose yt is second-order stationary and fy(ν ) is nonnegative, bounded, and fy(1/2) = 0. Let wt = yt −yt−1.Find fw(ν ) in terms of fy(ν ).
Exercise 6.9.14. For ν ∈ [0,π], define fy(ν) =50 ν ∈ [14−.01, 14+.01]0 otherwise and fy(−ν) = fy(ν ). Find σ (0), σ (1), σ (2).
Exercise 6.9.15.(a) Use the relationships sin(a+b) = sin(a) cos(b)+cos(a) sin(b)and References 245 cos(a+b) = cos(a) cos(b)−sin(a) sin(b)to show the following.cos(a) cos(b) = 1 2
Exercise 6.9.16. Apply the relationship in Exercise 6.9.15a on the sine of a sum to the process in Exercise 6.9.2 to show that the process can be rewritten as yt = Asin(2πνt +φ ), where A is the
Exercise 7.9.1. Which of the following processes are stationary? Which are invertible?(a) (1−1.5B+0.54B2)yt = (1−0.5B)et .(b) (1− 58 B)yt = (1+0.1B−0.56B2)et .(c) (1−1.6B+0.55B2)yt =
Exercise 7.9.2. Consider the AR(1) model yt = 10−0.7yt−1+et , σ 2 = 3.(a) Find μ, σ (0), σ (1), σ (2), and σ (3).(b) If y4 = 12, will y5 tend to be less than or greater than μ?(c) Is the
Exercise 7.9.3. Consider the MA(1) model yt = 10+et −0.4et−1, σ 2 = 3.(a) Find μ, σ (0), σ (1), σ (2), and σ (3).(b) If y4 = 12, will y5 tend to be less than or greater than μ?(c) Is the
Exercise 7.9.4. Consider the ARMA(1,1) model yt = 10−0.7yt−1+et −0.4et−1, σ 2 = 3(a) Find μ, σ (0), σ (1), σ (2), and σ (3).(b) If y4 = 12, will y5 tend to be less than or greater than
Exercise 7.9.5. Do a time domain analysis of the sunspot data in Exercise 6.9.5.This should include estimation, model fitting, and checking of assumptions.Exercise 7.9.6. Use a multiplicative
Exercise 7.9.7. For an MA(1) process, find an estimate of θ1 in terms of ρˆ (1).Are any restrictions on ρˆ (1) needed?
Exercise 7.9.8. For an AR(2) process, use the Yule–Walker equations, with σˆy(·)replacing σy(·) to obtain estimatesφˆ1 =ρˆ (1) (1−ρˆ (1))1−ρˆ (1)2 andφˆ2 =ρˆ (2)−ρˆ (1)2
Exercise 7.9.9. Show that the variance for predicting k steps ahead in an AR(1)process isσ 2 1−φ 2k 11−φ 2 1.
Exercise 7.9.10. Show, for an MA(1) process with parameter θ1, thatˆE(yn+1|Y∞) = −∞Σs=0θ s 1yn−s
Exercise 7.9.11. Let et be a second-order stationary process. Show that yt = et −θ1et−1 and wt = et − 1θ1 et−1 have the same correlation function.
Exercise 7.9.12. What are the largest possible values of |ρ(1)| and |ρ(2)| for an MA(2) process?
Exercise 7.9.13. Find the ψis, μ, and σ (k) for the following processes.(a) The AR(3) process(1−0.9B)(1+0.8B)(1−0.6B)yt = et .(b) The ARMA(2,1) process(1−0.9B)(1−0.6B)yt = (1−0.5)et .
Exercise 7.9.14. Show that if all the roots x0 of Φ(x) have |x0| > 1, then there exists a polynomialΨ(x) =∞Σi=0ψixi with∞Σi=0|ψi| < ∞and, for x ∈ [−1,1],[Ψ(x)] [Φ(x)] = 1.Hint: Do a
Exercise 7.9.15. Generate 25 realizations of an ARMA(1,1) process for φ1 =−0.8,−0.2,0, .2, .8 and θ1 = −0.8,−0.2,0, .2, .8.
Exercise 7.9.16. Prove PA-V’s Exercise 6.9.9, that(a) ρ12·3 =ρ12− 4 ρ13ρ23 1−ρ2 13 41−ρ2 23(b) ρ12·34 =4ρ12·4−ρ13·4ρ23·4 1−ρ2 13·4 41−ρ2 23·4
Exercise 9.10.1. Show that ifW ∼W(n,Σ ,0), then AWA ∼W(n,AΣ A,0).
Exercise 9.10.2. Show that if W1, · · ·Wr are independent with Wi ∼W(ni,Σ ,0), then rΣi=1 Wi ∼W)rΣi=1 ni,Σ ,0*.
Exercise 9.10.3. Show that ifW ∼W(n,Σ ,0), thenλ Wλλ Σλ∼ χ2(n,0).
Exercise 9.10.4. For i = 1,2, 3, let yi ∼ N (μ +(i−2)ξ ,Σ ), where Σ is known and y1, y2, and y3 are independent. Find the maximum likelihood estimates of μand ξ .
Exercise 9.10.5. Based on the multivariate linear model Y = XB+e, E(e) = 0, Cov(εi,ε j) =δi jΣ , find a 99% prediction interval for y0ξ , where y0 is an independent observation that is
Exercise 9.10.6. Let y1, y2, · · · , yn be i.i.d. N(Xβ ,Σ ), where Σn×n is unknown.Show that the maximum likelihood estimate of Xβ is X ˆβ = X(XE−1X)−XE−1 ¯ y· .
Exercise 9.10.7. Consider the multivariate linear model Y = XB+e and the parametric function ΛBW, where W is a q×r matrix of rank r. Find simultaneous confidence intervals for all parameters of
Exercise 9.10.8. Use Lemma 9.6.2 to show that if A is nonnegative definite and B is positive definite, then AB−1 is nonnegative definite. Hint: Show that the nonnegative definite matrix
Exercise 9.10.9. Rewrite the multivariate linear model in terms of Vec(Y).Write it similarly to both (9.1.2) with covariance (9.1.3) and using the Vec operator with Kronecker products as in (9.1.4)
Exercise 9.10.10. In the multivariate linear model Y = XB+e the likelihood equations reduce to trΣ −1[dθ jΣ ]+= trΣ −1[dθ jΣ ]Σ −1Σˆ+, for j = 1, . . . , s where Σˆ ≡Y(I−M)Y/n
Exercise 9.10.11. Use the univariate linear model form of the multivariate linear model and the likelihood equations from Sect. 4.3 to obtain the MLEs for the multivariate linear model.
Exercise 10.6.1. Jolicoeur and Mosimann (1960) give data on the length, width, and height of painted turtle shells. The carapace dimensions of 24 females and 24 males are given in Table 10.3. Use
Exercise 10.6.2. Smith, Gnanadesikan, and Hughes (1962) provide data on characteristics of the urine of young men. The men are categorized into four groups based on their degree of obesity. The four
Exercise 10.6.3. Analyze the repeated measures data given by Danford, Hughes, and McNee (1960) in Biometrics on pages 562 and 563.Exercise 10.6.4. Box (1950) gives data on the weights of three groups
Exercise 11.8.1. Box (1950) gives data on the weights of three groups of rats.One group was given thyroxin in their drinking water, one thiouracil, and the third group was a control. Weights are
Exercise 11.8.2. Box (1950) presents data on the weight loss of a fabric due to abrasion. Two fillers were used in three proportions. Some of the fabric was given a surface treatment. Weight loss was
Exercise 12.9.1. Consider the data of Example 10.3.1. Suppose a person has heart rate measurements of y = (84,82,80,69).(a) Using normal theory linear discrimination, what is the estimated maximum
Exercise 12.9.2. In the motion picture Diary of a Mad Turtle the main character, played by Richard Benjamin Kingsley, claims to be able to tell a female turtle by a quick glance at her carapace.
Exercise 12.9.3. Using the data of Exercise 10.6.3, do a stepwise discriminant analysis to distinguish among the thyroxin, thiouracil, and control rat populations based on their weights at various
Exercise 12.9.4. Lachenbruch (1975) presents information on four groups of junior technical college students from greater London. The information consists of summary statistics for the performance of
Exercise 12.9.6. Show that the Mahalanobis distance is invariant under affine transformations z = Ay+b of the random vector y when A is nonsingular.
Exercise 12.9.7. Let y be an observation from one of two normal populations that have means of μ1 and μ2 and common covariance matrix Σ . Define λ = (μ1−μ2)Σ −1.(a) Show that, under
Exercise 12.9.8. Consider a two group allocation problem in which the prior probabilities are π(1) =π(2) = 0.5 and the sampling distributions are exponential, namely f (y|i) =θie−θiy, y ≥
Exercise 12.9.6. Show that the Mahalanobis distance is invariant under affine transformations z = Ay+b of the random vector y when A is nonsingular.
Exercise 12.9.7. Let y be an observation from one of two normal populations that have means of μ1 and μ2 and common covariance matrix Σ . Define λ = (μ1−μ2)Σ −1.(a) Show that, under
Exercise 12.9.8. Consider a two group allocation problem in which the prior probabilities are π(1) =π(2) = 0.5 and the sampling distributions are exponential, namely f (y|i) =θie−θiy, y ≥
Exercise 12.9.9. Suppose that the distributions for two populations are bivariate normal with the same covariance matrix. For π(1) =π(2) = 0.5, find the value of the correlation coefficient that
Exercise 14.5.1.(a) Find the vector b that minimizes qΣi=1yi−μi−b(x−μx)2.(b) For given weights wi, i = 1, . . . ,q, find the vector b that minimizes qΣi=1
Exercise 14.5.2. In a population of large industrial corporations, the covariance matrix for y1 = assets/106 and y2 = net income/106 isΣ =75 5 5 1.(a) Determine the principal components.(b) What
Exercise 14.5.3. What are the principal components associated withΣ =⎡⎢⎣5 0 0 0 0 3 0 0 0 0 3 0 0 0 0 2⎤⎥⎦?Discuss the problem of reducing the variables to a two-dimensional space.
Exercise 14.5.4. Let v1 = (2,1,1,0), v2 = (0,1,−1,0), v3 = (0,0,0,2), andΣ =3Σi=1 vivi .(a) Find the principal components of Σ .(b) What is the predictive variance of each principal
Exercise 14.5.5. Do a principal components analysis of the female turtle carapace data of Exercise 10.6.1.
Exercise 14.5.6. The data in Table 14.1 are a subset of the Chapman data reported by Dixon and Massey (1983). It contains the age, systolic blood pressure, diastolic blood pressure, cholesterol,
Exercise 14.5.7. Assume a two-factor model withΣ =⎡⎣0.15 0.00 0.05 0.00 0.20 −0.01 0.05 −0.01 0.05⎤⎦and B =0.3 0.2 0.1 0.2 −0.3 0.1.What isΨ? What are the communalities?
Exercise 14.5.8. Using the vectors v1 and v2 from Exercise 14.5.4, letΛ = v1v1+v2v2.Give the eigenvector solution for B and another set of loadings that generates Λ.
Exercise 14.5.9. Given thatΣ =⎡⎣1.00 0.30 0.09 0.30 1.00 0.30 0.09 0.30 1.00⎤⎦andΨ = D(0.1,0.2,0.3), find Λ and two choices of B.
Exercise 14.5.10. Find definitions for the well-known factor loading matrix rotations varimax, direct quartimin, quartimax, equamax, and orthoblique. What is each rotation specifically designed to
Exercise 14.5.11. Do a factor analysis of the female turtle carapace data of Exercise 10.6.1. Include tests for the numbers of factors and examine various factorloading rotations.
Exercise 14.5.12. Do a factor analysis of the Chapman data discussed in Exercise 14.5.6.
Exercise 14.5.13. Show the following determinant equality.|Ψ +BB| = |I+BΨ−1B||Ψ|.
Exercise 14.5.14. Find the likelihood ratio test for H0 : Σ =σ 2(1−ρ)I+ρJJagainst the general alternative.
Suppose that yi has a N(????i, ????2) distribution, i = 1,…, n. Formulate the normal linear model as a GLM, specifying the random component, linear predictor, and link function.
Show the exponential dispersion family representation for the gamma distribution (4.29). When do you expect it to be a useful distribution for GLMs?
Show that the t distribution is not in the exponential dispersion family.(Although GLM theory works out neatly for family (4.1), in practice it is sometimes useful to use other distributions, such as
Show that an alternative expression for the GLM likelihood equations is∑n i=1(yi − ????i)var(yi)????????i????????j= 0, j = 1, 2,…, p.Show that these equations result from the generalized least
For a GLM with canonical link function, explain how the likelihood equations imply that the residual vector e = (y − ????̂) is orthogonal with C(X).
Suppose yi has a Poisson distribution with g(????i) = ????0 + ????1xi, where xi = 1 for i = 1,…, nA from group A and xi = 0 for i = nA + 1, ..., nA + nB from group B, and with all observations
Refer to the previous exercise. Using the likelihood equations, show that the same result holds for (a) any link function for this Poisson model, (b) any GLM of the form g(????i) = ????0 + ????1xi
For the two-way layout with one observation per cell, consider the model whereby yij ∼ N(????ij, ????2) with????ij = ????0 + ????i + ????j + ????????i????j.For independent observations, is this a
Consider the expression for the weight matrix W in var(????̂) = (XT WX)−1 for a GLM. Find W for the ordinary normal linear model, and show how var(????̂)follows from the GLM formula.
For the normal bivariate linear model, the asymptotic variance of the correlation r is (1 − ????2)2∕n. Using the delta method, show that the transform 12 log[(1 + r)∕(1 − r)] is variance
For a binomial random variable ny with parameter ????, consider the null model.a. Explain how to invert the Wald, likelihood-ratio, and score tests of H0:???? = ????0 against H1: ???? ≠ ????0 to
For the normal linear model, Section 3.3.2 showed how to construct a confidence interval for E(y) at a fixed x0. Explain how to do this for a GLM.
For a GLM assuming yi ∼ N(????i, ????2), show that the Pearson chi-squared statistic is the same as the deviance. Find the form of the difference between the deviances for nested models M0 and M1.
In a GLM that uses a noncanonical link function, explain why it need not be true that ∑i ̂????i = ∑i yi. Hence, the residuals need not have a mean of 0.Explain why a canonical link GLM needs an
For a binomial GLM, explain why the Pearson residual for observation i, ei = (yi − ̂????i)∕√ ̂????i(1 − ̂????i)∕ni, does not have an approximate standard normal distribution, even for a
Find the form of the deviance residual (4.21) for an observation in (a) a binomial GLM, (b) a Poisson GLM.
Suppose x is uniformly distributed between 0 and 100, and y is binary with log[????i∕(1 − ????i)] = −2.0 + 0.04xi. Randomly generate n = 25 independent observations from this model. Fit the
Derive the formula var(????̂j) = ????2∕{(1 − R2 j )[∑i(xij − x̄j)2]}.
Consider the value ????̂ that maximizes a function L(????). This exercise motivates the Newton–Raphson method by focusing on the single-parameter case.a. Using L′(????̂) = L′(????(0)) +
For n independent observations from a Poisson distribution with parameter????, show that Fisher scoring gives ????(t+1) = ȳ for all t > 0. By contrast, what happens with the Newton–Raphson method?
For an observation y from a Poisson distribution, write a short computer program to use the Newton–Raphson method to maximize the likelihood. With y = 0, summarize the effects of the starting value
For noncanonical link functions in a GLM, show that the observed information matrix may depend on the data and hence differs from the expected information matrix. Thus, the Newton–Raphson method
The bias–variance tradeoff: Before an election, a polling agency randomly samples n = 100 people to estimate ???? = population proportion who prefer candidate A over candidate B. You estimate ????
In selecting explanatory variables for a linear model, what is inadequate about the strategy of selecting the model with largest R2 value?
For discrete probability distributions of {pj} for the “true” model and {pMj} for a model M, prove that the Kullback–Leibler divergence E{log[p(y)∕pM(y)]}≥0.
For a normal linear model M1 with p + 1 parameters, namely, {????j} and ????2, which has ML estimator ̂????2 = [∑n i=1(yi − ̂????i)2]∕n, show that AIC = n[log(2???? ̂????2) + 1] + 2(p +
Section 4.7.2 mentioned that using a gamma GLM with log-link function gives similar results to applying a normal linear model to log(y).a. Use the delta method to show that when y has standard
Download the Houses.dat data file from www.stat.ufl.edu/~aa/glm/data. Summarize the data with descriptive statistics and plots. Using a forward selection procedure with all five predictors together
Refer to the previous exercise. Use backward elimination to select a model.a. Use an initial model containing the two-factor interactions. When you reach the stage at which all terms are
Refer to the previous two exercises. Conduct a model-selection process assuming a gamma distribution for y, using (a) identity link, (b) log link. For each, interpret the final model.
For the Scottish races data of Section 2.6, the Bens of Jura Fell Race was an outlier for an ordinary linear model with main effects of climb and distance in predicting record times. Alternatively
Exercise 1.21 presented a study comparing forced expiratory volume after 1 hour of treatment for three drugs (a,b, and p = placebo), adjusting for a baseline measurement x1. Table 4.1 shows the
Refer to Exercise 2.45 and the study for comparing instruction methods. Write a report summarizing a model-building process. Include instruction type in the chosen model, because of the study goals
The horseshoe crab dataset Crabs2.dat at the text website comes from a study of factors that affect sperm traits of males. One response variable is ejaculate size, measured as the log of the amount
The MASS package of R contains the Boston data file, which has several predictors of the median value of owner-occupied homes, for 506 neighborhoods in the suburbs near Boston. Describe a
For x between 0 and 100, suppose the normal linear model holds with E(y) = 45 + 0.1x + 0.0005x2 + 0.0000005x3 + 0.0000000005x4+0.0000000000005x5 and ???? = 10.0. Randomly generate 25 observations
What does the fit of the “correct” model in the previous exercise illustrate about collinearity?
Randomly generate 100 observations (xi, yi) that are independent uniform random variables over [0, 100]. Fit a sequence of successively more complex polynomial models for using x to predict y, of

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