Questions and Answers of Investments Analysis And Management

13.2 Using the classification procedure established in Exercise 13.1, classify the vectors (Assume equal misclassification costs, and equal prior probabilities.)
13.1 Use the data of Exercise 7.6 to establish a sampling theory classification procedure for classifying new observations into π1 = N(θ, 1), or π2 = N(φ, 2). [Hint: Use the approach of Section
12.9 Suppose you had a large sample of stable distribution data available. How would you estimate the characteristic exponent graphically?
12.8 What is meant by the restricted problem of stable portfolio analysis?
12.6 Suppose it is desired to minimize risk in a portfolio while maximizing median return, and the returns for prospective assets follow a multivariate symmetric stable distribution with
12.5 Explain the difference between an efficient portfolio in a restricted sense and an unrestricted efficient portfolio.
12.4 Let X ≡ (X1,X2,Xa)’ denote the single share vector of returns on American Tobacco Company, Consolidated Cigar, and Liggett and Myers; let Y ≡(Y1, Y2)’ denote the single share returns
12.3 The single share returns vector (hypothetical) for Associated Dry Goods, Gimbel Brothers, and Sears, Roebuck and Company is found to follow a multivariate symmetric stable law of order 1, with
12.2 Suppose the vector of single share returns on the portfolio in Exercise 12.1 followed the law where Find the efficient portfolio in a restricted Model I sense.
12.1 A simple investment portfolio is to consist of shares of American Snuff, General Cigar, and P. Lorrilard Company. Let Y ≡ (Y1, Y2, Y3)’, where Yj denotes the return on one share of Company
11.10 Explain the use of canonical correlations in studying goodness-of-fit in generalized regression.
11.9 Discuss the meaning of canonical correlations applied to jointly dependent discrete random variables.
11.8 Compare the distribution of the sample canonical correlations with that of the population canonical correlations given the sample, with respect to a diffuse (vague) prior distribution. [Hint:
11.7 How could inferential methods of canonical correlations analysis be used to study the meaningfulness of a small change in an index number such as the Department of Labor’s “Cost of Living”
11.6 Relate the notion of a canonical correlation coefficient to that of a multiple correlation coefficient.
11.5 How would you carry out a canonical correlations analysis on a set of data vectors collected weekly for N weeks, and which exhibits serial correlation?
11.4 How would you apply the ideas of canonical correlations analysis to the problem of testing hypotheses in the multivariate analysis of variance model?
11.3 Suppose the covariance matrix used in Exercise 11.1 were a sample covariance matrix based upon a sample of size 100 (instead of a population covariance matrix). Test the first canonical
11.2 Consider carrying out the analysis of Exercise 11.1 on the implied correlation matrix instead of on the covariance matrix. To what extent is the difference in results of the two analyses
11.1 The population covariance matrix for changes in the numbers of users of four different illicit drugs was given in Exercise 3.1. Suppose it is desired to try to predict changes in marijuana and
10.15 Suppose you carry out a factor analysis and find that the factors have no natural interpretation. Would you still defend the results? Why?
10.14 It is often useful to carry out a factor analysis with no distributional assumptions on the data. MINRES is such a procedure. Explain the MINRES procedure. [Hint: see Harman, 1976.]
10.13 Principal components analysis is often used by researchers instead of factor analysis. What would be the advantages?
10.12 Using the data contained in the covariance matrix in Example (9.3.1): Carry out a maximum likelihood factor analysis separately for a one-factor model, two-factor model, three-factor model and
10.11 Explain the difference between the Factor Analysis and Latent Structure Analysis models.
10.10 It is hypothesized that there are two distinct schools of thought about strategic policy of superpowers in the nuclear age: those who feel a superpower should hold a “flexible response”
10.9 Suppose on the basis of a sample of data from M subjects the latent probabilities and recruitment probabilities of a latent class model of latent structure analysis are estimated. Assume there
10.8 Devise a group of five questions (each with two possible answers) one might ask (items) of each of N subjects in order to classify them into their degree of discrimination against people of
10.7 Suppose in the latent class model of latent structure analysis it is assumed there is lack of correlation among the response variables for each item instead of independence. How would the model
10.6 By consulting some of the references, suggest two distinct methods for estimating communalities in a factor analysis model.
10.5 Explain the circumstances under which the orthogonal factor model of factor analysis should be used, as opposed to the circumstances under which the oblique factor model should be used.
10.4 Suppose it is desired to carry out a Bayesian analysis of the latent class model of latent structure analysis. Find the joint natural conjugate prior density for the latent parameters.
10.3 Compare the reasons for using a maximum likelihood method of solution of a factor analysis problem with those for using a principal factor solution. Are there conditions under which one might be
10.2 In a factor analysis problem, compare the advantages and disadvantages in terms of required assumptions, interpretation, and so on, of carrying out the analysis on the data correlation matrix
10.1 Suppose the fractions of “yes” votes and “no” votes on a congressional bill are recorded for every bill voted upon in a particular session of the United States Congress. It is of
9.8 Let :p × p have latent roots λ1 ≥ λ2 ≥ . . . ≥ λp > 0. Explain why where is a good measure of whether or not a principal components analysis would be useful in a given situation; the
9.7 Let X1, . . . , XN denote bivariate observations from a covariance stationary time series, so of course, the observations are not independent (in general). If we wish to find the principal
9.6 Suppose X :p × 1 follows a general multivariate t-distribution. Show that the principal components have the same geometric interpretation as they would if X were normally distributed.
9.5 For the covariance matrix given in Example (8.4.1), find the principal components and their variances.
9.4 Find 95 percent confidence intervals for the variances of the principal components in Exercise 9.1.
9.3 In Exercise 9.1, assume the population covariance matrix has intraclass covariance structure.Find the maximum likelihood estimates of the distinct latent roots. Find a 95 percent confidence
9.2 In Exercise 9.1 compute the sample correlation matrix and determine the answer to Part (a) for the sample correlation matrix. Why is there a difficulty in interpreting this result unless the
9.1 Suppose that for a sample of size 101 the sample covariance matrix for a set of three variables which are purported to measure administrative ability (in the same units) is given by Find the
8.21 For the polytomous response model in Sect. (8.10.4), adopt a Dirichlet prior distribution on the category probabilities and develop the posterior distribution for the category probabilities.
8.20 Generalize the two-way layout MANOVA model of Sect. 8.7.2 to the multivariate three-way layout with K-observations per cell. Give the model, the restrictions, the sum-of-squares matrices, and
8.19 Theorem (8.6.6) gives the predictive distribution for a new observation y*. Find the predictive distribution for E(y*) under the same assumptions.
8.17 Show that at least one of the latent roots of a correlation matrix, R: p × p, must exceed one for R ≠ I. 8.18 Let a system of linear simultaneous stochastic equations be denoted by the model
8.16 Show that the addition of a variable to a linear regression cannot decrease the value of the squared multiple correlation coefficient.
8.15 Suppose the production of certain model automobile is summarized by the data below.Fit a regression line to the data by ordinary least squares, adopting the model yt = a + bt + et, t = 1, ... ,
8.14 Suppose x1, . . . ,xN are independent bivariate vectors denoting changes in verbal and quantitative scores on achievement tests for people in a job training center who take the pair of tests
8.13 When the cells have unequal numbers of observations in a MANOVA, the number of replications for each cell might be taken to be the harmonic average of the numbers of observations for all the
8.12 Let yi = α + βxi + e1i, zi = a + bxi + e2i, i = 1, . . . , n, denote standard regressions of y on x and z on x, respectively; that is, E(e1i) = E(e2i) = 0, all errors are independent and
8.11 Show that if y ≡ (yj) denotes a vector of n observations on a dependent variable Y, and if for k = rank (X), k < n,denotes a standard univariate regression model, the sample multiple
8.10 Parameterize the Normal multivariate regression model in (8.6.14) in terms A ≡ –1. Then, using the generalized natural conjugate prior approach of Section 8.6.2, find the joint generalized
8.9 Explain why the likelihood ratio, trace, and maximum root criteria for hypothesis testing in MANOVA, are all equivalent when rank (HG–1) = 1, where G and H are defined in Section 8.7.3, and H
8.8 The country of Moralia has decided that to defend itself from its enemies it will need to purchase some fighter airplanes. Three different aircraft types are in contention for possible purchase
8.7 Let πj ≡ N (θj, ), j = 1, 2, 3 denote three p-variate Normal populations and let {x1(j), . . . , xNi (j)} denote samples of p-variate observations of size Nj for each of the populations.
8.6 For the data given in Example (8.4.1), assume a diffuse prior on (B, ) in the Normal standard Multivariate regression model and test the hypothesis that H: β11, = 0 (use a 1 percent level of
8.5 Explain the conditions under which generalized multivariate regression methods yield more efficient slope coefficient estimators than those obtainable by standard multivariate regression
8.4 For the data given in Example (8.4.1), test the hypothesis that β11 = β12 = β13 = 0, where B = (βij), i = 1, 2, j = 1, 2, 3. (Use the likelihood ratio criterion and the 5 percent level of
8.3 In Exercise 8.2 suppose instead of the shown, were unknown but with the structure Estimate .
8.2 Let yj denote the change in the number of people residing in Chicago from year j to year (j + 1) who would prefer to live in integrated neighborhoods; j = 1, 2, 3. Suppose y = (y1,y2,y3)’, with
8.1 Consider a Model III multivariate regression model (stochastic regression). Formulate the model giving appropriate assumptions. Are least squares estimators unbiased? Why? [Hint: See Table 8.2.1.]
7.15 For the data given in Exercise 7.14, test the hypothesis that the covariance matrices are equal (at the 5% level).
7.14 A pharmaceutical company claims to have developed a drug for simultaneously reducing blood vessel clogging cholesterol levels in the human body, and for inducing body weight reduction. To test
7.13 Score on a sales examination, X was claimed to be related to years of education, Y. The following data was collected.Assume (X, Y) follows a bivariate normal distribution and test the hypothesis
7.12 Suppose x: p × 1, and L(x|θ) = N(θ,σ1¹⁵I), for σ1¹⁵ known. Show that if L(θ Φ) = N(Φe,σ2¹⁵I), where Φ is a scalar, e is a p-vector of ones, and σ2¹⁵ is known, and if the
7.11 The following statement appeared in Time magazine: “As early as the 12th century, Hebrew scholars began to question whether the entire ‘Book of Isaiah’ was written by the same author.
7.10 How would a Bayesian handle the problem in Example (7.3.1) if he used a diffuse prior?
7.9 For the sample covariance matrix given in Exercise 7.3, test the hypothesis at the 10 percent level that the components are independent (assuming Normality of the observations, and that 10
7.8 Suppose zα ≡ (xα,yα)’ denotes the scores of psychotics, and of normals, respectively, on a verbal achievement test, α = 1,...,11. Suppose also that L(zα) = N(θ, ). On the basis of the
7.7 For the data given in Exercise 7.6, test the hypothesis (at the 5 percent level) that 1 = 2 assuming the samples are sufficiently large for Bayesian asymptotic theory to be a useful
7.6 Suppose x1, . . . ,xN denote independent bivariate vectors of observations representing reported number of robberies and reported number of car thefts at N different times in a given police
7.5 Suppose x1, . . . ,xN are independent p-variate observations from N(θi, 0), where xα denotes a vector whose components are reported numbers of crimes of various types, and 0 = diag (4,2,7).
7.4 For the observed data given in Exercise 7.3, compute the Stein estimator of θ and compare it with the maximum likelihood estimator.
7.3 Suppose xα denotes a vector of the three scores on a battery of three tests for the αth child in a class, and L(xα) = N(θ, ), > 0. On testing ten children (α = 1, . . . ,10) it is found that
7.2 Let xj : p × 1 denote the responses to p questions asked, of the jth person in a sample survey; j = 1,...,N. Assume L(Xj) = N(θ, ) for all j. Suppose stratified random sampling has been used so
7.1 Let x1,...,xN denote N independent observations from the multivariate logNormal distribution with density Find sufficient statistics for θ and .
6.13 Let Z:m × m denote a random matrix following the generalized Wishart distribution (see Sect. 6.6.3).
6.12 Let T denote a random p × q matrix following the law given by eqn. (6.2.7). Define Find E(U). [Hint: see Theorem (6.2.6).]
6.11 Let Y : 2 × 1 have log-characteristic function logyφY(t) = -|2t1 + 3t2 |–|6t1—5t2|. Show that øy(t) is the characteristic function of a multivariate symmetric stable distribution not
6.10 Let Y be a random p-vector, and let φY(t) denote its characteristic function. Suppose log , for some positive integer m, where Ωj : p × p, Ωj ≥ 0 for every j, and 0 < α ≤ 2. Show that
6.9 Suppose Y: p × 1, and define for 0 < α ≤ 2, log φy(t) = ia(t) - γ (t)[1 +iβ(t)ω(1,α)], where ω(μ,α) is defined in (6.5.4); γ(t) ≡ and Ωj: p × p, m = 1, 2,...; β(t) ≡ ≤ 1;
6.8 Suppose that in an experiment on behavior, k mutually exclusive responses are possible on a single trial. Suppose that out of n independent multinomial type trials [see Example (4.4.1)], r1
6.7 Let Yj denote the return one period from now on the jth oil security in an investment portfolio, j = 1,...,4 and Y ≡ (Y1,...,Y4)’; and let Zj denote the one period return on the jth office
6.6 Let X denote the time between successive arrivals of Douglas DC-10 commercial airplanes at a busy airport, and let Y denote the time between successive arrivals of Boeing 747 commercial airplanes
6.5 Let X and Y denote response times for the same individual in a pair of psychological experiments so that X and Y are correlated. Suppose it is found that log (X) and log (Y) follow the bivariate
6.4 Let X, : p × 1 denote a vector representing the response of subject j to a question posed in a sample survey that has p possible response categories, j = 1,...,N. The numbers of subjects in each
6.3 In Exercise 6.2, find the predictive distribution for a new observation, X(tN+1).
6.2 Let Xj(t) denote the change in the number of families considered to be in the poverty class (the “poverty” level changes annually) in State j, j = 1,... ,p, in the United States in two
6.1 Suppose on the basis of interviews with 100 consumers and retailers, it is found that average price changes for two commodities are given by x’ = (4,–3), and the sample covariance matrix for
5.12 Suppose . Find the kth moment of |V|. Suppose . Find the kth moment of |V|.
5.11 Show that if , and = W(Σ2,p,n), Σ2 > 0, and V1 and V2 are independent, is Wishart if and only if Σ1 = Σ2 .
5.10 Show that if , and if U = V–1, then L(U) = W—1 (Σ—1,p,n + p + 1).
5.9 Show that the characteristic function of the distribution of V: p × p, where , is given by:φ(T) = I–2i T –n/2, for T: p × p, and i =√–1.
5.8 For the three equation simultaneous equation system given in (5.3.3), the density of the structural coefficient estimator of β13 is given in (5.3.8). Discuss the moments of . What is the
5.7 Suppose V: p × p is a random symmetric matrix with (V) = W( ,p,n). Calculate the value off which maximizes the density of V, and show that it yields a maximum.
5.6 For Example (5.2.1) involving price changes for commodities during a postwar period compared with a prewar period, find the predictive density for a new observation given n previous observations,
5.5 Let : p × p denote the covariance matrix of the p × 1 vector X = (xi), i = 1, ... ,p, where xi denotes the location in a coordinate system of the ith fire out of the first p fires recorded in a
5.4 Let V be defined as in Exercise 5.1, and let where y(t) is also q × 1; y(t) is based upon q securities in another industry; ; and y(1),... ,y(M) are mutually independent and independent of the

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